Problems in Invariant Differential Operators on Homogeneous Manifolds

Abstract

In this paper, we consider six homogeneous manifolds $GL(n,\mathbb{R})/O(n,\mathbb{R})$, $SL(n,\mathbb{R})/SO(n,\mathbb{R})$, $Sp(2n,\mathbb{R})/U\left(n\right),$$(GL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)})/O(n,\mathbb{R})$, $(SL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)})/SO(n,\mathbb{R}),$$(Sp(2n,\mathbb{R})⋉H_{\mathbb{R}}^{(n,m)})/(U\left(n\right)\times S(m,\mathbb{R})).$ They are homogeneous manifolds which are important geometrically and number theoretically. These first three spaces are well-known symmetric spaces and the other three are not symmetric spaces. It is well known that the algebra of invariant differential operators on a symmetric space is commutative. The algebras of invariant differential operators on these three non-symmetric spaces are not commutative and have complicated generators. We discuss invariant differential operators on these non-symmetric spaces and provide natural but difficult problems about invariant theory.

1. Introduction

We consider the following six homogeneous manifolds which are important geometrically and number theoretically. We list them below.

(I)	$GL(n,\mathbb{R})/O(n,\mathbb{R});$
(II)	$SL(n,\mathbb{R})/SO(n,\mathbb{R});$
(III)	$Sp(2n,\mathbb{R})/U\left(n\right);$
(IV)	$(GL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)})/O(n,\mathbb{R});$
(V)	$(SL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)})/SO(n,\mathbb{R});$
(VI)	$(Sp(2n,\mathbb{R})⋉H_{\mathbb{R}}^{(n,m)})/(U\left(n\right)\times S(m,\mathbb{R})).$

Here, $H_{\mathbb{R}}^{(n,m)}$ is the Heisenberg group defined by Formula (8) and $S(m,\mathbb{R})$ denotes the additive group consisting of all $m\times m$ real symmetric matrices. The above three (I), (II), and (III) are symmetric spaces of real dimension ${\displaystyle \frac{n(n+1)}{2}},{\displaystyle \frac{n(n+1)}{2}}-1$ and $n(n+1)$, respectively. In particular, the symmetric space (III) is an Einstein–Kähler Hermitian symmetric manifold. The theory of automorphic forms on these spaces was developed by Selberg, Maass, Siegel, and outstanding number theorists. The adelic version of the theory of automorphic forms on these spaces was developed by the Langlands school. It is well known that the algebras $\mathbb{D}$(I), $\mathbb{D}$(II) and $\mathbb{D}$(III) of all invariant differential operators on these three symmetric spaces, respectively, are finitely generated and commutative. Those algebras are polynomial algebras. A set of algebraic independent generators of $\mathbb{D}$(I) was first found by Maass and Selberg explicitly (cf. [1,2,3]).Later, Helgason [4] provided another set of algebraically independent generators of $\mathbb{D}$(I). A set of explicit algebraic independent generators of $\mathbb{D}$(II) was recently constructed by Brennecken, Ciardo, and Hilgert [5] using the so-called Maass–Selberg operators. An explicit set of algebraic independent generators of $\mathbb{D}$(III) was found by Maass (cf. [2]).

The remaining three (IV), (V), and (VI) are not symmetric homogeneous manifolds of real dimension ${\displaystyle \frac{n(n+1)}{2}}+mn,{\displaystyle \frac{n(n+1)}{2}}+mn-1$ and $n(n+1)+2mn$, respectively. The homogeneous space (VI) is a Kähler manifold and so is a symplectic manifold. The theory of automorphic forms including Jacobi forms on the homogeneous space (VI) was developed in the past three decades but is still not well established. So far, nobody has developed the theory of automorphic forms on the homogeneous spaces (IV) and (V). The algebras $\mathbb{D}$(IV), $\mathbb{D}$(V), and $\mathbb{D}$(VI) of all invariant differential operators on these three non-symmetric homogeneous spaces, respectively, are $\mathsf{not} \mathsf{commutative}$. Recently, it was shown that $\mathbb{D}$(IV), $\mathbb{D}$(V), and $\mathbb{D}$(VI) are finitely generated. So, any set of a finite system of generators of each of the algebras $\mathbb{D}$(IV), $\mathbb{D}$(V), and $\mathbb{D}$(VI) has algebraic relations (the Second Funddamental Theorem of Invariant Theory). Unfortunately, nobody found explicit generators of algebras $\mathbb{D}$(IV), $\mathbb{D}$(V), and $\mathbb{D}$(VI) as of now.

The aim of this article is to study invariant differential operators on the homogeneous manifolds (IV), (V), and (VI) and provide some problems of the classical invariant theory. The paper is organized as follows. In Section 2, we briefly review some properties on differential operators on homogeneous manifolds following Chapter II of Helgason’s book [4]. In Section 3, we review $GL(n,\mathbb{R})$-invariant differential operators on the symmetric space (I) following the works of Selberg and Maass. We note that the symmetric space (I) is diffeomorphic to the open convex cone ${\mathcal{P}}_n$ in the Euclidean space ${\mathbb{R}}^N$ with $N={\displaystyle \frac{n(n+1)}{2}}$ given by

$${\mathcal{P}}_n:=\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0\}.$$

(1)

We providethe notion of automorphic forms on ${\mathcal{P}}_n$ defined by Selberg, Maass, and Terras using the algebra $\mathbb{D}$(I) of invariant differential operators on ${\mathcal{P}}_n$ (cf. [6], p. 234). In Section 4, we review $SL(n,\mathbb{R})$-invariant differential operators on the symmetric space (II). We note that the symmetric space (II) is diffeomorphic to the following symmetric space:

$${\mathfrak{P}}_n:=\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0,det\left(Y\right)=1\}.$$

(2)

We provide the notion of Maass forms on ${\mathfrak{P}}_n$ defined by Goldfeld [7] (Definition 5.1.3, pp. 115–116). In Section 5, we review $Sp(2n,\mathbb{R})$-invariant differential operators on the symmetric space (III). Maass found the explicit algebraically independent generators of $\mathbb{D}$(III) (cf. [2], pp. 112–118) and Shimura [8] also found algebraically independent generators of $\mathbb{D}$(III) using the universal enveloping algebra of the Lie algebra of the symplectic group $Sp(2n,\mathbb{R})$. We note that the symmetric space (III) is biholomorphic to the so-called Siegel upper half plane ${\mathbb{H}}_n$ given by

$${\mathbb{H}}_n:=\{\Omega \in {\mathbb{C}}^{(n,n)}|\Omega ={}^t\Omega , \mathrm{Im}\Omega >0\}.$$

(3)

It is known that ${\mathbb{H}}_n$ is an Einstein–Kähler Hermitian manifold of complex dimension ${\displaystyle \frac{n(n+1)}{2}}$ which is biholomorphic to the generalized unit disk ${\mathbb{D}}_n$ given by

$${\mathbb{D}}_n:=\{W\in {\mathbb{C}}^{(n,n)}|W={}^{t}W, I_n-W\overline{W}>0\}.$$

(4)

The symmetric complex manifold ${\mathbb{H}}_n$ provides the rich, deep, and beautiful theory in algebraic geometry (e.g., Satake compactification, toroidal compactifications, moduli of abelian varieties, etc.) and number theory (e.g., Siegel modular forms, L-functions, etc.). We provide the notion of Maass–Siegel functions using the algebra $\mathbb{D}\left({\mathbb{H}}_n\right).$ We also provide the notion of Siegel–Maass forms defined by Kramer and Mandal (cf. [9]). In Section 6, we study $G{L}_{n,m}\left(\mathbb{R}\right)$-invariant differential operators on the non-symmetric homogeneous space (IV). We show that the semidirect product $G{L}_{n,m}\left(\mathbb{R}\right)$ of $GL(n,\mathbb{R})$ and the additive group ${\mathbb{R}}^{(m,n)}$ given by

$$G{L}_{n,m}\left(\mathbb{R}\right):=GL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)}$$

acts naturally and transitively on the following space:

$${\mathcal{P}}_{n,m}:={\mathcal{P}}_n\times {\mathbb{R}}^{(m,n)}=\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0\}\times {\mathbb{R}}^{(m,n)}.$$

(5)

We refer to Formula (61) for the precise action of $G{L}_{n,m}\left(\mathbb{R}\right)$ on ${\mathcal{P}}_{n,m}.$ We see that the homogeneous space (IV) is diffeomorphic to the homogeneous space ${\mathcal{P}}_{n,m}.$ It is shown that the algebra $\mathbb{D}$(IV) of $G{L}_{n,m}\left(\mathbb{R}\right)$-invariant differential operators on the homogeneous space (IV) is not commutative. So far, nobody has found a set of generators of $\mathbb{D}$(IV). We provide some examples of explicit invariant differential operators on ${\mathcal{P}}_{n,m}$ and investigate invariant differential operators on (IV). We provide some open problems that should be solved in the future. In Section 7, we study $S{L}_{n,m}\left(\mathbb{R}\right)$-invariant differential operators on the non-symmetric homogeneous space (V). The semidirect product $S{L}_{n,m}\left(\mathbb{R}\right)$ of $SL(n,\mathbb{R})$ and the additive group ${\mathbb{R}}^{(m,n)}$ given by

$$S{L}_{n,m}\left(\mathbb{R}\right):=SL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)}$$

acts naturally and transitively on the following space:

$${\mathfrak{P}}_{n,m}:={\mathfrak{P}}_n\times {\mathbb{R}}^{(m,n)}=\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0,det\left(Y\right)=1\}\times {\mathbb{R}}^{(m,n)}.$$

(6)

See Formula (81) for the precise action of $S{L}_{n,m}\left(\mathbb{R}\right)$ on ${\mathfrak{P}}_{n,m}.$ We see that the homogeneous space (V) is diffeomorphic to the homogeneous space ${\mathfrak{P}}_{n,m}.$ It is shown that the algebra $\mathbb{D}$(V) of $S{L}_{n,m}\left(\mathbb{R}\right)$-invariant differential operators on the homogeneous space (V) is not commutative. So far, nobody has found a set of generators of $\mathbb{D}$(V). We provide some examples of explicit invariant differential operators on ${\mathfrak{P}}_{n,m}$ and investigate invariant differential operators on (V). We provide some open problems that should be solved in the future. In the final section, we study $G^J$-invariant differential operators on the non-symmetric homogeneous space (VI). The homogeneous space (VI) is biholomorphic to the so-called $\mathrm{Siegel}–\mathrm{Jacobi} \mathrm{space} {\mathbb{H}}_{n,m}$ given by

$${\mathbb{H}}_{n,m}:={\mathbb{H}}_n\times {\mathbb{C}}^{(m,n)}=\{\Omega \in {\mathbb{C}}^{(n,n)}|\Omega ={}^t\Omega , \mathrm{Im}\Omega >0\}\times {\mathbb{C}}^{(m,n)}.$$

(7)

The Jacobi group

$$G^J:=Sp(2n,\mathbb{R})⋉H_{\mathbb{R}}^{(n,m)}\hspace{1em}\left(\mathrm{semidirect}\mathrm{product}\right)$$

acts naturally and transitively on the Siegel–Jacobi space ${\mathbb{H}}_{n,m}$. See Formula (96) for the action of $G^J$ on ${\mathbb{H}}_{n,m}.$ Here,

$$H_{\mathbb{R}}^{(n,m)}=\left\{(\lambda ,\mu ;\kappa )\right|\lambda ,\mu \in {\mathbb{R}}^{(m,n)},\kappa \in {\mathbb{R}}^{(m,m)},\kappa +\mu {}^t\lambda \mathrm{symmetric}\}$$

(8)

denotes the Heisenberg group endowed with the following multiplication:

$$(\lambda ,\mu ;\kappa )\circ ({\lambda }^{\prime },{\mu }^{\prime };{\kappa }^{\prime })=(\lambda +{\lambda }^{\prime },\mu +{\mu }^{\prime };\kappa +{\kappa }^{\prime }+\lambda {}^t\mu ^{\prime }-\mu {}^t\lambda ^{\prime })$$

with $(\lambda ,\mu ;\kappa ),({\lambda }^{\prime },{\mu }^{\prime };{\kappa }^{\prime })\in H_{\mathbb{R}}^{(n,m)}.$ It is shown that the algebra $\mathbb{D}$(VI) of $G^J$-invariant differential operators on the homogeneous space (VI) is $\mathsf{not}$ commutative. So far, nobody has found a set of generators of $\mathbb{D}$(VI). We provide some examples of explicit invariant differential operators on ${\mathbb{H}}_{n,m}$ and investigate invariant differential operators on (VI). We provide some open problems that should be solved in the future. Using the commutative subalgebra of $\mathbb{D}$(VI) containing the Laplace operator of ${\mathbb{H}}_{n,m},$ we introduce the notion of Maass–Jacobi functions.

Notations. 

We denote by $\mathbb{Q},\mathbb{R}$, and $\mathbb{C}$ the field of rational numbers, the field of real numbers, and the field of complex numbers, respectively. We denote by $\mathbb{Z}$ and ${\mathbb{Z}}^{+}$ the ring of integers and the set of all positive integers, respectively. ${\mathbb{R}}^{\times }$ (resp. ${\mathbb{C}}^{\times }$) denotes the group of nonzero real (resp. complex) numbers. The symbol “:=” means that the expression on the right is the definition of that on the left. For two positive integers k and l, $F^{(k,l)}$ denotes the set of all $k\times l$ matrices with entries in a commutative ring F. For a square matrix $A\in F^{(k,k)}$ of degree k, $\mathrm{Tr}\left(A\right)$ denotes the trace of A. For any $M\in F^{(k,l)},{}^{t}M$ denotes the transpose of M. For a positive integer n, $I_n$ denotes the identity matrix of degree n. For $A\in F^{(k,l)}$ and $B\in F^{(k,k)}$, we set $B\left[A\right]={}^{t}ABA$ (Siegel’s notation). For a complex matrix A, $\overline{A}$ denotes the complex conjugate of A. $\mathrm{diag}(a_1,\cdots ,a_n)$ denotes the $n\times n$ diagonal matrix with diagonal entries $a_1,\cdots ,a_n$. For a square matrix $\Omega$, Im $\Omega$ denotes the imaginary part of $\Omega$. For a smooth manifold X, we denote by $C^{\infty }\left(X\right)$ (resp. $C_c^{\infty }\left(X\right))$ the algebra of all infinitely differentiable functions (resp. with compact support) on X. $O\left(n\right):=O(n,\mathbb{R})=\{g\in GL(n,\mathbb{R})|g{}^{t}g={}^{t}gg=I_n\}$ is the real orthogonal matrix of degree n. $SO\left(n\right):=SO(n,\mathbb{R})=O\left(n\right)\cap SL(n,\mathbb{R}).$

We denote

$$\begin{array}{cc}\hfill G{L}_{n,m}\left(\mathbb{R}\right)& =GL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)},\hspace{1em}G{L}_{n,m}\left(\mathbb{Z}\right)=GL(n,\mathbb{Z})⋉{\mathbb{Z}}^{(m,n)},\hfill \\ \hfill S{L}_{n,m}\left(\mathbb{R}\right)& =SL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)},\hspace{1em}S{L}_{n,m}\left(\mathbb{Z}\right)=SL(n,\mathbb{Z})⋉{\mathbb{Z}}^{(m,n)},\hfill \\ \hfill {\mathcal{P}}_n& =\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0\}\cong GL(n,\mathbb{R})/O(n,\mathbb{R}),\hfill \\ \hfill {\mathfrak{P}}_n& =\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0,det\left(Y\right)=1\}\cong SL(n,\mathbb{R})/SO(n,\mathbb{R}),\hfill \\ \hfill {\mathfrak{H}}_n& \left(\mathrm{see} \mathrm{Definition} 4\right),\hfill \\ \hfill {\mathcal{P}}_{n,m}& ={\mathcal{P}}_n\times {\mathbb{R}}^{(m,n)}\cong G{L}_{n,m}\left(\mathbb{R}\right)/O(n,\mathbb{R}),\hfill \\ \hfill {\mathfrak{P}}_{n,m}& ={\mathfrak{P}}_n\times {\mathbb{R}}^{(m,n)}\cong S{L}_{n,m}\left(\mathbb{R}\right)/SO(n,\mathbb{R}),\hfill \\ \hfill {\mathfrak{H}}_{n,m}& ={\mathfrak{H}}_n\times {\mathbb{R}}^{(m,n)}\cong S{L}_{n,m}\left(\mathbb{R}\right)/SO(n,\mathbb{R}),\hfill \\ \hfill {\Gamma }_n& =GL(n,\mathbb{Z}),\hspace{1em}{\Gamma }^n=SL(n,\mathbb{Z}),\hfill \\ \hfill {\mathfrak{R}}_n& =GL(n,\mathbb{Z})\setminus {\mathcal{P}}_n\cong GL(n,\mathbb{Z})\setminus GL(n,\mathbb{R})/O(n,\mathbb{R}),\hfill \\ \hfill {\mathfrak{S}}_n& =SL(n,\mathbb{Z})\setminus {\mathfrak{P}}_n\cong SL(n,\mathbb{Z})\setminus SL(n,\mathbb{R})/SO(n,\mathbb{R}),\hfill \\ \hfill {\mathfrak{R}}_{n,m}& =G{L}_{n,m}\left(\mathbb{Z}\right)\setminus {\mathcal{P}}_{n,m}\cong G{L}_{n,m}\left(\mathbb{Z}\right)\setminus G{L}_{n,m}\left(\mathbb{R}\right)/O(n,\mathbb{R}),\hfill \\ \hfill {\mathfrak{S}}_{n,m}& =S{L}_{n,m}\left(\mathbb{Z}\right)\setminus {\mathfrak{P}}_{n,m}\cong S{L}_{n,m}\left(\mathbb{Z}\right)\setminus S{L}_{n,m}\left(\mathbb{R}\right)/SO(n,\mathbb{R}).\hfill \end{array}$$

Here, $“\cong ”$ denotes the diffeomorphism.

$$J_n=\left(\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right)$$

denotes the symplectic matrix of degree $2n$.

$${\mathbb{H}}_n=\{\Omega \in {\mathbb{C}}^{(n,n)}|\Omega ={}^t\Omega ,\hspace{1em}\mathrm{Im}\Omega >0\}$$

denotes the Siegel upper half plane of degree n.

$$Sp(2n,\mathbb{R})=\{M\in {\mathbb{R}}^{(2n,2n)}|{}^{t}M{J}_{n}M=J_n\}$$

denotes the symplectic group of degree n and

$${\Gamma }_n^{♭}=Sp(2n,\mathbb{Z})=\{\gamma \in {\mathbb{Z}}^{(2n,2n)}|{}^t\gamma J_n\gamma =J_n\}\subset Sp(2n,\mathbb{R})$$

denotes the Siegel modular group of degree n. We let

$$G^J=Sp(2n,\mathbb{R})⋉H_{\mathbb{R}}^{(n,m)}$$

be the Jacobi group. Here, $H_{\mathbb{R}}^{(n,m)}$ is the Heisenberg group (see Formula (8) for the precise definition). We put ${\Gamma }^J:={\Gamma }_n^{♭}⋉H_{\mathbb{Z}}^{(n,m)}.$

$${\mathfrak{X}}_n:={\Gamma }_n^{♭}\setminus {\mathbb{H}}_n=Sp(2n,\mathbb{Z})\setminus Sp(2n,\mathbb{R})/U\left(n\right)$$

and

$${\mathfrak{X}}_{n,m}:={\Gamma }^J\setminus {\mathbb{H}}_{n,m}=(Sp(2n,\mathbb{Z})⋉H_{\mathbb{Z}}^{(n,m)})\setminus G^J/(U\left(n\right)\times S(m,\mathbb{R})).$$

$\mathbb{D}\left({\mathcal{P}}_n\right)$ denotes the algebra of all $GL(n,\mathbb{R})$-invariant differential operators on ${\mathcal{P}}_n$. $\mathbb{D}\left({\mathfrak{P}}_n\right)$ denotes the algebra of all $SL(n,\mathbb{R})$-invariant differential operators on ${\mathfrak{P}}_n$. $\mathbb{D}\left({\mathfrak{H}}_n\right)$ denotes the algebra of all $SL(n,\mathbb{R})$-invariant differential operators on ${\mathfrak{H}}_n$. $\mathbb{D}\left({\mathbb{H}}_n\right)$ denotes the algebra of all $Sp(2n,\mathbb{R})$-invariant differential operators on ${\mathbb{H}}_n$. $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$ denotes the algebra of all $G{L}_{n,m}\left(\mathbb{R}\right)$-invariant differential operators on ${\mathcal{P}}_{n,m}$. $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$ denotes the algebra of all $S{L}_{n,m}\left(\mathbb{R}\right)$-invariant differential operators on ${\mathfrak{P}}_{n,m}$. $\mathbb{D}\left({\mathfrak{H}}_{n,m}\right)$ denotes the algebra of all $S{L}_{n,m}\left(\mathbb{R}\right)$-invariant differential operators on ${\mathfrak{H}}_{n,m}$. $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$ denotes the algebra of all $G^J$-invariant differential operators on ${\mathbb{H}}_{n,m}$. ${\mathcal{Z}}_{n,m}$ denotes the center of $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$. ${\mathfrak{Z}}_{n,m}$ denotes the center of $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$. ${\tilde{\mathfrak{Z}}}_{n,m}$ denotes the center of $\mathbb{D}\left({\mathfrak{H}}_{n,m}\right)$. ${\mathfrak{C}}_{n,m}$ denotes the center of $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$.

2. Preliminaries

Throughout this section, we let G be a connected real Lie group of finite dimension n and let K be a subgroup of G. We let $\mathfrak{g}$ (resp. $\mathfrak{k}$) be the Lie algebra of G (resp. K). The symmetric algebra $S\left(\mathfrak{g}\right)$ is defined to be the algebra of complex-valued polynomial functions on the dual space ${\mathfrak{g}}^{\ast }$. If $X_1,\cdots ,X_n$ is a basis of $\mathfrak{g}$, $S\left(\mathfrak{g}\right)$ can be identified with the algebra of all polynomials

$$\sum _{\left(k\right)}{a}_{k_1\dots k_n}{X}_1^{k_1}\cdots X_n^{k_n},\hspace{1em}\left(k\right)=(k_1,\cdots ,k_n)\in {\left({\mathbb{Z}}_{+}\right)}^n.$$

Here, ${\mathbb{Z}}_{+}$ denotes the set of all non-negative integers. For an element $g\in G$, $L_g$ (resp. $R_g$) is the left (resp. right) translation by g defined by

$$L_g\left(h\right)=gh (\mathrm{resp}.R_g\left(h\right)=hg)\hspace{1em}\mathrm{for} \mathrm{all} h\in G.$$

If $X\in \mathfrak{g}$, we let $\tilde{X}$ denote a differential operator on G defined by

$$(\tilde{X}f)(g):={\displaystyle \frac{\partial }{\partial t}}{|}_{t=0}f(g·exp\left(tX\right)),\hspace{1em}f\in C^{\infty }\left(G\right),g\in G.$$

$\mathbb{E}\left(G\right)$ denotes the algebra of all differential operators on G. A differential operator $D\in \mathbb{E}\left(G\right)$ is said to be $\mathrm{left-invariant}$ if

$$D(f\circ L_g)=D\left(f\right)\circ L_g\hspace{1em}\mathrm{for} \mathrm{all} f\in C^{\infty }\left(G\right)\mathrm{and}g\in G.$$

We let $\mathbb{D}\left(G\right)$ be the algebra of all left-invariant differential operators on G and let $\mathbf{Z}\left(G\right)$ be the center of $\mathbb{D}\left(G\right).$ It is easily seen that

$${\displaystyle \widetilde{\mathrm{ad}\left(X\right)Y}}=\tilde{X}\tilde{Y}-\tilde{Y}\tilde{X}\hspace{1em}\mathrm{for} \mathrm{all} X,Y\in \mathfrak{g}.$$

Now, for any $X\in \mathfrak{g}$, we define the map $\mathrm{ad}\left(X\right):\mathbb{D}\left(G\right)⟶\mathbb{D}\left(G\right)$ by

$$\mathrm{ad}\left(X\right)D:=\tilde{X}D-D\tilde{X}\hspace{1em}\mathrm{for} \mathrm{all} D\in \mathbb{D}\left(G\right).$$

(9)

Obviously $\mathrm{ad}\left(X\right)$ is a derivation of the algebra $\mathbb{D}\left(G\right).$ We define

$$e^{\mathrm{ad}\left(X\right)}\left(D\right):=\sum _{k=0}^{\infty }{\displaystyle \frac{1}{k!}}{\left(\mathrm{ad}\left(X\right)\right)}^k\left(D\right),\hspace{1em}D\in \mathbb{D}\left(G\right).$$

(10)

Definition 1. 

The coset space $G/K$ is said to be$\mathsf{reductive}$if there exists a subspace $\mathfrak{p}$ such that

$$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}\hspace{1em}\mathrm{and}\hspace{1em}{\mathrm{Ad}}_G\left(k\right)\mathfrak{p}\subset \mathfrak{p}\hspace{1em}\mathrm{for} \mathrm{all} k\in K.$$

(11)

Here ${\mathrm{Ad}}_G:G⟶GL\left(\mathfrak{g}\right)$ denotes the adjoint representation of G on $\mathfrak{g}.$

Theorem 1. 

We assume the coset space $G/K$ is reductive. Then there exists a unique linear bijection (called the$\mathsf{symmetrization}$)

$$\lambda :S\left(\mathfrak{g}\right)⟶\mathbb{D}\left(G\right)$$

(12)

such that $\lambda \left(X^m\right)={\tilde{X}}^m(X\in \mathfrak{g}, m\in {\mathbb{Z}}^{+}).$ If $X_1,\cdots ,X_n$ is any basis of $\mathfrak{g}$ and $P\in S\left(\mathfrak{g}\right);$ then, for any $f\in C^{\infty }\left(G\right)$,

$$\left(\lambda \left(P\right)f\right)\left(g\right)={\left[P({\partial }_1,\cdots ,{\partial }_n)f(g·exp(t_1{X}_1+\cdots +t_n{X}_n)\right]}_{t=0},$$

(13)

where ${\partial }_i=\partial /\partial t_i(1\le i\le n)$ and $t=(t_1,\cdots ,t_n)\in {\mathbb{R}}^n.$ Here, the suffix ${[·]}_{t=0}$ means the evaluation at $t=0$ after differentiation.

Proof. 

The proof can be found in [4] (Theorem 4.3, pp. 280–281). □

Definition 2. 

We fix an element $g\in G$. The mapping

$$\mathrm{Ad}\left(g\right):\mathbb{D}\left(G\right)⟶\mathbb{D}\left(G\right)$$

is defined by

$$\mathrm{Ad}\left(g^{-1}\right)D=D^{R_g}\hspace{1em}\mathrm{for} \mathrm{all} D\in \mathbb{D}\left(G\right).$$

(14)

Here, $D^{R_g}:C^{\infty }\left(G\right)⟶C^{\infty }\left(G\right)$ is a differential operator on G defined by

$$D^{R_g}f=D(f\circ R_{g^{-1}})\circ R_{g^{-1}}\hspace{1em}\mathrm{for} \mathrm{all} f\in C^{\infty }\left(G\right).$$

We let

$$\tilde{\mathfrak{g}}:=\{\tilde{X}\in \mathbb{E}\left(G\right)|X\in \mathfrak{g}\}.$$

Since $\tilde{\mathfrak{g}}$ generates $\mathbb{D}\left(G\right)$, we have

$$\mathrm{Ad}(expX)D=e^{\mathrm{ad}\left(X\right)}\left(D\right)\hspace{1em}\mathrm{for} \mathrm{all} X\in \mathfrak{g}\mathrm{and}D\in \mathbb{D}\left(G\right).$$

(15)

We let $I\left(\mathfrak{g}\right)$ be the space of all ${\mathrm{Ad}}_G\left(G\right)$-invariants in $S\left(\mathfrak{g}\right)$, i.e.,

$$I\left(\mathfrak{g}\right)=\{P\in S\left(\mathfrak{g}\right)|{\mathrm{Ad}}_G\left(g\right)P=P\hspace{1em}\mathrm{for} \mathrm{all} g\in G\}.$$

Theorem 2. 

We assume the coset space $G/K$ is reductive. Then

$$\lambda \left(I\right(\mathfrak{g}\left)\right)=\mathbf{Z}\left(G\right).$$

(16)

Moreover, $\mathbf{Z}\left(G\right)$ consists of all bi-invariant differential operators on G.

Proof. 

The proof can be found in [4] (Corollary 4.5, pp. 283–284). □

If $G/K$ is a reductive homogeneous manifold in the sense of Definition 1, we let $\pi :G⟶G/K$ be the projection map and we put $\tilde{f}=f\circ \pi$ for a function f on G.

Theorem 3. 

We assume the coset space $G/K$ is reductive. We put

$${\mathbb{D}}_K\left(G\right):=\{D\in \mathbb{D}\left(G\right)|D^{R_k}=D\hspace{1em}\mathrm{for} \mathrm{all} k\in K\}.$$

Then, the mapping

$$\mu :{\mathbb{D}}_K\left(G\right)⟶\mathbb{D}(G/K)$$

defined by

$$\widetilde{\mu \left(D\right)f}=D\tilde{f},\hspace{1em}f\in C^{\infty }(G/K)$$

is a surjective homomorphism. The kernel is given by

$$\ker \mu ={\mathbb{D}}_K\left(G\right)\cap \mathbb{D}\left(G\right)\mathfrak{k}$$

and hence we have the isomorphism

$$\mathbb{D}(G/K)\cong {\mathbb{D}}_K\left(G\right)/({\mathbb{D}}_K\left(G\right)\cap \mathbb{D}\left(G\right)\mathfrak{k}).$$

Proof. 

The proof can be found in [4] (Theorem 4.6, pp. 285–286). □

Corollary 1. 

We assume the coset space $G/K$ is reductive. We let $I\left(\mathfrak{p}\right)$ denote the set of all ${\mathrm{Ad}}_G\left(K\right)$-invariants in $S\left(\mathfrak{p}\right)$. Then

$${\mathbb{D}}_K\left(G\right)=\left({\mathbb{D}}_K\left(G\right)\cap \mathbb{D}\left(G\right)\mathfrak{k}\right)\oplus \lambda (I\left(\mathfrak{p}\right).$$

Proof. 

See [4], p. 286. □

Theorem 4. 

We let $G/K$ be a reductive homogeneous space. The mapping

$$\Theta :I\left(\mathfrak{p}\right)⟶\mathbb{D}(G/K)$$

defined by

$$\Theta \left(P\right):=D_{\lambda \left(P\right)}\hspace{1em}\mathrm{for} \mathrm{all} P\in I\left(\mathfrak{p}\right)$$

is a linear bijection. Explicitly, for any function $f\in C^{\infty }(G/K),$

$$\left(D_{\lambda \left(D\right)}f\right)\left(gK\right)={\left[P({\partial }_1,\cdots ,{\partial }_n)\tilde{f}(g·exp(t_1{X}_1+\cdots +t_r{X}_r)\right]}_{t=0},$$

(17)

where ${\partial }_i=\partial /\partial t_i(1\le i\le r),t=(t_1,\cdots ,t_r)\in {\mathbb{R}}^r$ and $\{X_1,\cdots ,X_r\}$ is a basis of $\mathfrak{p}$.

Proof. 

The proof can be found in [4], Theorem 4.6, pp. 285–286. □

Remark 1. 

(1) Θ is not multiplicative in general. In fact, we have

$$\Theta \left(P_1{P}_2\right)=\Theta \left(P_1\right)\Theta \left(P_2\right)+\Theta \left(Q\right)\hspace{1em}\mathrm{for} \mathrm{all} P_1,P_2\in I\left(\mathfrak{p}\right),$$

• where $Q\in I\left(\mathfrak{p}\right)$ has degree < degree($P_1$) + degree($P_2$).

(2)

If $P_1,\cdots ,P_d$ are generators of $I\left({\mathfrak{p}}_{\mathbb{C}}\right)$, then $\Theta \left(P_1\right),\cdots ,\Theta \left(P_d\right)$ are generators of $\mathbb{D}(G/K).$ Here, ${\mathfrak{p}}_{\mathbb{C}}$ denotes the complexification of $\mathfrak{p}.$

(3)

If $I\left(\mathfrak{p}\right)$ has a finite system of generators $P_1,\cdots ,P_d$ and we put $D_i=\Theta \left(P_i\right)(1\le i\le d)$, then each $D\in \mathbb{D}(G/K)$ can be written

$$D=\sum _{\left(n\right)}{a}_{n_1\cdots n_d}{D}_1^{n_1}\cdots D_d^{n_d},$$

where $\left(n\right)=(n_1,\cdots ,n_d)\in {\mathbb{Z}}_{+}^d.$

Theorem 5. 

(1) We let $(G,K)$ be a symmetric pair (i.e., $G/K$ is a symmetric space), G semisimple, and K a maximal compact subgroup of G. Then $\mathbb{D}(G/K)$ is a commutative algebra. Here, $\mathbb{D}(G/K)$ denotes the algebra of all invariant differential operators on $G/K$. (2) We let H be a connected Lie group of finite dimension and let $H^{◊}$ be the diagonal in $H\times H$. Under the bijection

$$(h_1, h_2)H^{◊}\mapsto h_1{h}_2^{-1},\hspace{1em}{h}_1,h_2\in H$$

of $(H\times H)/H^{◊}$ onto H, we have the identification

$$\mathbb{D}((H\times H)/H^{◊})=\mathbf{Z}\left(H\right).$$

Here, $\mathbb{D}((H\times H)/H^{◊})$ (resp. $\mathbb{D}\left(H\right)$) denotes the algebra of invariant differential operators on $(H\times H)/H^{◊}$ (resp. H) and $\mathbf{Z}\left(H\right)$ denotes the center of $\mathbb{D}\left(H\right)$.

Proof. 

The proof can be found in [4] (Theorem 5.7, pp. 294–295). □

We let $M=G/K$ be a symmetric space of the noncompact type, i.e., G is a connected semisimple Lie group with finite center and K a maximal compact subgroup of G. We let $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ be the Cartan decomposition of the Lie algebra $\mathfrak{g}$ of G. We let $\mathfrak{a}\subset \mathfrak{p}$ be a maximal abelian subspace of $\mathfrak{p}$ and let ${\mathfrak{a}}^{+}\subset \mathfrak{a}$ be a fixed Weyl chamber. We let $G=KAN$ be an Iwasawa decomposition of G. We denote by $\mathbb{D}\left(A\right)$ the algebra of invariant differential operators on $A·o$. Here, $o=e·K$ is the origin of $G/K$ (e is the identity element of G) and $A·o:=\{a·o|a\in A\}$ denotes the A-orbit of o in $G/K$. We let W be the Weyl group of G, that is, the Weyl group of the root system of G.

We recall the linear bijection $\lambda :S\left(\mathfrak{g}\right)⟶\mathbb{D}\left(G\right)$ in Theorem 1. We see that $S\left(\mathfrak{a}\right)$ can be identified with $\mathbb{D}\left(A\right)$. We let ${\mathbb{D}}_W\left(A\right)$ be the set of all W-invariant differential operators on the orbit $A·o$ and $I\left(\mathfrak{a}\right)$ the set of all W-invariants in $S\left(\mathfrak{a}\right)$. Then, there exists a bijection of $\mathbb{D}(G/K)$ onto ${\mathbb{D}}_W\left(A\right)$ (cf. see [4] (Theorem 5.13, pp. 300–302)). Furthermore, there exists a surjective homomorphism of ${\mathbb{D}}_K\left(G\right)$ onto ${\mathbb{D}}_W\left(A\right)$ with kernel

$${\mathbb{D}}_K\left(G\right)\cap \mathbb{D}\left(G\right)\mathfrak{k}.$$

We refer to [4] (Theorem 5.18, p. 306) for more detail. Combining all these results, we conclude that if $G/K$ is a symmetric space of the noncompact type, then $\mathbb{D}(G/K)$ is a polynomial algebra in r algebraically independent generators ${\delta }_1,\cdots ,{\delta }_r$ whose degrees $d_1,\cdots ,d_r$ are canonically determined by G. We note that $r=dimA$ is the rank of G or the rank of $G/K$.

3. Invariant Differential Operators on $\mathit{GL}(\mathit{n},\mathbb{R})/\mathit{O}(\mathit{n},\mathbb{R})$

For any positive integer $n\ge 1$, we let

$${\mathcal{P}}_n:=\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0\}$$

be the open convex cone in the Euclidean space ${\mathbb{R}}^N$ with $N={\displaystyle \frac{n(n+1)}{2}}.$ Then, $GL(n,\mathbb{R})$ acts ${\mathcal{P}}_n$ transitively by

$$g·Y=gY{}^{t}g,\hspace{1em}\mathrm{where}g\in GL(n,\mathbb{R})\hspace{1em}\mathrm{and}\hspace{1em}Y\in {\mathcal{P}}_n.$$

(18)

Since $O\left(n\right)$ is the isotopic subgroup of $GL(n,\mathbb{R})$ at $I_n$, the symmetric space $GL(n,\mathbb{R})/O\left(n\right)$ is diffeomorphoc to ${\mathcal{P}}_n$.

For $Y=\left(y_{ij}\right)\in {\mathcal{P}}_n,$ we put

$$dY=\left(d{y}_{ij}\right)\hspace{1em}\mathrm{and}\hspace{1em}{\displaystyle \frac{\partial }{\partial Y}}=\left({\displaystyle \frac{1+{\delta }_{ij}}{2}}{\displaystyle \frac{\partial }{\partial y_{ij}}}\right).$$

For a fixed element $A\in GL(n,\mathbb{R})$, we put

$$Y_{\ast }=A·Y=AY{}^{t}A,\hspace{1em}Y\in {\mathcal{P}}_n.$$

Then

$$d{Y}_{\ast }=AdY{}^{t}A\hspace{1em}\mathrm{and}\hspace{1em}{\displaystyle \frac{\partial }{\partial Y_{\ast }}}={}^{t}A^{-1}{\displaystyle \frac{\partial }{\partial Y}}{A}^{-1}.$$

(19)

We can see easily that for any positive real number $C>0$,

$$d{s}_{n;C}^2=C·\mathrm{Tr}\left({\left(Y^{-1}dY\right)}^2\right)$$

is a Riemannian metric on ${\mathcal{P}}_n$ invariant under Action (18) and its Laplace operator is given by

$${\Delta }_{n;C}={\displaystyle \frac{1}{C}}·\mathrm{Tr}\left({\left(Y{\displaystyle \frac{\partial }{\partial Y}}\right)}^2\right),$$

where $\mathrm{Tr}\left(M\right)$ denotes the trace of a square matrix M. We also can see that

$$d{\mu }_n\left(Y\right)={(detY)}^{-{\displaystyle \frac{n+1}{2}}}\prod _{i\le j}d{y}_{ij}$$

(20)

is a $GL(n,\mathbb{R})$-invariant volume element on ${\mathcal{P}}_n$.

Theorem 6. 

A geodesic $\alpha \left(t\right)$ joining $I_n$ and $Y\in {\mathcal{P}}_n$ has the form

$$\alpha \left(t\right)=exp\left(tA\right[V\left]\right),\hspace{1em}t\in [0,1],$$

where

$$Y=(expA)\left[V\right]=exp\left(A\left[V\right]\right)=exp\left({}^{t}VAV\right)$$

is the spectral decomposition of Y, where $V\in O(n,\mathbb{R}),A=\mathrm{diag}(a_1,\cdots ,a_n)$ with all $a_j\in \mathbb{R}.$ The distance of $\alpha \left(t\right)(0\le t\le 1)$ between $I_n$ and Y is

$${\left(\sum _{j=1}^n{a}_j^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

Proof. 

The proof can be found in [6] (pp. 16–17). □

We consider the following $\mathsf{Maass}–\mathsf{Selberg} \left(\mathsf{differential}\right) \mathsf{operators}$ ${\delta }_1,{\delta }_2,\cdots ,{\delta }_n$ on ${\mathcal{P}}_n$ defined by

$${\delta }_k=\mathrm{Tr}\left({\left(Y{\displaystyle \frac{\partial }{\partial Y}}\right)}^k\right),\hspace{1em}k=1,2,\cdots ,n,$$

(21)

By Formula (19), we obtain

$${\left(Y_{\ast }{\displaystyle \frac{\partial }{\partial Y_{\ast }}}\right)}^i=A{\left(Y{\displaystyle \frac{\partial }{\partial Y}}\right)}^i{A}^{-1}$$

for any $A\in GL(n,\mathbb{R})$, so each ${\delta }_i(1\le i\le n)$ is invariant under Action (18) of $GL(n,\mathbb{R})$.

Maass [1,2] and Selberg [3] proved the following.

Theorem 7. 

The algebra $\mathbb{D}\left({\mathcal{P}}_n\right)$ of all $GL(n,\mathbb{R})$-invariant differential operators on ${\mathcal{P}}_n$ is generated by ${\delta }_1,{\delta }_2,\cdots ,{\delta }_n.$ Furthermore, ${\delta }_1,{\delta }_2,\cdots ,{\delta }_n$ are algebraically independent and $\mathbb{D}\left({\mathcal{P}}_n\right)$ is isomorphic to the commutative ring $\mathbb{C}[x_1,x_2,\cdots ,x_n]$ with n indeterminates $x_1,x_2,\cdots ,x_n.$

Proof. 

The proof can be found in [2] (pp. 64–66) and [6] (pp. 29–30). The last statement follows immediately from the work of Harish–Chandra [10,11] or [4] (p. 294). □

Remark 2. 

A different description of $\mathbb{D}\left({\mathcal{P}}_n\right)$ was given by Helgason [4] (Chapter II, Exercise C.1, p. 337; Solution pp. 571–572). See also [4] (Chapter II, Exercise C.8, pp. 339–340) for a related topic.

We let $={\mathbb{R}}^{(n,n)}$ be the Lie algebra of $GL(n,\mathbb{R})$. The adjoint representation $\mathrm{Ad}$ of $GL(n,\mathbb{R})$ is given by

$$\mathrm{Ad}\left(g\right)=gX{g}^{-1},\hspace{1em}g\in GL(n,\mathbb{R}),X\in \mathfrak{g}.$$

The Killing form B of $\mathfrak{g}$ is given by

$$B(X,Y)=2n\mathrm{Tr}\left(XY\right)-2\mathrm{Tr}\left(X\right)\mathrm{Tr}\left(Y\right),\hspace{1em}X,Y\in \mathfrak{g}.$$

Since $B(a{I}_n,X)=0$ for all $a\in \mathbb{R}$ and $X\in \mathfrak{g}, B$ is degenerate. So the Lie algebra $\mathfrak{g}$ of $GL(n,\mathbb{R})$ is not semisimple.

We put $K=O\left(n\right).$ The Lie algebra $\mathfrak{k}$ of K is given by

$$\mathfrak{k}=\left\{X\in \mathfrak{g}|X+{}^{t}X=0\right\}.$$

We let $\mathfrak{p}$ be the subspace of defined by

$$\mathfrak{p}=\left\{X\in |X={}^{t}X\in {\mathbb{R}}^{(n,n)}\right\}.$$

Then

$$\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$$

is the direct sum of $\mathfrak{k}$ and $\mathfrak{p}$ with respect to the Killing form B. Since $\mathrm{Ad}\left(k\right)\mathfrak{p}\subset \mathfrak{p}$ for any $k\in K,K$ acts on $\mathfrak{p}$ via the adjoint representation by

$$k·X=\mathrm{Ad}\left(k\right)X=kX{}^{t}k,\hspace{1em}k\in K,X\in \mathfrak{p}.$$

(22)

Action (22) induces the action of K on the polynomial algebra $\mathrm{Pol}\left(\mathfrak{p}\right)$ of $\mathfrak{p}$ and the symmetric algebra $S\left(\mathfrak{p}\right)$. We denote by $\mathrm{Pol}{\left(\mathfrak{p}\right)}^K$ (resp. $S{\left(\mathfrak{p}\right)}^K$) the subalgebra of $\mathrm{Pol}\left(\mathfrak{p}\right)$ (resp. $S\left(\mathfrak{p}\right)$) consisting of all K-invariants. The following inner product $(,)$ on $\mathfrak{p}$ defined by

$$(X,Y)=B(X,Y),\hspace{1em}X,Y\in \mathfrak{p}$$

gives an isomorphism as vector spaces

$$\mathfrak{p}\cong {\mathfrak{p}}^{\ast },\hspace{1em}X\mapsto f_X,\hspace{1em}X\in \mathfrak{p},$$

(23)

where ${\mathfrak{p}}^{\ast }$ denotes the dual space of $\mathfrak{p}$ and $f_X$ is the linear functional on $\mathfrak{p}$ defined by

$$f_X\left(Y\right)=(Y,X),\hspace{1em}Y\in \mathfrak{p}.$$

It is known that there is a canonical linear bijection of $S{\left(\mathfrak{p}\right)}^K$ onto $\mathbb{D}\left({\mathcal{P}}_n\right)$. Identifying $\mathfrak{p}$ with ${\mathfrak{p}}^{\ast }$ by the above isomorphism (23), we obtain a canonical linear bijection

$${\Phi }_n:\mathrm{Pol}{\left(\mathfrak{p}\right)}^K⟶\mathbb{D}\left({\mathcal{P}}_n\right)$$

(24)

of $\mathrm{Pol}{\left(\mathfrak{p}\right)}^K$ onto $\mathbb{D}\left({\mathcal{P}}_n\right)$. The map ${\Phi }_n$ is described explicitly as follows. We put $N=n(n+1)/2$. We let $\left\{{\xi }_{\alpha }|1\le \alpha \le N\right\}$ be a basis of $\mathfrak{p}$. If $P\in \mathrm{Pol}{\left(\mathfrak{p}\right)}^K$, then

$$\left({\Phi }_n\left(P\right)f\right)\left(gK\right)={\left[P\left({\displaystyle \frac{\partial }{\partial t_{\alpha }}}\right)f\left(gexp\left(\sum _{\alpha =1}^N{t}_{\alpha }{\xi }_{\alpha }\right)K\right)\right]}_{\left(t_{\alpha }\right)=0},$$

(25)

where $f\in C^{\infty }\left({\mathcal{P}}_n\right)$. We refer to [4] (Theorem 4.6, pp. 285–286) or Formula (17) for more detail. In general, it is very hard to express ${\Phi }_n\left(P\right)$ explicitly for a polynomial $P\in \mathrm{Pol}{\left(\mathfrak{p}\right)}^K$.

We let

$$q_i\left(X\right)=\mathrm{Tr}\left(X^i\right),\hspace{1em}i=1,2,\cdots ,n$$

(26)

be the polynomials on $\mathfrak{p}$. Here, we take a coordinate $x_{11},x_{12},\cdots ,x_{nn}$ in $\mathfrak{p}$ given by

$$X=\left(\begin{array}{cccc}{x}_{11}& {\displaystyle \frac{1}{2}}{x}_{12}& \dots & {\displaystyle \frac{1}{2}}{x}_{1n}\\ {\displaystyle \frac{1}{2}}{x}_{12}& x_{22}& \dots & {\displaystyle \frac{1}{2}}{x}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{1}{2}}{x}_{1n}& {\displaystyle \frac{1}{2}}{x}_{2n}& \dots & x_{nn}\end{array}\right).$$

For any $k\in K$,

$$(k·q_i)\left(X\right)=q_i\left(k^{-1}Xk\right)=\mathrm{Tr}\left(k^{-1}{X}^{i}k\right)=q_i\left(X\right),\hspace{1em}i=1,2,\cdots ,n.$$

Thus, $q_i\in \mathrm{Pol}{\left(\mathfrak{p}\right)}^K$ for $i=1,2,\cdots ,n.$ By a classical invariant theory (cf. [12,13]), we can prove that the algebra $\mathrm{Pol}{\left(\mathfrak{p}\right)}^K$ is generated by the polynomials $q_1,q_2,\cdots ,q_n$ and that $q_1,q_2,\cdots ,q_n$ are algebraically independent. Using Formula (26), we can show without difficulty that

$${\Phi }_n\left(q_1\right)=\mathrm{Tr}\left(2Y{\displaystyle \frac{\partial }{\partial Y}}\right).$$

However, ${\Phi }_n\left(q_i\right)(i=2,3,\cdots ,n)$ are still not known explicitly.

We propose the following conjecture.

Conjecture 8. 

For any $n,$

$${\Phi }_n\left(q_i\right)=\mathrm{Tr}\left({\left(2Y{\displaystyle \frac{\partial }{\partial Y}}\right)}^i\right),\hspace{1em}i=1,2,\cdots ,n.$$

Remark 3. 

The above conjecture is true for $n=1,2.$

The fundamental domain ${\mathfrak{M}}_n$ for $GL(n,\mathbb{Z})$ in ${\mathcal{P}}_n$ which was found by H. Minkowski [14] is defined as a subset of ${\mathcal{P}}_n$ consisting of $Y=\left(y_{ij}\right)\in {\mathcal{P}}_n$ satisfying the following conditions (M.1)–(M.2) (cf. [2], (p. 123)):

(M.1)

${aY}^{t}a\ge y_{kk}$ for every $a=\left(a_i\right)\in {\mathbb{Z}}^n$ in which $a_k,\cdots ,a_n$ are relatively prime for $k=1,2,\cdots ,n$.

(M.2)

$y_{k,k+1}\ge 0$ for $k=1,\cdots ,n-1.$

We say that a point of ${\mathfrak{M}}_n$ is Minkowski reduced or simply M-reduced. ${\mathfrak{M}}_n$ is a convex cone through the origin bounded by a finite number of hyperplanes and is closed in ${\mathcal{P}}_n$ (cf. [2], pp. 123–124). Thus, we see that ${\mathfrak{M}}_n$ is a semi-algebraic set with real analytic structure.

We let

$${\mathfrak{R}}_n:=GL(n,\mathbb{Z})\setminus GL(n,\mathbb{R})/O(n,\mathbb{R})=GL(n,\mathbb{Z})\setminus {\mathcal{P}}_n$$

be the locally symmetric space. ${\mathcal{P}}_n$ parameterizes principally polarized real tori of dimension n (cf. [15]). The arithmetic quotient ${\mathfrak{R}}_n$ is the moduli space of isomorphism classes of principally polarized real tori of dimension n. Unfortunately ${\mathfrak{R}}_n$ does not admit the structure of a real algebraic variety and does not admit a compactification which is defined over the rational number field $\mathbb{Q}$ (cf. [16] or [17]).

We offer the definition of automorphic forms for $GL(n,\mathbb{Z})$ given by A. Terras (cf. [6], (p. 182)).

For $s=(s_1,\cdots ,s_n)\in {\mathbb{C}}^n$, Atle Selberg [3] (pp. 57–58) introduced the power function $p_s:{\mathcal{P}}_n⟶\mathbb{C}$ defined by

$$p_s\left(Y\right):=\prod _{j=1}^n{(det{Y}_j)}^{s_j},\hspace{1em}Y\in {\mathcal{P}}_n,$$

(27)

where $Y_j\in {\mathcal{P}}_j(1\le j\le n)$ is the $j\times j$ upper left corner of Y. It is known that $p_s\left(Y\right)$ is a joint eigenfunction of $\mathbb{D}\left({\mathcal{P}}_n\right)$, i.e., $p_s\left(Y\right)$ is an eigenfunction of each invariant differential operator in $\mathbb{D}\left({\mathcal{P}}_n\right)$ (cf. [6], (pp. 39–40)).

Definition 3. 

A real analytic function $f:{\mathcal{P}}_n⟶\mathbb{C}$ is said to be a$\mathsf{automorphic} \mathsf{form}$for $GL(n,\mathbb{Z})$ if it satisfies the following Conditions (A1)–(A3):

(A1)

$f\left(\gamma Y{}^t\gamma \right)=f\left(Y\right)$ for all $Y\in {\mathcal{P}}_n$ and all $\gamma \in GL(n,\mathbb{Z})$;

(A2)

f is an eigenfunction of all $D\in \mathbb{D}\left({\mathcal{P}}_n\right)$, i.e., $Df={\lambda }_{D}f$ for some eigenvalue ${\lambda }_D$;

(A3)

f has at most polynomial growth at infinity, i.e.,

$$\left|f\left(Y\right)\right|\le C|p_s\left(Y\right)|\hspace{1em}\mathrm{for} \mathrm{some}s\in {\mathbb{C}}^n\mathrm{and}C>0.$$

We set ${\Gamma }_n=GL(n,\mathbb{Z})$. We denote by $\mathbf{A}({\Gamma }_n,\lambda )$ the space of all automorphic forms for ${\Gamma }_n$ with a given eigenvalue system $\lambda$. An automorphic form f in $\mathbf{A}({\Gamma }_n,\lambda )$ is called a $\mathsf{cusp} \mathsf{form}$ for ${\Gamma }_n$ if for any k with $1\le k\le n-1.$ We have

$${\int }_{X\in T^{(k,n-k)}}f\left({}^t\left(\begin{array}{cc}{I}_k& x\\ 0& I_{n-k}\end{array}\right)Y\left(\begin{array}{cc}{I}_k& x\\ 0& I_{n-k}\end{array}\right)\right)dx=0\hspace{1em}\mathrm{for} \mathrm{all} Y\in {\mathcal{P}}_n.$$

(28)

Here, $T=\mathbb{R}/\mathbb{Z}$ denotes a circle, that is, a one-dimensional torus, and $T^{(k,n-k)}$ denotes the set of all $k\times (n-k)$ matrices with entries in T. Condition (A3) implies the vanishing of the constant terms in some Fourier expansions of $f\left(Y\right)$ as a periodic function in the x-variable in partial Iwasawa coordinates.

Remark 4. 

Borel and Jacquet defined automorphic forms for a connected reductive group over $\mathbb{Q}$ (cf. [18], (pp. 199–200) and [19], (pp. 189–190)). The definition given by Borel and Jacquet is slightly different from Definition 3 given by Terras.

One of the motivations to study automorphic forms for $GL(n,\mathbb{Z})$ is the need to study various kinds of L-functions with many gamma factors in their functional equations. Another motivation for the study of automorphic forms for $GL(n,\mathbb{Z})$ is to develop the theory of harmonic analysis on $L^2(GL(n,\mathbb{Z})\setminus {\mathcal{P}}_n)$ and $L^2(GL(n,\mathbb{Z})\setminus GL(n,\mathbb{R}))$ which involves the unitary representations of $GL(n,\mathbb{R})$.

Remark 5. 

Grenier investigated a fundamental domain for $GL(n,\mathbb{Z})$ on ${\mathcal{P}}_n$ and constructed a compactification of $GL(n,\mathbb{Z})\setminus {\mathcal{P}}_n$ (cf. [20,21]).

Remark 6. 

Using the Grenier operator defined by Douglas Grenier (cf. [22]), we can define the notion of $\mathsf{stable\; automorphic} \mathsf{forms}$ for $GL(n,\mathbb{Z})$.

4. Invariant Differential Operators on $\mathit{SL}(\mathit{n},\mathbb{R})/\mathit{SO}(\mathit{n},\mathbb{R})$

First of all, we provide some geometric properties on $SL(n,\mathbb{R})/SO(n,\mathbb{R}).$

We let

$${\mathfrak{P}}_n:=\left\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0,det\left(Y\right)=1\right\}$$

be a symmetric space associated to $SL(n,\mathbb{R})$. Indeed, $SL(n,\mathbb{R})$ acts on ${\mathfrak{P}}_n$ transitively by

$$g\circ Y=gY{}^{t}g,\hspace{1em}g\in SL(n,\mathbb{R}),Y\in {\mathfrak{P}}_n.$$

(29)

Thus, ${\mathfrak{P}}_n$ is a smooth manifold diffeomorphic to the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$ through the bijective map

$$SL(n,\mathbb{R})/SO(n,\mathbb{R})⟶{\mathfrak{P}}_n,\hspace{1em}g·SO(n,\mathbb{R})\mapsto g\circ I_n=g{}^{t}g,\hspace{1em}g\in SL(n,\mathbb{R}).$$

For $Y\in {\mathfrak{P}}_n$, we have a partial Iwasawa decomposition

$$Y=\left(\begin{array}{cc}{v}^{-1}& 0\\ 0& v^{1/(n-1)}W\end{array}\right)\left[\left(\begin{array}{cc}1& {}^{t}x\\ 0& I_{n-1}\end{array}\right)\right]=\left(\begin{array}{cc}{v}^{-1}& v^{-1}{}^{t}x\\ v^{-1}x& v^{-1}x{}^{t}x+v^{1/(n-1)}W\end{array}\right)$$

(30)

where $v>0,x\in {\mathbb{R}}^{(n-1,1)}$ and $W\in {\mathfrak{P}}_{n-1}.$ From now on, for brevity, we write $Y=[v,x,W]$ instead of Decomposition (30). In these coordinates, $Y=[v,x,W]$,

$$d{s}_Y^2={\displaystyle \frac{n}{n-1}}{v}^{-2}d{v}^2+2v^{-n/(n-1)}{W}^{-1}\left[dx\right]+d{s}_W^2$$

is a $SL(n,\mathbb{R})$-invariant metric on ${\mathfrak{P}}_n$, where $dx={}^t(d{x}_1,\cdots ,d{x}_{n-1})$ and $d{s}_W^2$ is a $SL(n-1,\mathbb{R})$-invariant metric on ${\mathfrak{P}}_{n-1}$. The Laplace operator ${\Delta }_n$ of $({\mathfrak{P}}_n,d{s}_Y^2)$ is given by

$${\Delta }_n={\displaystyle \frac{n-1}{n}}{v}^2{\displaystyle \frac{{\partial }^2}{\partial v^2}}-{\displaystyle \frac{1}{n}}{\displaystyle \frac{\partial }{\partial v}}+v^{n/(n-1)}W\left[{\displaystyle \frac{\partial }{\partial x}}\right]+{\Delta }_{n-1}$$

inductively, where if $x={}^t(x_1,\cdots ,x_{n-1})\in {\mathbb{R}}^{(n-1,1)}$,

$${\displaystyle \frac{\partial }{\partial x}}={}^t\left({\displaystyle \frac{\partial }{\partial x_1}},\cdots ,{\displaystyle \frac{\partial }{\partial x_{n-1}}}\right)$$

and ${\Delta }_{n-1}$ is the Laplace operator of $({\mathfrak{P}}_{n-1},d{s}_W^2)$,

$$d{\nu }_n=v^{-(n+2)/2}dvdxd{\nu }_{n-1}$$

is a $SL(n,\mathbb{R})$-invariant volume element on ${\mathfrak{P}}_n$ where $dx=d{x}_1\cdots d{x}_{n-1}$ and $d{\nu }_{n-1}$ is a $SL(n-1,\mathbb{R})$-invariant volume element on ${\mathfrak{P}}_{n-1}$.

Following earlier work of Minkowski [14], Siegel [23] showed that the volume of the arithmetic quotient $SL(n,\mathbb{Z})\setminus {\mathfrak{P}}_n$ is given as follows:

$$\mathrm{Vol}(SL(n,\mathbb{Z})\setminus {\mathfrak{P}}_n)={\int }_{SL(n,\mathbb{Z})\setminus {\mathfrak{P}}_n}d{\nu }_n=n{2}^{n-1}\prod _{k=2}^n{\displaystyle \frac{\zeta \left(k\right)}{\mathrm{Vol}\left(S^{k-1}\right)}},$$

(31)

where

$$\mathrm{Vol}(S^{k-1})={\displaystyle \frac{2{\left(\sqrt{\pi }\right)}^k}{\Gamma (k/2)}}$$

denotes the volume of the $(k-1)$-dimensional sphere $S^{k-1}$, $\Gamma \left(x\right)$ denotes the usual Gamma function, and $\zeta \left(k\right)={\sum }_{m=1}^{\infty }{m}^{-k}$ denotes the Riemann zeta function. The proof of (31) can be found in [24] or [7] (pp. 27–37).

We let $\mathbb{D}\left({\mathfrak{P}}_n\right)$ be the algebra of all differential operators on ${\mathfrak{P}}_n$ invariant under the action (29) of $SL(n,\mathbb{R})$. It is well known (cf. [4,10,11]) that $\mathbb{D}\left({\mathfrak{P}}_n\right)$ is commutative and is isomorphic to the polynomial algebra $\mathbb{C}[x_1,x_2,\cdots ,x_{n-1}]$ with n indeterminates $x_1,x_2,\cdots ,x_{n-1}$. We observe that $n-1$ is the rank of $SL(n,\mathbb{R})$, i.e., the rank of the symmetric space $SL(n,\mathbb{R})/SO(n,\mathbb{R})$.

In [5], using the Maass–Selberg operators ${\delta }_1,{\delta }_2,\cdots ,{\delta }_n$ (see Formula (21)), Brennecken, Ciardo, and Hilgert found explicit generators $E_1,E_2,\cdots ,E_{n-1}$ of $\mathbb{D}\left({\mathfrak{P}}_n\right)$. Obviously, $E_1,E_2,\cdots ,E_{n-1}$ are algebraically independent. We briefly sketch their method of finding generators $E_1,E_2,\cdots ,E_{n-1}$ of $\mathbb{D}\left({\mathfrak{P}}_n\right)$.

We denote by $\mathbb{D}\left(GL\right(n,\mathbb{R})/O(n,\mathbb{R}\left)\right)$ (resp. $\mathbb{D}\left(SL\right(n,\mathbb{R})/SO(n,\mathbb{R}\left)\right)$) the algebra of all $GL(n,\mathbb{R})$ (resp. $SL(n,\mathbb{R}))$-invariant differential operators on $GL(n,\mathbb{R})/O(n,\mathbb{R})$ (resp. $SL(n,\mathbb{R})/SO(n,\mathbb{R})$). Let us consider the following two mappings,

$$\varphi :SL(n,\mathbb{R})/SO(n,\mathbb{R})⟶GL(n,\mathbb{R})/O(n,\mathbb{R})$$

(32)

defined by

$$\varphi (g·SO(n,\mathbb{R}\left)\right):=g·O(n,\mathbb{R})\hspace{1em}\mathrm{for} \mathrm{all} g\in SL(n,\mathbb{R})$$

and

$$p:GL{(n,\mathbb{R})}^{+}/SO(n,\mathbb{R})⟶SL(n,\mathbb{R})/SO(n,\mathbb{R})$$

(33)

defined by

$$p(g·SO(n,\mathbb{R})):={(det\left(g\right))}^{-1/n}g·SO(n,\mathbb{R})\hspace{1em}\mathrm{for} \mathrm{all} g\in GL{(n,\mathbb{R})}^{+}.$$

Here,

$$GL{(n,\mathbb{R})}^{+}:=\{g\in GL(n,\mathbb{R})|det\left(g\right)>0\}$$

is a subgroup of $GL(n,\mathbb{R}).$ It is easily seen that the mapping

$$q:GL{(n,\mathbb{R})}^{+}/SO(n,\mathbb{R})⟶GL(n,\mathbb{R})/O(n,\mathbb{R})$$

(34)

defined by

$$q(g·SO(n,\mathbb{R})):=g·O(n,\mathbb{R})\hspace{1em}\mathrm{for} \mathrm{all} g\in GL{(n,\mathbb{R})}^{+}$$

is a diffeomorphism. We use this fact to identify $GL{(n,\mathbb{R})}^{+}/SO(n,\mathbb{R})$ with $GL(n,\mathbb{R})/O(n,\mathbb{R})$. Now, $L_g$ denotes the left translation by g on $GL(n,\mathbb{R})/O(n,\mathbb{R})$ and $SL(n,\mathbb{R})/SO(n,\mathbb{R})$. Brennecken, Ciardo, and Hilgert [5] show the following properties (BCH1)–(BCH4):

(BCH1)

The maps $\varphi$ and p are $SL(n,\mathbb{R})$-equivariant, i.e.,

$$\varphi \circ L_g=L_g\circ \varphi \hspace{1em}\mathrm{and}\hspace{1em}p\circ L_g=L_g\circ p\hspace{1em}\mathrm{for} \mathrm{all} g\in SL(n,\mathbb{R})$$

(BCH2)

The mapping

$$\mathcal{L}:\mathbb{D}\left(GL\right(n,\mathbb{R})/O(n,\mathbb{R}\left)\right)⟶\mathbb{D}\left(SL\right(n,\mathbb{R})/SO(n,\mathbb{R}\left)\right)$$

(35)

defined by

$$\mathcal{L}\left(D\right)f:=D(f\circ p)\circ \varphi$$

for all $D\in \mathbb{D}\left(GL\right(n,\mathbb{R})/O(n,\mathbb{R}\left)\right)$ and $f\in C^{\infty }(SL(n,\mathbb{R})/SO(n,\mathbb{R}))$ is a morphism of algebras. Here, we note that we identified $GL{(n,\mathbb{R})}^{+}/SO(n,\mathbb{R})$ with $GL(n,\mathbb{R})/O(n,\mathbb{R})$. Furthermore,

$$\left(\mathcal{L}\right(D\left)f\right)\circ \varphi =D(f\circ p)$$

for all $D\in \mathbb{D}\left(GL\right(n,\mathbb{R})/O(n,\mathbb{R}\left)\right)$ and $f\in C^{\infty }(SL(n,\mathbb{R})/SO(n,\mathbb{R}))$.

(BCH3)

We let

$$S_n\left(\mathbb{R}\right):=\{X\in {\mathbb{R}}^{(n,n)}|X={}^{t}X\}.$$

According to Theorem 4 or [4] (Theorem 4.6, pp. 285–286), for each $D\in \mathbb{D}\left(GL\right(n,\mathbb{R})/O(n,\mathbb{R}\left)\right),$ there exists a polynomial $Q_D$ on $S_n\left(\mathbb{R}\right)$ such that

$$\left(Df\right)(g·O(n,\mathbb{R}))=Q_D\left({\displaystyle \frac{\partial }{\partial X}}\right){|}_{X=0}f(g·exp\left(X\right)·O(n,\mathbb{R})),$$

where $X=\left(x_{ij}\right)\in {\mathbb{R}}^{(n,n)}$ with $x_{ij}=x_{ji}(1\le i,j\le n)$ and

$${\displaystyle \frac{\partial }{\partial X}}=\left({\displaystyle \frac{1+{\delta }_{ij}}{2}}{\displaystyle \frac{\partial }{\partial x_{ij}}}\right).$$

Here, ${\delta }_{ij}$ is the Kronecker delta symbol. They show that each $f\in C^{\infty }(SL(n,\mathbb{R})/SO(n,\mathbb{R}))$,

$$\left(\mathcal{L}\left(D\right)f\right)(g·SO(n,\mathbb{R}))=Q_D\left({\displaystyle \frac{\partial }{\partial X}}\right){|}_{X=0}f\left((g·exp(X-n^{-1}\mathrm{Tr}\left(X\right)I_n)·SO(n,\mathbb{R}))\right).$$

(BCH4)

The morphism $\mathcal{L}$ is surjective and $\mathcal{L}\left({\delta }_1\right)=0.$

Combining the above properties (BCH1)–(BCH4), proved the following theorem was proven (cf. [5], (Theorem 3.5)):

Theorem 9. 

We let ${\delta }_1,{\delta }_2,\cdots ,{\delta }_n$ be the Maass–Selberg operators. Then $\mathcal{L}\left({\delta }_1\right)=0$ and $\mathcal{L}\left({\delta }_k\right)(2\le k\le n)$ are given by

$$\begin{array}{ccc}\hfill & & \mathcal{L}\left({\delta }_k\right)f(g·SO(n,\mathbb{R}))\hfill \\ \hfill & =& \mathrm{Tr}\left({\left({\displaystyle \frac{\partial }{\partial X}}\right)}^k\right){|}_{X=0}f\left((g·exp(X-n^{-1}\mathrm{Tr}\left(X\right)I_n)·SO(n,\mathbb{R}))\right),\hfill \end{array}$$

where $X=\left(x_{ij}\right)\in {\mathbb{R}}^{(n,n)}$ with $x_{ij}=x_{ji}(1\le i,j\le n)$ and

$${\displaystyle \frac{\partial }{\partial X}}=\left({\displaystyle \frac{1+{\delta }_{ij}}{2}}{\displaystyle \frac{\partial }{\partial x_{ij}}}\right).$$

The differential operators $\mathcal{L}\left({\delta }_2\right),\mathcal{L}\left({\delta }_3\right),\cdots ,\mathcal{L}\left({\delta }_n\right)$ are algebraically independent generators of $\mathbb{D}\left(SL\right(n,\mathbb{R})/SO(n,\mathbb{R}\left)\right).$

Corollary 2. 

We let

$$S_{n;0}\left(\mathbb{R}\right):=\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y,\mathrm{Tr}\left(Y\right)=0\}.$$

Then, for each $f\in C^{\infty }(SL(n,\mathbb{R})/SO(n,\mathbb{R}))$ and $Y\in S_{n;0}\left(\mathbb{R}\right)$, we have

$$\mathcal{L}\left({\delta }_k\right)f(g·SO(n,\mathbb{R}))=\mathrm{Tr}\left({\left({\displaystyle \frac{\partial }{\partial Y}}\right)}^k\right){|}_{Y=0}f(g·exp\left(Y\right))·SO(n,\mathbb{R}),\hspace{1em}2\le k\le n.$$

We let be the Lie algebra of $SL(n,\mathbb{R})$. The adjoint representation $\mathrm{Ad}$ of $SL(n,\mathbb{R})$ is given by

$$\mathrm{Ad}\left(g\right)=gX{g}^{-1},\hspace{1em}g\in SL(n,\mathbb{R}),X\in \mathfrak{g}.$$

The Killing form B of is given by

$$B(X,Y)=2n\mathrm{Tr}\left(XY\right),\hspace{1em}X,Y\in \mathfrak{g}.$$

For brevity, we put $K=SO(n,\mathbb{R}).$ The Lie algebra $\mathfrak{k}$ of K is

$$\mathfrak{k}=\left\{X\in \mathfrak{g}|X+{}^{t}X=0\right\}.$$

We let ${\mathfrak{p}}_0$ be the subspace of defined by

$${\mathfrak{p}}_0=\left\{X\in \mathfrak{g}|X={}^{t}X\in {\mathbb{R}}^{(n,n)},\hspace{1em}\mathrm{Tr}\left(X\right)=0\right\}.$$

Then,

$$\mathfrak{g}=\mathfrak{k}\oplus {\mathfrak{p}}_0$$

is the direct sum of $\mathfrak{k}$ and ${\mathfrak{p}}_0$ with respect to the Killing form B, since $\mathrm{Ad}\left(k\right){\mathfrak{p}}_0\subset {\mathfrak{p}}_0$ for any $k\in K,K$ acts on ${\mathfrak{p}}_0$ via the adjoint representation by

$$k·X=\mathrm{Ad}\left(k\right)X={kX}^{t}k,\hspace{1em}k\in K,X\in {\mathfrak{p}}_0.$$

(36)

Action (36) induces the action of K on the polynomial algebra $\mathrm{Pol}\left({\mathfrak{p}}_0\right)$ of ${\mathfrak{p}}_0$ and the symmetric algebra $S\left({\mathfrak{p}}_0\right)$. We denote by $\mathrm{Pol}{\left({\mathfrak{p}}_0\right)}^K$ (resp. $S{\left({\mathfrak{p}}_0\right)}^K$) the subalgebra of $\mathrm{Pol}\left({\mathfrak{p}}_0\right)$ (resp. $S\left({\mathfrak{p}}_0\right)$) consisting of all K-invariants. The following inner product ${(,)}_0$ on ${\mathfrak{p}}_0$ defined by

$${(X,Y)}_0=B(X,Y),\hspace{1em}X,Y\in {\mathfrak{p}}_0$$

gives an isomorphism as vector spaces

$${\mathfrak{p}}_0\cong {\mathfrak{p}}_0^{\ast },\hspace{1em}X\mapsto g_X,\hspace{1em}X\in {\mathfrak{p}}_0,\hspace{1em}$$

(37)

where ${\mathfrak{p}}_0^{\ast }$ denotes the dual space of ${\mathfrak{p}}_0$ and $g_X$ is the linear functional on ${\mathfrak{p}}_0$ defined by

$$g_X\left(Y\right)={(Y,X)}_0,\hspace{1em}Y\in {\mathfrak{p}}_0.$$

It is known that there is a canonical linear bijection of $S{\left({\mathfrak{p}}_0\right)}^K$ onto $\mathbb{D}\left(\mathfrak{P}\right)$. Identifying ${\mathfrak{p}}_0$ with ${\mathfrak{p}}_0^{\ast }$ by the above isomorphism (37), we obtain a canonical linear bijection

$${\Psi }_n:\mathrm{Pol}{\left({\mathfrak{p}}_0\right)}^K⟶\mathbb{D}\left({\mathfrak{P}}_n\right)$$

(38)

of $\mathrm{Pol}{\left({\mathfrak{p}}_0\right)}^K$ onto $\mathbb{D}\left({\mathfrak{P}}_n\right)$. The map ${\Psi }_n$ is described explicitly as follows. We put $N_0=n(n+1)/2-1$. We let $\left\{{\xi }_{\alpha }|1\le \alpha \le N_0\right\}$ be a basis of ${\mathfrak{p}}_0$. If $P\in \mathrm{Pol}{\left({\mathfrak{p}}_0\right)}^K$, then

$$\left({\Psi }_n\left(P\right)f\right)\left(gK\right)={\left[P\left({\displaystyle \frac{\partial }{\partial t_{\alpha }}}\right)f\left(g·exp\left(\sum _{\alpha =1}^{N_0}{t}_{\alpha }{\xi }_{\alpha }\right)K\right)\right]}_{\left(t_{\alpha }\right)=0},$$

(39)

where $f\in C^{\infty }\left({\mathfrak{P}}_n\right)$. We refer to [4] (Theorem 4.6, pp. 285–286) or Formula (17) for more detail. In general, it is very hard to express ${\Psi }_n\left(P\right)$ explicitly for a polynomial $P\in \mathrm{Pol}{\left({\mathfrak{p}}_0\right)}^K$.

If we repeat a partial decomposition process for $Y\in {\mathfrak{P}}_n$, we obtain the Iwasawa decomposition

$$Y=y^{-1}\mathrm{diag}\left(1,y_1^2,{\left(y_1{y}_2\right)}^2,\cdots ,{(y_1{y}_2\cdots y_{n-1})}^2\right)\left[\left(\begin{array}{cccc}1& x_{12}& \cdots & x_{1n}\\ 0& 1& \cdots & x_{2n}\\ 0& 0& \ddots & ⋮\\ 0& 0& 0& 1\end{array}\right)\right],$$

(40)

where $y>0,y_j\in \mathbb{R}(1\le j\le n-1)$ and $x_{ij}\in \mathbb{R}(1\le i<j\le n).$ Here, $y=y_1^{2(n-1)}\cdots y_{n-1}^2$ and $\mathrm{diag}(a_1,\cdots ,a_n)$ denotes the $n\times n$ diagonal matrix with diagonal entries $a_1,\cdots ,a_n$. In this case, we denote $Y=(y_1,\cdots ,y_{n-1},x_{12},\cdots ,x_{n-1,n}).$

We define ${\Gamma }_n=GL(n,\mathbb{Z})/\{\pm I_n\}.$ We observe that ${\Gamma }_n=SL(n,\mathbb{Z})/\{\pm I_n\}$ if n is even and ${\Gamma }_n=SL(n,\mathbb{Z})$ if n is odd. An automorphic form for ${\Gamma }_n$ is defined to be a real analytic function $f:{\mathfrak{P}}_n⟶\mathbb{C}$ satisfying the following conditions (AF1)–(AF3):

(AF1)

f is an eigenfunction for all $SL(n,\mathbb{R})$-invariant differential operators on ${\mathfrak{P}}_n$.

(AF2)

$f\left(\gamma Y{}^t\gamma \right)=f\left(Y\right)\hspace{1em}\mathrm{for} \mathrm{all} \gamma \in {\Gamma }_n\hspace{1em}\mathrm{and}\hspace{1em}Y\in {\mathfrak{P}}_n.$

(AF3)

There exist a constant $C>0$ and $s\in {\mathbb{C}}^{n-1}$ with $s=(s_1,\cdots ,s_{n-1})$

such that $\left|f\left(Y\right)\right|\le C|p_{-s}\left(Y\right)|$ as the upper left determinants $det{Y}_j⟶\infty ,$$j=1,2,\cdots ,n-1$, where

$$p_{-s}\left(Y\right):=\prod _{j=1}^{n-1}{(det{Y}_j)}^{-s_j}$$

is Selberg’s power function (cf. [3,6]).

We denote by $\mathbf{A}\left({\Gamma }_n\right)$ the space of all automorphic forms for ${\Gamma }_n.$ A $\mathrm{cusp} \mathrm{form}$ $f\in \mathbf{A}\left({\Gamma }_n\right)$ is an automorphic form for ${\Gamma }_n$ satisfying the following conditions:

$${\int }_{X\in {(\mathbb{R}/\mathbb{Z})}^{(j,n-j)}}f\left(Y\left[\left(\begin{array}{cc}{I}_j& X\\ 0& I_{n-j}\end{array}\right)\right]\right)dX=0,\hspace{1em}1\le j\le n-1.$$

Here, ${(\mathbb{R}/\mathbb{Z})}^{(j,n-j)}$ denotes the set of all $j\times (n-j)$ matrices with entries in the one-dimensional real torus $\mathbb{R}/\mathbb{Z}$. We denote by ${\mathbf{A}}_0\left({\Gamma }_n\right)$ the space of all cusp forms for ${\Gamma }_n.$

Definition 4. 

For any positive integer $n\ge 2$, we define ${\mathfrak{H}}_n$ to be the set of all $n\times n$ real matrices of the form $z=x·y$, where

$$x=\left(\begin{array}{ccccc}1& x_{12}& x_{13}& \cdots & x_{1n}\\ 0& 1& x_{22}& \cdots & x_{2n}\\ 0& 0& \ddots & ⋮& ⋮\\ 0& 0& 0& 0& 1\end{array}\right)$$

and

$$y=\mathrm{diag}(y_1{y}_2\cdots y_{n-1},y_1{y}_2\cdots y_{n-2},\cdots ,y_1,1)$$

with $x_{ij}\in \mathbb{R}$ for $1\le i<j\le n$ and $y_i\ge 0$ for $1\le i\le n-1.$

We can show that ${\mathfrak{H}}_n$ is diffeomorphic to ${\mathfrak{P}}_n$. In fact, we have the Iwasawa decomposition

$$GL(n,\mathbb{R})={\mathfrak{H}}_n·O\left(n\right)·Z_n,$$

where $Z_n(\cong {\mathbb{R}}^{\times })$ is the center of $GL(n,\mathbb{R})$ (cf. [7], (Proposition 1.2.6, pp. 11–12)). Here,

$$O\left(n\right):=O(n,\mathbb{R})=\{k\in GL(n,\mathbb{R})|{}^{t}kk=k{}^{t}k=I_n\}$$

denotes the real orthogonal group of degree n. We see easily that

$${\mathfrak{H}}_n\cong GL(n,\mathbb{R})/(O\left(n\right)·{\mathbb{R}}^{\times }),$$

where ≅ denotes the diffeomorphism.

It is seen that $GL(n,\mathbb{R})$ acts on ${\mathfrak{H}}_n$ by left translation (cf. [7], (Proposition 1.2.10, p. 14)). Then, we obtain

$$SL(n,\mathbb{Z})\setminus SL(n,\mathbb{R})/SO\left(n\right)\cong SL(n,\mathbb{Z})\setminus GL(n,\mathbb{R})/(O\left(n\right)·{\mathbb{R}}^{\times }),$$

where $SO\left(n\right):=SO(n,\mathbb{R})=SL(n,\mathbb{R})\cap O\left(n\right).$ We let

$${\mathfrak{S}}_n:=SL(n,\mathbb{Z})\setminus SL(n,\mathbb{R})/SO\left(n\right)=SL(n,\mathbb{Z})\setminus {\mathfrak{P}}_n,{\mathfrak{P}}_n:=SL(n,\mathbb{R})/SO\left(n\right)$$

be the locally symmetric space. Therefore, we obtain the following isomorphism:

$${\mathfrak{S}}_n\cong SL(n,\mathbb{Z})\setminus {\mathfrak{H}}_n.$$

${\mathfrak{P}}_n$ parameterizes special principally polarized real tori of dimension n (cf. [15]). The arithmetic quotient ${\mathfrak{S}}_n$ is the moduli space of isomorphism classes of special principally polarized real tori of dimension n. Unfortunately, ${\mathfrak{S}}_n$ does not admit the structure of a real algebraic variety and does not admit a compactification which is defined over the rational number field $\mathbb{Q}$ (cf. [16] or [17]).

Remark 7. 

In [25,26], Borel and Ji constructed the geodesic compactification, the standard compactification, and a maximal Satake compactification of the locally symmetric space ${\mathfrak{S}}_n$.

Remark 8. 

Müller [27] studied Weyl’s law for the cuspidal spectrum of $SL(n,\mathbb{R})$. In [28], Lapid and Müller studied the cuspidal spectrum of ${\mathfrak{S}}_n$. In [29], Matz and Müller introduced the analytic torsion for ${\mathfrak{S}}_n$.

Proposition 1. 

We let $n\ge 2.$ Following the coordinates of Definition 4, we put

$$d^{\ast }x=\prod _{1\le i<j\le n}d{x}_{ij}\hspace{1em}\mathrm{and}\hspace{1em}d{y}^{\ast }=\prod _{k=1}^{n-1}{y}_k^{-n(n-k)-1}d{y}_k.$$

Then,

$$d^{\ast }z=d^{\ast }x·d^{\ast }y$$

is the left $SL(n,\mathbb{R})$-invariant volume element on ${\mathfrak{H}}_n.$

Proof. 

The proof can be found in [7] (Proposition 1.5.3, pp. 25–26). □

Theorem 10. 

We let $n\ge 2.$ Then, volume $\mathbf{Vol}({\Gamma }^n\setminus {\mathfrak{H}}_n)$ of ${\Gamma }^n\setminus {\mathfrak{H}}_n$ is given by

$$\mathbf{Vol}({\Gamma }^n\setminus {\mathfrak{H}}_n)={\int }_{{\Gamma }^n\setminus {\mathfrak{H}}_n}{d}^{\ast }z=n·2^{n-1}·\prod _{k=2}^n{\displaystyle \frac{\zeta \left(k\right)}{\mathbf{Vol}\left(S^{k-1}\right)}},$$

where ${\Gamma }^n=SL(n,\mathbb{Z})$ and

$$\mathbf{Vol}\left(S^{k-1}\right)={\displaystyle \frac{2{\left(\sqrt{\pi }\right)}^k}{\Gamma (k/2)}}$$

denotes the volume of the $(k-1)$-dimensional sphere $S^{k-1}$, $\zeta \left(k\right)={\sum }_{n=1}^{\infty }{n}^{-k}$ is the Riemann zeta function, and $\Gamma \left(p\right)$ denotes the usual Gamma function.

Proof. 

The proof can be found in [7] (Theorem 1.6.1, pp. 27–37). □

Remark 9. 

Since ${\mathfrak{H}}_n$ is diffeomorphic to ${\mathfrak{P}}_n$,

$$\mathbf{Vol}({\Gamma }^n\setminus {\mathfrak{H}}_n)=\mathbf{Vol}({\Gamma }^n\setminus {\mathfrak{P}}_n)\hspace{1em}(\mathrm{see} \left(31\right)).$$

The calculation of Goldfeld [7] (Theorem 1.6.1, pp. 27–37) is different from that of Garret [24].

For any $\nu =({\nu }_1,{\nu }_2,\dots ,{\nu }_{n-1}),$ we define the function $I_{\nu }:{\mathfrak{H}}_n⟶\mathbb{C}$ by

$$I_{\nu }\left(z\right):=\prod _{i=1}^{n-1}\prod _{j=1}^{n-1}{y}_i^{b_{ij}{\nu }_j},$$

(41)

where

$$b_{ij}:=\left\{\begin{array}{cc} ij,\hfill & \mathrm{if} i+j\le n\hfill \\ (n-i)(n-j),\hfill & \mathrm{if} i+j\ge n.\hfill \end{array}\right.$$

Then, we see that $I_{\nu }\left(z\right)$ is an eigenfunction of $\mathbb{D}\left({\mathfrak{H}}_n\right).$ Let us write

$$D{I}_{\nu }\left(z\right)={\lambda }_D·I_{\nu }\left(z\right)\hspace{1em}\mathrm{for} \mathrm{every} D\in \mathbb{D}\left({\mathfrak{H}}_n\right)$$

(42)

since

$${\lambda }_{D_1{D}_2}={\lambda }_{D_1}{\lambda }_{D_2}\hspace{1em}\mathrm{for} \mathrm{all} D_1,D_2\in \mathbb{D}\left({\mathfrak{H}}_n\right).$$

Function ${\lambda }_D$ (viewed as a function of D) is a character of $\mathbb{D}\left({\mathfrak{H}}_n\right)$ which is called the $\mathsf{Harish}\mathsf{–}\mathsf{Chandra} \mathsf{character}$.

Following Goldfeld (cf. [7], (Definition 5.1.3, pp. 115–116), the notion of a Maass form is defined in the following way.

Definition 5. 

We let $n\ge 2.$ We put ${\Gamma }^n=SL(n,\mathbb{Z}).$ For any $\nu =({\nu }_1,{\nu }_2,\cdots ,{\nu }_{n-1})\in {\mathbb{C}}^{n-1},$ a smooth $f:{\Gamma }^n\setminus {\mathfrak{H}}_n⟶\mathbb{C}$ is said to be a$\mathsf{Maass} \mathsf{form}$for ${\Gamma }^n$ of type ν if it satisfies the following conditions(M1)–(M3):

(M1)

$F\left(\gamma z\right)=f\left(z\right) \mathrm{for} \mathrm{all} \gamma \in {\Gamma }^n\mathrm{and}z\in {\mathfrak{H}}_n.$

(M2)

$Df\left(z\right)={\lambda }_{D}f\left(z\right) \mathrm{for} \mathrm{all} D\in \mathbb{D}\left({\mathfrak{H}}_n\right)\mathrm{given} \mathrm{by} \left(42\right)$.

(M3)

${\int }_{{\Gamma }^n\cap U\setminus U}f\left(uz\right)du=0$ for all upper triangular groups U of the form

$$U=\left\{\left(\begin{array}{cccc}{I}_{r_1}& \ast & \ast & \ast \\ 0& I_{r_2}& \ast & \ast \\ 0& 0& \ddots & \ast \\ 0& 0& 0& I_{r_b}\end{array}\right)\right\}$$

with $r_1+r_2+\cdots +r_b=n.$ Here, $I_r$ denotes the $r\times r$ identity matrix and ∗ denotes arbitrary real matrices.

Remark 10. 

In [7], Dorian Goldfeld studied Whittaker functions associated with Maass forms, Hecke operators for ${\Gamma }^n$, the Godement-Jacquet L-function for ${\Gamma }^n$, Eisenstein series for ${\Gamma }^n$, and Poincaré series for ${\Gamma }^n$.

5. Invariant Differential Operators on $\mathit{S}\mathit{p}(\mathbf{2}\mathit{n},\mathbb{R})/\mathit{U}\left(\mathit{n}\right)$

The first part of this section is based on the author’s paper [30] (pp. 279–281). Throughout this section, we let $G:=Sp(2n,\mathbb{R})$ and $K=U\left(n\right).$ We let

$${\mathbb{H}}_n:=\{\Omega \in {\mathbb{C}}^{(n,n)}|\Omega ={}^t\Omega ,\hspace{1em}\mathrm{Im}\Omega >0\}$$

be the Siegel upper half plane of degree n. Then, G acts on ${\mathbb{H}}_n$ transitively by

$$M·\Omega =(A\Omega +B){(C\Omega +D)}^{-1},$$

(43)

where $M=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in G$ and $\Omega \in {\mathbb{H}}_n.$ The stabilizer of Action (43) at $i{I}_n$ is

$$\left\{\left(\begin{array}{cc}A& B\\ -B& A\end{array}\right)|A+iB\in U\left(n\right)\right\}\cong U\left(n\right).$$

Thus, we obtain the biholomorphic map

$$G/K⟶{\mathbb{H}}_n,\hspace{1em}gK\mapsto g·i{I}_n,\hspace{1em}g\in G.$$

${\mathbb{H}}_n$ is a Hermitian symmetric manifold. In fact, it is known that ${\mathbb{H}}_n$ is an Einstein–Kähler Hermitian symmetric space.

For $\Omega =\left({\omega }_{ij}\right)\in {\mathbb{H}}_n,$ we write $\Omega =X+iY$ with $X=\left(x_{ij}\right),Y=\left(y_{ij}\right)$ real. We put $d\Omega =\left(d{\omega }_{ij}\right)$ and $d\overline{\Omega }=\left(d{\overline{\omega }}_{ij}\right)$. We also put

$${\displaystyle \frac{\partial }{\partial \Omega }}=\left({\displaystyle \frac{1+{\delta }_{ij}}{2}}{\displaystyle \frac{\partial }{\partial {\omega }_{ij}}}\right)\hspace{1em} \mathrm{and}\hspace{1em} {\displaystyle \frac{\partial }{\partial {\overline{\Omega }}_{ij}}}=\left({\displaystyle \frac{1+{\delta }_{ij}}{2}}{\displaystyle \frac{\partial }{\partial {\overline{\omega }}_{ij}}}\right).$$

C. L. Siegel [31] introduced the symplectic metric $d{s}_{n;A}^2$ on ${\mathbb{H}}_n$ invariant under the action (43) of $Sp(2n,\mathbb{R})$ that is given by

$$d{s}_{n;A}^2=A·\mathrm{Tr}(Y^{-1}d\Omega Y^{-1}d\overline{\Omega }),\hspace{1em}A>0.$$

(44)

It is known that the metric $d{s}_{n;A}^2$ is a Kähler–Einstein metric. H. Maass [32] proved that its Laplace operator ${\Delta }_{n;A}$ is given by

$${\Delta }_{n;A}={\displaystyle \frac{4}{A}}·\mathrm{Tr}\left(Y{}^t\left(Y{\displaystyle \frac{\partial }{\partial \overline{\Omega }}}\right){\displaystyle \frac{\partial }{\partial \Omega }}\right)$$

(45)

and

$$d{v}_n(\Omega )={(detY)}^{-(n+1)}\prod _{1\le i\le j\le n}d{x}_{ij}\prod _{1\le i\le j\le n}d{y}_{ij}$$

(46)

is a $Sp(2n,\mathbb{R})$-invariant volume element on ${\mathbb{H}}_n$ (cf. [33], (p. 130)).

We let $\mathbb{D}\left({\mathbb{H}}_n\right)$ be the algebra of all differential operators on ${\mathbb{H}}_n$ invariant under Action (43). Then, according to Harish–Chandra [10,11],

$$\mathbb{D}\left({\mathbb{H}}_n\right)=\mathbb{C}[D_1,\cdots ,D_n],$$

where $D_1,\cdots ,D_n$ are algebraically independent invariant differential operators on ${\mathbb{H}}_n$. That is, $\mathbb{D}\left({\mathbb{H}}_n\right)$ is a commutative algebra that is finitely generated by n algebraically independent invariant differential operators on ${\mathbb{H}}_n$. Maass [2] found the explicit $D_1,\cdots ,D_n$. We let ${\mathfrak{g}}_{\mathbb{C}}$ be the complexification of the Lie algebra of G. It is known that $\mathbb{D}\left({\mathbb{H}}_n\right)$ is isomorphic to the center of the universal enveloping algebra of ${\mathfrak{g}}_{\mathbb{C}}$ (cf. [4]).

Now, we review differential operators on the Siegel upper half plane ${\mathbb{H}}_n$ invariant under Action (43). The isotropy subgroup K at $i{I}_n$ for Action (43) is a maximal compact subgroup given by

$$K=\left\{\left(\begin{array}{cc}A& -B\\ B& A\end{array}\right)|A{}^{t}A+B{}^{t}B=I_n, A{}^{t}B=B{}^{t}A, A,B\in {\mathbb{R}}^{(n,n)}\right\}\cong U\left(n\right).$$

We let $\mathfrak{k}$ be the Lie algebra of K. Then, the Lie algebra of G has a Cartan decomposition $=\mathfrak{k}\oplus \mathfrak{m}$, where

$$\mathfrak{g}=\left\{\left(\begin{array}{cc}{X}_1& X_2\\ X_3& -{}^{t}X_1\end{array}\right)|X_1,X_2,X_3\in {\mathbb{R}}^{(n,n)},X_2={}^{t}X_2,X_3={}^{t}X_3\right\},$$

$$\mathfrak{k}=\left\{\left(\begin{array}{cc}X& -Y\\ Y& X\end{array}\right)\in {\mathbb{R}}^{(2n,2n)}|{}^{t}X+X=0,Y={}^{t}Y\right\},$$

$$\mathfrak{m}=\left\{\left(\begin{array}{cc}X& Y\\ Y& -X\end{array}\right)|X={}^{t}X,Y={}^{t}Y,X,Y\in {\mathbb{R}}^{(n,n)}\right\}.$$

The subspace $\mathfrak{m}$ of may be regarded as the tangent space of ${\mathbb{H}}_n$ at $i{I}_n.$ The adjoint representation of G on induces the action of K on $\mathfrak{m}$ given by

$$k\circ Z=kZ{}^{t}k,\hspace{1em}\mathrm{where} k\in K \mathrm{and} Z\in \mathfrak{m}.$$

(47)

We let $T_n$ be the vector space of $n\times n$ symmetric complex matrices. We let $\Psi :\mathfrak{m}⟶T_n$ be the map defined by

$$\Psi \left(\left(\begin{array}{cc}X& Y\\ Y& -X\end{array}\right)\right)=X+iY,\hspace{1em}\left(\begin{array}{cc}X& Y\\ Y& -X\end{array}\right)\in \mathfrak{m}.$$

(48)

We let $\delta :K⟶U\left(n\right)$ be the isomorphism defined by

$$\delta \left(\left(\begin{array}{cc}A& -B\\ B& A\end{array}\right)\right)=A+iB,\hspace{1em}\left(\begin{array}{cc}A& -B\\ B& A\end{array}\right)\in K,$$

(49)

where $U\left(n\right)$ denotes the unitary group of degree n. We identify $\mathfrak{m}$ (resp. K) with $T_n$ (resp. $U\left(n\right)$) through the map $\Psi$ (resp. $\delta$). We consider the action of $U\left(n\right)$ on $T_n$ defined by

$$h·\omega =h\omega {}^{t}h,\hspace{1em}h\in U\left(n\right),\omega \in T_n.$$

(50)

Then, the adjoint Action (47) of K on $\mathfrak{m}$ is compatible with Action (50) of $U\left(n\right)$ on $T_n$ through map $\Psi .$ Precisely for any $k\in K$ and $Z\in \mathfrak{m}$, we obtain

$$\Psi \left(kZ{}^{t}k\right)=\delta \left(k\right)\Psi \left(Z\right){}^t\delta \left(k\right).$$

(51)

Action (50) induces the action of $U\left(n\right)$ on the polynomial algebra $\mathrm{Pol}\left(T_n\right)$ and the symmetric algebra $S\left(T_n\right)$, respectively. We denote by $\mathrm{Pol}{\left(T_n\right)}^{U\left(n\right)}$ $\left(\mathrm{resp}.S{\left(T_n\right)}^{U\left(n\right)}\right)$ the subalgebra of $\mathrm{Pol}\left(T_n\right)$ $\left(\mathrm{resp}.S\left(T_n\right)\right)$ consisting of $U\left(n\right)$-invariants. The following inner product $(,)$ on $T_n$ defined by

$$(Z,W)=\mathrm{Tr}\left(Z\overline{W}\right),\hspace{1em}Z,W\in T_n$$

gives an isomorphism as vector spaces

$$T_n\cong T_n^{\ast },\hspace{1em}Z\mapsto h_Z,\hspace{1em}Z\in T_n,$$

(52)

where $T_n^{\ast }$ denotes the dual space of $T_n$ and $f_Z$ is the linear functional on $T_n$ defined by

$$h_Z\left(W\right)=(W,Z),\hspace{1em}W\in T_n.$$

It is known that there is a canonical linear bijection of $S{\left(T_n\right)}^{U\left(n\right)}$ onto the algebra $\mathbb{D}\left({\mathbb{H}}_n\right)$ of differential operators on ${\mathbb{H}}_n$ invariant under Action (43) of G. Identifying $T_n$ with $T_n^{\ast }$ by the above isomorphism (52), we obtain a canonical linear bijection

$${\Theta }_n:\mathrm{Pol}{\left(T_n\right)}^{U\left(n\right)}⟶\mathbb{D}\left({\mathbb{H}}_n\right)$$

(53)

of $\mathrm{Pol}{\left(T_n\right)}^{U\left(n\right)}$ onto $\mathbb{D}\left({\mathbb{H}}_n\right)$. The map ${\Theta }_n$ is described explicitly as follows. Similarly, Action (47) induces the action of K on the polynomial algebra $\mathrm{Pol}\left(\mathfrak{m}\right)$ and the symmetric algebra $S\left(\mathfrak{m}\right)$, respectively. Through map $\Psi$, the subalgebra $\mathrm{Pol}{\left(\mathfrak{m}\right)}^K$ of $\mathrm{Pol}\left(\mathfrak{m}\right)$ consisting of K-invariants is isomorphic to $\mathrm{Pol}{\left(T_n\right)}^{U\left(n\right)}$. We put $N_{\ast }=n(n+1)$. We let $\left\{{\xi }_{\alpha }|1\le \alpha \le N_{\ast }\right\}$ be a basis of a real vector space $\mathfrak{m}$. If $P\in \mathrm{Pol}{\left(\mathfrak{m}\right)}^K$, then

$$\left({\Theta }_n\left(P\right)f\right)\left(gK\right)={\left[P\left({\displaystyle \frac{\partial }{\partial t_{\alpha }}}\right)f\left(g·\exp \left(\sum _{\alpha =1}^{N_{\ast }}{t}_{\alpha }{\xi }_{\alpha }\right)K\right)\right]}_{\left(t_{\alpha }\right)=0},$$

(54)

where $f\in C^{\infty }\left({\mathbb{H}}_n\right)$. We refer to [4] (Theorem 4.6, pp. 285–286) for more detail. In general, it is hard to express ${\Theta }_n\left(P\right)$ explicitly for a polynomial $P\in \mathrm{Pol}{\left(\mathfrak{m}\right)}^K$.

According to the work of Harish–Chandra [10,11], the algebra $\mathbb{D}\left({\mathbb{H}}_n\right)$ is generated by n algebraically independent generators and is isomorphic to the commutative algebra $\mathbb{C}[x_1,\cdots ,x_n]$ with n indeterminates. We note that n is the real rank of G. We let ${\mathfrak{g}}_{\mathbb{C}}$ be the complexification of $\mathfrak{g}$. It is known that $\mathbb{D}\left({\mathbb{H}}_n\right)$ is isomorphic to the center of the universal enveloping algebra of ${\mathfrak{g}}_{\mathbb{C}}$. H. Maass found algebraically independent generators $D_1,D_2,\cdots ,D_n$ of $\mathbb{D}\left({\mathbb{H}}_n\right)$ ([2], (pp. 112–118)). In fact, we see that

$$-D_1={\Delta }_{n;1}=4\mathrm{Tr}\left(Y{}^t\left(Y{\displaystyle \frac{\partial }{\partial \overline{\Omega }}}\right){\displaystyle \frac{\partial }{\partial \Omega }}\right)$$

(55)

is the Laplace operator for the invariant metric $d{s}_{n;1}^2$ on ${\mathbb{H}}_n$. Shimura [8] found another algebraically independent generators of $\mathbb{D}\left({\mathbb{H}}_n\right)$.

Example 1. 

We consider the case when $n=1.$ The algebra $\mathrm{Pol}{\left(T_1\right)}^{U\left(1\right)}$ is generated by the polynomial

$$q\left(\omega \right)=\omega \overline{\omega },\hspace{1em}\omega =x+iy\in \mathbb{C} \mathrm{with} x,y \mathrm{real}.$$

Using Formula (54), we obtain

$${\Theta }_1\left(q\right)=4y^2\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}\right).$$

Therefore, $\mathbb{D}\left({\mathbb{H}}_1\right)=\mathbb{C}\left[{\Theta }_1\left(q\right)\right]=\mathbb{C}\left[D_1\right].$

For two complex numbers $\alpha ,\beta \in \mathbb{C}$, Maass considered the following matrix-valued differential operator given by

$${\Omega }_{\alpha ,\beta }:={\Lambda }_{\beta -{\displaystyle \frac{n+1}{2}}}{K}_{\alpha }+\alpha \left(\beta -{\displaystyle \frac{n+1}{2}}\right)·I_n,$$

(56)

where

$$K_{\alpha }:=(\Omega -\overline{\Omega }){\displaystyle \frac{\partial }{\partial \Omega }}+\alpha ·I_n$$

and

$${\Lambda }_{\beta -{\displaystyle \frac{n+1}{2}}}:=(\Omega -\overline{\Omega }){\displaystyle \frac{\partial }{\partial \overline{\Omega }}}-\left(\beta -{\displaystyle \frac{n+1}{2}}\right)·I_n.$$

That is,

$$\begin{array}{ccc}\hfill {\Omega }_{\alpha ,\beta }& =& (\Omega -\overline{\Omega }){}^t\left((\Omega -\overline{\Omega }){\displaystyle \frac{\partial }{\partial \overline{\Omega }}}\right){\displaystyle \frac{\partial }{\partial \Omega }}-\beta (\Omega -\overline{\Omega }){\displaystyle \frac{\partial }{\partial \Omega }}+\alpha (\Omega -\overline{\Omega }){\displaystyle \frac{\partial }{\partial \overline{\Omega }}}\hfill \\ \hfill & =& -4·Y{}^t\left(Y{\displaystyle \frac{\partial }{\partial \overline{\Omega }}}\right){\displaystyle \frac{\partial }{\partial \Omega }}-2\beta iY{\displaystyle \frac{\partial }{\partial \Omega }}+2\alpha iY{\displaystyle \frac{\partial }{\partial \overline{\Omega }}}.\hfill \end{array}$$

We refer to [32] (p. 49), [34] (p. 176), and [2], (p. 119). Then,

$$\mathrm{Tr}\left({\Omega }_{\alpha ,\beta }\right)=-{\Delta }_{n;1}-2\beta i\mathrm{Tr}\left(Y{\displaystyle \frac{\partial }{\partial \Omega }}\right)+2\alpha i\mathrm{Tr}\left(Y{\displaystyle \frac{\partial }{\partial \overline{\Omega }}}\right),$$

(57)

where ${\Delta }_{n;1}$ is the Laplace operator of $({\mathbb{H}}_n,d{s}_{n;1}^2)$ (see Formulas (44), (45) and (55)).

Definition 6. 

The differential operator

$${\mathcal{L}}_{\alpha ,\beta }:=-\mathrm{Tr}\left({\Omega }_{\alpha ,\beta }\right)$$

is called the$\mathsf{Siegel}\mathsf{–}\mathsf{Maass} \mathsf{Laplacian}$of weight $(\alpha ,\beta )$.

Remark 11. 

We note that ${\mathcal{L}}_{0,0}={\Delta }_{n;1}\in \mathbb{D}\left({\mathbb{H}}_n\right)$ but ${\mathcal{L}}_{\alpha ,\beta }\notin \mathbb{D}\left({\mathbb{H}}_n\right)$ if $(\alpha ,\beta )\ne (0,0)$.

The following definition is given by Kramer and Mandal (cf. [9], (Definition 4.7]).

Definition 7. 

We set ${\Gamma }_n^{♭}=Sp(2n,\mathbb{Z}).$ We let $\Gamma \subset Sp(2n,\mathbb{R})$ be a subgroup of $Sp(2n,\mathbb{R})$ commensurable with ${\Gamma }_n^{♭}$, i.e., the intersection $\Gamma \cap {\Gamma }_n^{♭}$ is a finite index subgroup of Γ as well as of ${\Gamma }_n^{♭}$. We let ${\gamma }_j\in {\Gamma }_n^{♭}(1\le j\le h)$ denote a set of representatives for the left cosets of $\Gamma \cap {\Gamma }_n^{♭}$ in ${\Gamma }_n^{♭}.$ For two complex numbers $\alpha ,\beta \in \mathbb{C}$, we then let ${\mathfrak{V}}_{\alpha ,\beta }^n(\Gamma )$ denote the space of all functions $\phi :{\mathbb{H}}_n⟶\mathbb{C}$ satisfying the following conditions (KM1)–(KM3):

(KM1)

φ is real analytic;

(KM2)

$\phi (\gamma ·\Omega )=det{(C\Omega +D)}^{\alpha }det{(C\overline{\Omega }+D)}^{\beta }\phi (\Omega )$$\mathrm{for} \mathrm{all} \gamma =\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in \Gamma$;

(KM3)

given $Y_0\in {\mathbb{R}}^{(n,n)}$ with $Y_0={}^{t}Y_0>0,$ there exist a positive real number $M\in {\mathbb{R}}^{+}$ and a positive integer $N\in {\mathbb{Z}}^{+}$ such that the inequalities

$$|det{(C_j\Omega +D_j)}^{-\alpha }det{(C_j\overline{\Omega }+D_j)}^{-\beta }\phi ({\gamma }_j·\Omega )|\le M·\mathrm{Tr}{\left(Y\right)}^N$$

holds in the region $\{\Omega =X+iY\in {\mathbb{H}}_n|Y\ge Y_0\}$ for the set of representatives

$${\gamma }_j=\left(\begin{array}{cc}{A}_j& B_j\\ C_j& D_j\end{array}\right)\in {\Gamma }_n^{♭}(1\le j\le h).$$

Remark 12. 

For $\phi \in {\mathfrak{V}}_{\alpha ,\beta }^n(\Gamma )$, we set

$${\Vert \phi \Vert }_{\alpha ,\beta }={\int }_{\Gamma \setminus {\mathbb{H}}_n}det{\left(Y\right)}^{\alpha +\beta }{\left|\phi (\Omega )\right|}^{2}d{v}_n(\Omega ),$$

where $d{v}_n(\Omega )$ is a $Sp(2n,\mathbb{R})$-invariant volume element on ${\mathbb{H}}_n$ (see Formula (46)). In this way, we obtain the Hilbert space

$${\mathcal{H}}_{\alpha ,\beta }^n(\Gamma ):=\{\phi \in {\mathfrak{V}}_{\alpha ,\beta }^n{(\Gamma )|\Vert \phi \Vert }_{\alpha ,\beta }<\infty \}$$

equipped with the inner product

$${\langle \phi ,\psi \rangle }_{\alpha ,\beta }:={\int }_{\Gamma \setminus {\mathbb{H}}_n}det{\left(Y\right)}^{\alpha +\beta }\phi (\Omega )\overline{\psi (\Omega )}d{v}_n(\Omega )$$

for all $\phi ,\psi \in {\mathcal{H}}_{\alpha ,\beta }^n(\Gamma ).$ We note that in order to enable $\parallel \phi \parallel <\infty$, the exponent $N\in {\mathbb{Z}}^{+}$ in part $\left(\mathrm{KM}3\right)$ of Definition 7 has to be 0.

Remark 13. 

Kramer and Mandal showed that the Siegel–Maass Laplacian ${\mathcal{L}}_{\alpha ,\beta }$ acts as a symmetric operator on ${\mathcal{H}}_{\alpha ,\beta }^n(\Gamma )$ (cf. [9], (Theorem 5.1, pp. 11–15). That is,

$${\langle {\mathcal{L}}_{\alpha ,\beta }\phi ,\psi \rangle }_{\alpha ,\beta }={\langle \phi ,{\mathcal{L}}_{\alpha ,\beta }\psi \rangle }_{\alpha ,\beta }$$

for all $\phi ,\psi \in {\mathcal{H}}_{\alpha ,\beta }^n(\Gamma )$.

Definition 8. 

We let $\Gamma \subset Sp(2n,\mathbb{R})$ be a subgroup of $Sp(2n,\mathbb{R})$ commensurable with ${\Gamma }_n^{♭}$. The elements of ${\mathcal{H}}_{\alpha ,\beta }^n(\Gamma )$ are called$\mathsf{automorphic} \mathsf{forms} \mathsf{of} \mathsf{weight}$($\alpha ,\beta$) and degree n for Γ. Moreover, if $\phi \in {\mathcal{H}}_{\alpha ,\beta }^n(\Gamma )$ is an eigenfunction of ${\mathcal{L}}_{\alpha ,\beta }$, it is called a$\mathsf{Siegel}\mathsf{–}\mathsf{Maass} \mathsf{form} \mathsf{of} \mathsf{weight}$($\alpha ,\beta$) and degree n for Γ.

For the present being, we assume that $\Gamma \subset Sp(2n,\mathbb{R})$ is a subgroup of $Sp(2n,\mathbb{R})$ commensurable with ${\Gamma }_n^{♭}$. The case that $\alpha ={\displaystyle \frac{k}{2}}$ and $\beta =-{\displaystyle \frac{k}{2}}$ with $k\in {\mathbb{Z}}^{+}$ provides an application to the study of Siegel cusp forms of weight k for $\Gamma$. We recall the notion of Siegel modular forms.

Definition 9. 

We let $\Gamma \subset Sp(2n,\mathbb{R})$ be a subgroup of $Sp(2n,\mathbb{R})$ commensurable with ${\Gamma }_n^{♭}$. We let ${\gamma }_j\in {\Gamma }_n^{♭}(1\le j\le h)$ denote a set of representatives for the left cosets of of $\Gamma \cap {\Gamma }_n^{♭}$ in ${\Gamma }_n^{♭}.$ Function $f:{\mathbb{H}}_n⟶\mathbb{C}$ is called a$\mathsf{Siegel} \mathsf{modular} \mathsf{form} \mathsf{of} \mathsf{weight}$k and degree n for Γ if it satisfies the following conditions $\left(\mathrm{SI}1\right)$–$\left(\mathrm{SI}3\right)$:

(SI1)

f is holomorphic;

(SI2)

$f(\gamma ·\Omega )=det{(C\Omega +D)}^{k}f(\Omega ) \mathrm{for} \mathrm{all} \gamma =\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in \Gamma$;

(SI3)

given $Y_0\in {\mathbb{R}}^{(n,n)}$ with $Y_0={}^{t}Y_0>0,$ the quantities $det{(C_j\Omega +D_j)}^{-k}f({\gamma }_j·\Omega )$ are bounded in the region $\{\Omega =X+iY\in {\mathbb{H}}_n|Y\ge Y_0\}$ for the set of representatives ${\gamma }_j=\left(\begin{array}{cc}{A}_j& B_j\\ C_j& D_j\end{array}\right)\in {\Gamma }_n^{♭}(1\le j\le h)$.

We denote by $M_n(\Gamma ,k)$ the vector space of all Siegel modular form of weight k and degree n for Γ. It is known that $M_n(\Gamma ,k)$ is finite dimensional. Moreover, a Siegel modular form $f\in M_n(\Gamma ,k)$ is called a$\mathsf{Siegel} \mathsf{cusp} \mathsf{form} \mathsf{of} \mathsf{weight}$ k and degree n for Γ if the condition (SI3) is strengthened to the following condition ${\left(\mathrm{SI}3\right)}^{\ast }$:

(SI3)*

given $Y_0\in {\mathbb{R}}^{(n,n)}$ with $Y_0={}^{t}Y_0>0,$ the quantities $det{(C_j\Omega +D_j)}^{-k}f({\gamma }_j·\Omega )$ become arbitrarily small in the region $\{\Omega =X+iY\in {\mathbb{H}}_n|Y\ge Y_0\}$ for the set of representatives ${\gamma }_j=\left(\begin{array}{cc}{A}_j& B_j\\ C_j& D_j\end{array}\right)\in {\Gamma }_n^{♭}(1\le j\le h)$.

We denote by ${\mathfrak{C}}_k^n(\Gamma )$ the vector space of Siegel cusp form of weight k and degree n for Γ. The vector space ${\mathfrak{C}}_k^n(\Gamma )$ is a Hermitian inner product space equipped with the Petersson inner product given by

$$\langle f,g\rangle :={\int }_{\Gamma \setminus {\mathbb{H}}_n}det{\left(Y\right)}^{k}f(\Omega )\overline{g(\Omega )}d{v}_n(\Omega )\hspace{1em}(f,g\in {\mathfrak{C}}_k^n(\Gamma )).$$

We have

$${\Omega }_{{\displaystyle \frac{k}{2}},-{\displaystyle \frac{k}{2}}}=-4·Y{}^t\left(Y{\displaystyle \frac{\partial }{\partial \overline{\Omega }}}\right){\displaystyle \frac{\partial }{\partial \Omega }}+ikY{\displaystyle \frac{\partial }{\partial X}}\hspace{1em}\mathrm{and}\hspace{1em}{\mathcal{L}}_{{\displaystyle \frac{k}{2}},-{\displaystyle \frac{k}{2}}}=\mathrm{Tr}({\Omega }_{{\displaystyle \frac{k}{2}},-{\displaystyle \frac{k}{2}}}).$$

(58)

Theorem 11. 

We let $\Gamma \subset Sp(2n,\mathbb{R})$ be a subgroup of $Sp(2n,\mathbb{R})$ commensurable with ${\Gamma }_n^{♭}$ and let $\phi \in {\mathcal{H}}_{{\displaystyle \frac{k}{2}},-{\displaystyle \frac{k}{2}}}^n(\Gamma )$ be a Siegel–Maass form of weight $({\displaystyle \frac{k}{2}},-{\displaystyle \frac{k}{2}})$ for Γ. Then, if ${\mathcal{L}}_{{\displaystyle \frac{k}{2}},-{\displaystyle \frac{k}{2}}}\phi =\lambda \phi ,$ then $\lambda \in \mathbb{R}$ and

$$\lambda \ge {\displaystyle \frac{nk}{4}}(n+1-k).$$

The equality holds if and only if $\phi (\Omega )=det{\left(Y\right)}^{k/2}f(\Omega )$ for some Siegel cusp form $f\in {\mathfrak{C}}_k^n(\Gamma )$ of weight k for Γ. In other words,

$${\mathfrak{C}}_k^n(\Gamma )\cong \mathrm{Ker}\left({\mathcal{L}}_{{\displaystyle \frac{k}{2}},-{\displaystyle \frac{k}{2}}}+{\displaystyle \frac{nk}{4}}(n+1-k)·I_n\right)$$

(59)

of complex vector spaces induced by the assignment

$$f(\Omega )⟼det{\left(Y\right)}^{k/2}f(\Omega ),\hspace{1em}(\Omega =X+iY\in {\mathbb{H}}_n,f\in {\mathfrak{C}}_k^n(\Gamma )).$$

Proof. 

The proof can be found in [9] (pp. 15–19). □

Using the commutative algebra $\mathbb{D}\left({\mathbb{H}}_n\right)$, we introduce the notion of Maass–Siegel function for ${\Gamma }_n.$

Definition 10. 

Function $f:{\mathbb{H}}_n⟶\mathbb{C}$ is said to be a $\mathrm{Maass}–\mathrm{Siegel} \mathrm{function}$ for ${\Gamma }_n^{♭}$ if it satisfies the conditions (MS1)–(MS4):

(MS1)

f is real analytic;

(MS2)

$f(\gamma ·\Omega )=f(\Omega )$ for all $\gamma \in {\Gamma }_n^{♭}$ and $\Omega \in {\mathbb{H}}_n$;

(MS3)

f is an eigenfunction of all invariant differential operators in $\mathbb{D}\left({\mathbb{H}}_n\right)$;

(MS4)

given $Y_0\in {\mathbb{R}}^{(n,n)}$ with $Y_0={}^{t}Y_0>0,$ the quantities $f(\Omega )$ are bounded in the region $\{\Omega =X+iY\in {\mathbb{H}}_n|Y\ge Y_0\}$.

Function $f:{\mathbb{H}}_n⟶\mathbb{C}$ is said to be a$\mathsf{weak} \mathsf{Maass}\mathsf{–}\mathsf{Siegel} \mathsf{function}$for ${\Gamma }_n^{♭}$ if it satisfies the above conditions (MS1), (MS2) and (MS4) together with ${\left(\mathrm{MS}3\right)}^{\ast }$:

(MS3)*

f is an eigenfunction of the Laplace operator ${\Delta }_{n;A}$ of $({\mathbb{H}}_n,d{s}_{n;A}^2)$.

Problem 1. 

Develop the theory of harmonic analysis of $L^2({\Gamma }_n^{♭}\setminus {\mathbb{H}}_n).$ Develop the spectral theory of the Laplace operator ${\Delta }_{n;A}$ (see Formula (45)) on $L^2({\Gamma }_n^{♭}\setminus {\mathbb{H}}_n).$

6. Invariant Differential Operators on ${\mathbf{GL}}_{\mathbf{n},\mathbf{m}}/\mathbf{O}(\mathbf{n},\mathbb{R})$

This section is based on papers [15,35]. We recall that the group

$$G{L}_{n,m}\left(\mathbb{R}\right):=GL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)}$$

is the semidirect product of $GL(n,\mathbb{R})$ and the additive group ${\mathbb{R}}^{(m,n)}$ endowed with multiplication law

$$(g,\alpha )\circ (h,\beta ):=(gh,\alpha {}^{t}h^{-1}+\beta )$$

(60)

for all $g,h\in GL(n,\mathbb{R})$ and $\alpha ,\beta \in {\mathbb{R}}^{(m,n)}$. We also recall the Minkowski–Euclid space

$${\mathcal{P}}_{n,m}:={\mathcal{P}}_n\times {\mathbb{R}}^{(m,n)}=\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0\}\times {\mathbb{R}}^{(m,n)}.$$

Then, $G{L}_{n,m}\left(\mathbb{R}\right)$ acts on ${\mathcal{P}}_{n,m}$ naturally and transitively by

$$(g,\alpha )·(Y,V):=(gY{}^{t}g,(V+\alpha ){}^{t}g)$$

(61)

for all $(g,\alpha )\in G{L}_{n,m}\left(\mathbb{R}\right)$ and $(Y,V)\in {\mathcal{P}}_{n,m}.$ Since $O(n,\mathbb{R})$ is the stabilizer of the action (61) at $(I_n,0)$, the non-symmetric homogeneous space $G{L}_{n,m}\left(\mathbb{R}\right)/O(n,\mathbb{R})$ is diffeomorphic to the Minkowski–Euclid space ${\mathcal{P}}_{n,m}$. We denote by $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$ the algebra of all differential operators on ${\mathcal{P}}_{n,m}$ invariant under Action (6.2) of $G{L}_{n,m}\left(\mathbb{R}\right)$. We let

$$G{L}_{n,m}\left(\mathbb{Z}\right):=GL(n,\mathbb{Z})⋉{\mathbb{Z}}^{(m,n)}$$

denote the discrete subgroup of $G{L}_{n,m}\left(\mathbb{R}\right)$.

For a variable $(Y,V)\in {\mathcal{P}}_{n,m}$ with $Y\in {\mathcal{P}}_n$ and $V\in {\mathbb{R}}^{(m,n)}$, we put

$$Y=\left(y_{ij}\right) \mathrm{with} y_{ij}=y_{ji}, V=\left(v_{kl}\right),$$

$$dY=\left(d{y}_{ij}\right), dV=\left(d{v}_{kl}\right),$$

$$\left[dY\right]=\underset{i\le j}{\bigwedge }d{y}_{ij},\hspace{1em}\left[dV\right]=\underset{k,l}{\bigwedge }d{v}_{kl},$$

and

$${\displaystyle \frac{\partial }{\partial Y}}=\left({\displaystyle \frac{1+{\delta }_{ij}}{2}}{\displaystyle \frac{\partial }{\partial y_{ij}}}\right),\hspace{1em}{\displaystyle \frac{\partial }{\partial V}}=\left({\displaystyle \frac{\partial }{\partial v_{kl}}}\right),$$

where $1\le i,j,l\le n$ and $1\le k\le m.$

For a fixed element $(g,\alpha )\in G{L}_{n,m}\left(\mathbb{R}\right),$ we write

$$(Y_{★},V_{★})=(g,\alpha )·(Y,V)=\left(gY{}^{t}g,(V+\alpha ){}^{t}g\right),$$

where $(Y,V)\in {\mathcal{P}}_{n,m}$. Then, we obtain

$$Y_{★}=gY{}^{t}g,\hspace{1em}{V}_{★}=(V+\alpha ){}^{t}g$$

(62)

and

$${\displaystyle \frac{\partial }{\partial Y_{★}}}={}^{t}g^{-1}{\displaystyle \frac{\partial }{\partial Y}}{g}^{-1},\hspace{1em}{\displaystyle \frac{\partial }{\partial V_{★}}}={\displaystyle \frac{\partial }{\partial V}}{g}^{-1}.$$

(63)

Lemma 1. 

For all two positive real numbers a and b, the following metric $d{s}_{n,m;a,b}^2$ on ${\mathcal{P}}_{n,m}$ defined by

$$d{s}_{n,m;a,b}^2=a·\mathrm{Tr}\left(Y^{-1}dY{Y}^{-1}dY\right)+b·\mathrm{Tr}\left(Y^{-1}{}^t\left(dV\right)dV\right)$$

(64)

is a Riemannian metric on ${\mathcal{P}}_{n,m}$ which is invariant under Action (61) of $G{L}_{n,m}\left(\mathbb{R}\right)$. The Laplacian ${\Delta }_{n,m;a,b}$ of $({\mathcal{P}}_{n,m},d{s}_{n,m;a,b}^2)$ is given by

$${△}_{n,m;a,b}={\displaystyle \frac{1}{a}}·\mathrm{Tr}\left({\left(Y{\displaystyle \frac{\partial }{\partial Y}}\right)}^2\right)-{\displaystyle \frac{m}{2a}}\mathrm{Tr}\left(Y{\displaystyle \frac{\partial }{\partial Y}}\right)+{\displaystyle \frac{1}{b}}·\sum _{k\le p}{\left(\left({\displaystyle \frac{\partial }{\partial V}}\right)Y{}^t\left({\displaystyle \frac{\partial }{\partial V}}\right)\right)}_{kp}.$$

Moreover, ${△}_{n,m;a,b}$ is a differential operator of order 2 which is invariant under Action (61) of $G{L}_{n,m}\left(\mathbb{R}\right).$

Proof. 

The proof can be found in [15] (Lemma 8.1, p. 312). □

Lemma 2. 

The following volume element $d{\mu }_{n,m}(Y,V)$ on ${\mathcal{P}}_{n,m}$ defined by

$$d{\mu }_{n,m}(Y,V)={(detY)}^{-{\displaystyle \frac{n+m+1}{2}}}\left[dY\right]\left[dV\right]$$

(65)

is invariant under Action (61) of $G{L}_{n,m}\left(\mathbb{R}\right)$.

Proof. 

The proof can be found in [15] (Lemma 8.2, pp. 312–313). □

The Lie algebra ${\mathfrak{g}}_{★}$ of $G{L}_{n,m}\left(\mathbb{R}\right)$ is given by

$${\mathfrak{g}}_{★}=\left\{(X,Z)|X\in {\mathbb{R}}^{(n,n)},Z\in {\mathbb{R}}^{(m,n)}\right\}$$

equipped with the following Lie bracket:

$${\left[(X_1,Z_1),(X_2,Z_2)\right]}_{★}=\left({[X_1,X_2]}_0,Z_2{}^{t}X_1-Z_1{}^{t}X_2\right),$$

where ${[X_1,X_2]}_0=X_1{X}_2-X_2{X}_1$ denotes the usual matrix bracket and $(X_1,Z_1),(X_2,Z_2)\in {\mathfrak{g}}_{★}$. The adjoint representation ${\mathrm{Ad}}_{★}$ of $G{L}_{n,m}\left(\mathbb{R}\right)$ is given by

$${\mathrm{Ad}}_{★}\left((g,\lambda )\right)(X,Z)=\left(gX{g}^{-1},(Z-\lambda {}^{t}X){}^{t}g\right),$$

(66)

where $(g,\lambda )\in G{L}_{n,m}\left(\mathbb{R}\right)$ and $(X,Z)\in {\mathfrak{g}}_{★}$ and the adjoint representation ${\mathrm{ad}}_{★}$ of ${\mathfrak{g}}_{★}$ on ${\mathfrak{g}}_{★}$ is given by

$${\mathrm{ad}}_{★}\left((X,Z)\right)\left((X_1,Z_1)\right)={\left[(X,Z),(X_1,Z_1)\right]}_{★}.$$

We see that the Killing form $B_{★}$ of ${\mathfrak{g}}_{★}$ is given by

$$B_{★}\left((X_1,Z_1),(X_2,Z_2)\right)=(2n+m)\mathrm{Tr}\left(X_1{X}_2\right)-2\mathrm{Tr}\left(X_1\right)\mathrm{Tr}\left(X_2\right).$$

We let

$$K_{★}:=\{(k,0)\in G{L}_{n,m}\left(\mathbb{R}\right)|k\in O(n,\mathbb{R})\}\cong O(n,\mathbb{R}).$$

Then, the Lie algebra ${\mathfrak{k}}_{★}$ of $K_{★}$ is

$${\mathfrak{k}}_{★}=\left\{(X,0)\in {\mathfrak{g}}_{★}|X+{}^{t}X=0\right\}.$$

We let ${\mathfrak{p}}_{★}$ be the subspace of ${\mathfrak{p}}_{★}$ defined by

$${\mathfrak{p}}_{★}=\left\{(X,Z){\in }_{★}|X={}^{t}X\in {\mathbb{R}}^{(n,n)},Z\in {\mathbb{R}}^{(m,n)}\right\}.$$

Then, we have the following relation:

$$[{\mathfrak{k}}_{★},{\mathfrak{k}}_{★}]\subset {\mathfrak{k}}_{★}\hspace{1em}\mathrm{and}\hspace{1em}[{\mathfrak{k}}_{★},{\mathfrak{p}}_{★}]\subset {\mathfrak{p}}_{★}.$$

In addition, we have

$${\mathfrak{g}}^{★}={\mathfrak{k}}_{★}\oplus {\mathfrak{p}}_{★}\hspace{1em}(\mathrm{the} \mathrm{direct} \mathrm{sum}).$$

$K_{★}$ acts on ${\mathfrak{p}}_{★}$ via the adjoint representation ${\mathrm{Ad}}_{★}$ of $G{L}_{n,m}\left(\mathbb{R}\right)$ by

$$k_{★}·(X,Z)=\left(kX{}^{t}k,Z{}^{t}k\right),$$

(67)

where $k_{★}=(k,0)\in K_{★}$ with $k\in O(n,\mathbb{R})$ and $(X,Z)\in {\mathfrak{p}}_{★}.$

For brevity, we set $K=O(n,\mathbb{R}).$ Then, Action (68) induces the action of K on the polynomial algebra $\mathrm{Pol}\left({\mathfrak{p}}_{★}\right)$ of ${\mathfrak{p}}_{★}$ and the symmetric algebra $S\left({\mathfrak{p}}_{★}\right)$. We denote by $\mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K$ (resp. $S{\left({\mathfrak{p}}_{★}\right)}^K$) the subalgebra of $\mathrm{Pol}\left({\mathfrak{p}}_{★}\right)$ (resp. $S\left({\mathfrak{p}}_{★}\right)$) consisting of all K-invariants. The following inner product ${(,)}_{★}$ on ${\mathfrak{p}}_{★}$ defined by

$${\left((X_1,Z_1),(X_2,Z_2)\right)}_{★}=\mathrm{Tr}\left(X_1{X}_2\right)+\mathrm{Tr}\left(Z_1{}^{t}Z_2\right),(X_1,Z_1),(X_2,Y_2)\in {\mathfrak{p}}_{★}$$

gives an isomorphism as vector spaces

$${\mathfrak{p}}_{★}\cong {\mathfrak{p}}_{★}^{\ast },\hspace{1em}(X,Z)\mapsto f_{X,Z},\hspace{1em}(X,Z)\in {\mathfrak{p}}_{★},$$

(68)

where ${\mathfrak{p}}_{★}^{\ast }$ denotes the dual space of ${\mathfrak{p}}_{★}$ and $f_{X,Z}$ is the linear functional on ${\mathfrak{p}}_{★}$ defined by

$$f_{X,Z}\left((X_1,Z_1)\right)={\left((X,Z),(X_1,Z_1)\right)}_{★},\hspace{1em}(X_1,Z_1)\in {\mathfrak{p}}_{★}.$$

We let $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$ be the algebra of all differential operators on ${\mathcal{P}}_{n,m}$ that are invariant under Action (61) of $G{L}_{n,m}\left(\mathbb{R}\right)$. It is known that there is a canonical linear bijection of $S{\left({\mathfrak{p}}_{★}\right)}^K$ onto $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$. Identifying ${\mathfrak{p}}_{★}$ with ${\mathfrak{p}}_{★}^{\ast }$ by the above Isomorphism (68), we obtain a canonical linear bijection

$${\Phi }_{n,m}:\mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K⟶\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$$

(69)

of $\mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K$ onto $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$. The map ${\Phi }_{n,m}$ is described explicitly as follows. We put $N_{★}=n(n+1)/2+mn$. We let $\left\{{\eta }_{\alpha }|1\le \alpha \le N_{★}\right\}$ be a basis of ${\mathfrak{p}}_{★}$. If $P\in \mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K$, then

$$\left({\Phi }_{n,m}\left(P\right)f\right)\left(gK\right)={\left[P\left({\displaystyle \frac{\partial }{\partial t_{\alpha }}}\right)f\left(g·\exp \left(\sum _{\alpha =1}^{N_{★}}{t}_{\alpha }{\eta }_{\alpha }\right)K\right)\right]}_{\left(t_{\alpha }\right)=0},$$

(70)

where $f\in C^{\infty }\left({\mathcal{P}}_{n,m}\right)$. We refer to [4] (pp. 280–289). In general, it is very hard to express ${\Phi }_{n,m}\left(P\right)$ explicitly for a polynomial $P\in \mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K$.

We take a coordinate $(X,Z)$ in ${\mathfrak{p}}_{★}$ such that

$$X=\left(\begin{array}{cccc}{x}_{11}& {\displaystyle \frac{1}{2}}{x}_{12}& \dots & {\displaystyle \frac{1}{2}}{x}_{1n}\\ {\displaystyle \frac{1}{2}}{x}_{12}& x_{22}& \dots & {\displaystyle \frac{1}{2}}{x}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{1}{2}}{x}_{1n}& {\displaystyle \frac{1}{2}}{x}_{2n}& \dots & x_{nn}\end{array}\right)\in \mathfrak{p}\hspace{1em}\mathrm{and}\hspace{1em}Z=\left(z_{kl}\right)\in {\mathbb{R}}^{(m,n)}.$$

Here,

$$\mathfrak{p}:=\{X\in {\mathbb{R}}^{(n,n)}|X={}^{t}X\}.$$

We define the polynomials ${\alpha }_j,{\beta }_{pq}^{\left(k\right)},R_{jp}$, and $S_{jp}$ on ${\mathfrak{p}}_{★}$ by

$$\begin{array}{ccc}\hfill {\alpha }_j(X,Z)& =& \mathrm{Tr}\left(X^j\right),\hspace{1em} 1\le j\le n,\hfill \end{array}$$

(71)

$$\begin{array}{ccc}\hfill {\beta }_{pq}^{\left(k\right)}(X,Z)& =& {\left(Z{X}^k{}^{t}Z\right)}_{pq},\hspace{1em} 0\le k\le n-1, 1\le p\le q\le m,\hfill \end{array}$$

(72)

$$\begin{array}{ccc}\hfill R_{jp}(X,Z)& =& \mathrm{Tr}\left(X^j{\left({}^{t}ZZ\right)}^p\right),\hspace{1em} 0\le j\le n-1, 1\le p\le m,\hfill \end{array}$$

(73)

$$\begin{array}{ccc}\hfill S_{jp}(X,Z)& =& det\left(X^j{\left({}^{t}ZZ\right)}^p\right),\hspace{1em} 0\le j\le n-1, 1\le p\le m,\hfill \end{array}$$

(74)

where ${\left(ZX{}^{t}Z\right)}_{pq}$ denotes the $(p,q)$-entry of $ZX{}^{t}Z$.

We propose the following natural problems.

Problem 2. 

Find a complete list of explicit generators of $\mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K$.

Problem 3. 

Find all the relations among a set of generators of $\mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K$.

Problem 4. 

Find an easy or effective way to express the images of the above invariant polynomials under the Helgason map ${\Phi }_{n,m}$ explicitly.

Problem 5. 

Decompose $\mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K$ into $O(n,\mathbb{R})$-irreducibles.

Problem 6. 

Find a complete list of explicit generators of the algebra $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$. Or construct explicit $G{L}_{n,m}\left(\mathbb{R}\right)$-invariant differential operators on ${\mathcal{P}}_{n,m}.$

Problem 7. 

Find all the relations among a set of generators of $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$.

Problem 8. 

Is $\mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K$ finitely generated? Is $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$ finitely generated?

Problem 9. 

Find the center ${\mathcal{Z}}_{n,m}$ of $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$.

M. Itoh [36] proved the following theorem.

Theorem 12. 

$\mathrm{Pol}{\left({\mathfrak{p}}_{★}\right)}^K$ is generated by ${\alpha }_j(1\le j\le n)$ and ${\beta }_{pq}^{\left(k\right)}$ ($0\le k\le n-1,1\le p\le q\le m).$

Proof. 

We refer to [36] (Theorem 3.1). □

According to the above theorem, he solved Problem 2 and Problem 8. He also solved Problem 3 in [36] (Theorem 3.2).

We present some invariant differential operators on ${\mathcal{P}}_{n,m}.$ We define the differential operators $D_j,{\Omega }_{pq}$, and $L_p$ on ${\mathcal{P}}_{n,m}$ by

$$D_j=\mathrm{Tr}\left({\left(2Y{\displaystyle \frac{\partial }{\partial Y}}\right)}^j\right),\hspace{1em}1\le j\le n,$$

(75)

$${\Omega }_{pq}^{\left(k\right)}={\left\{{\displaystyle \frac{\partial }{\partial V}}{\left(2Y{\displaystyle \frac{\partial }{\partial Y}}\right)}^{k}Y{}^t\left({\displaystyle \frac{\partial }{\partial V}}\right)\right\}}_{pq},\hspace{1em}0\le k\le n-1,1\le p\le q\le m$$

(76)

and

$$L_p=\mathrm{Tr}\left({\left\{Y{}^t\left({\displaystyle \frac{\partial }{\partial V}}\right){\displaystyle \frac{\partial }{\partial V}}\right\}}^p\right),\hspace{1em}1\le p\le m.$$

(77)

Here, for matrix A we denote by $A_{pq}$ the $(p,q)$-entry of A.

Also, we define the invariant differential operators ${\mathfrak{S}}_{jp}$ by

$${\mathfrak{S}}_{jp}=\mathrm{Tr}\left({\left(2Y{\displaystyle \frac{\partial }{\partial Y}}\right)}^j{\left\{Y{}^t\left({\displaystyle \frac{\partial }{\partial V}}\right){\displaystyle \frac{\partial }{\partial V}}\right\}}^p\right),$$

(78)

where $1\le j\le n$ and $1\le p\le m.$

Remark 14. 

It is seen that $[D_1,{\Omega }_{pq}^{\left(0\right)}]=2{\Omega }_{pq}^{\left(0\right)}$ (cf. [35], (Theorem 8.1, p. 304)). Therefore, $\mathbb{D}\left({\mathcal{P}}_{n,m}\right)$ is not commutative. We refer to [35] for more details on invariant differential operators on the Minkowski–Euclid space ${\mathcal{P}}_{n,m}$.

We want to mention the special invariant differential operator on ${\mathcal{P}}_{n,m}$. In [37], the author studied the following differential operator $M_{n,m;\mathcal{M}}$ on ${\mathcal{P}}_{n,m}$ defined by

$$M_{n,m;\mathcal{M}}=\det \left(Y\right)·\det \left({\displaystyle \frac{\partial }{\partial Y}}+{\displaystyle \frac{1}{8\pi }}{}^t\left({\displaystyle \frac{\partial }{\partial V}}\right){\mathcal{M}}^{-1}\left({\displaystyle \frac{\partial }{\partial V}}\right)\right),$$

(79)

where $\mathcal{M}$ is a positive definite, symmetric half-integral matrix of degree m. This differential operator characterizes singular Jacobi forms. For more detail, we refer to [37]. According to (62) and (63), we see easily that the differential operator $M_{n,m;\mathcal{M}}$ is invariant under Action (61) of $G{L}_{n,m}\left(\mathbb{R}\right)$.

Question: 

Calculate the inverse ${\Phi }_{n,m}^{-1}\left(M_{n,m;\mathcal{M}}\right)$ of $M_{n,m;\mathcal{M}}$ under the Helgason map ${\Phi }_{n,m}$.

7. Invariant Differential Operators on ${\mathit{SL}}_{\mathit{n},\mathit{m}}\left(\mathbb{R}\right)/\mathit{SO}(\mathit{n},\mathbb{R})$

We recall that the group

$$S{L}_{n,m}\left(\mathbb{R}\right):=SL(n,\mathbb{R})⋉{\mathbb{R}}^{(m,n)}$$

is the semidirect product of $SL(n,\mathbb{R})$ and the additive group ${\mathbb{R}}^{(m,n)}$ endowed with multiplication law

$$(g,\alpha )\circ (h,\beta ):=(gh,\alpha {}^{t}h^{-1}+\beta )$$

(80)

for all $g,h\in SL(n,\mathbb{R})$, and $\alpha ,\beta \in {\mathbb{R}}^{(m,n)}$. We also recall the homogeneous space

$${\mathfrak{P}}_{n,m}:={\mathfrak{P}}_n\times {\mathbb{R}}^{(m,n)}=\{Y\in {\mathbb{R}}^{(n,n)}|Y={}^{t}Y>0,detY=1\}\times {\mathbb{R}}^{(m,n)}.$$

Then, $S{L}_{n,m}\left(\mathbb{R}\right)$ acts on ${\mathfrak{P}}_{n,m}$ naturally and transitively by

$$(g,\alpha )·(Y,V):=(gY{}^{t}g,(V+\alpha ){}^{t}g)$$

(81)

for all $(g,\alpha )\in S{L}_{n,m}\left(\mathbb{R}\right)$ and $(Y,V)\in {\mathfrak{P}}_{n,m}.$ Since $SO(n,\mathbb{R})$ is the stabilizer of Action (81) at $(I_n,0)$, the non-symmetric homogeneous space $S{L}_{n,m}\left(\mathbb{R}\right)/SO(n,\mathbb{R})$ is diffeomorphic to the non-symmetric space ${\mathfrak{P}}_{n,m}$. We denote by $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$ the algebra of all differential operators on ${\mathfrak{P}}_{n,m}$ invariant under Action (81) of $S{L}_{n,m}\left(\mathbb{R}\right)$. We let

$$S{L}_{n,m}\left(\mathbb{Z}\right):=SL(n,\mathbb{Z})⋉{\mathbb{Z}}^{(m,n)}$$

denote the discrete subgroup of $S{L}_{n,m}\left(\mathbb{R}\right)$.

From now on, we write $G_{\diamond }=S{L}_{n,m}\left(\mathbb{R}\right)$ for brevity. We let

$$\mathfrak{s}\mathfrak{l}(n,\mathbb{R})=\left\{X\in {\mathbb{R}}^{(n,n)}|\mathrm{Tr}\left(X\right)=0\right\}$$

be the Lie algebra of $SL(n,\mathbb{R}).$ Then, it is easy to see that the Lie algebra ${\mathfrak{g}}_{\diamond }$ of $G_{\diamond }$ is given by

$${\mathfrak{g}}_{\diamond }=\left\{(X,Z)|X\in \mathfrak{s}\mathfrak{l}(n,\mathbb{R}),Z\in {\mathbb{R}}^{(m,n)}\right\}$$

(82)

equipped with the following Lie bracket:

$${[(X_1,Z_1),(X_2,Z_2)]}_{\diamond }=({[X_1,X_2]}_0,Z_2{}^{t}X_1-Z_1{}^{t}X_2),$$

(83)

where ${[X_1,X_2]}_0:=X_1{X}_2-X_2{X}_1$ denotes the usual matrix bracket and $(X_1,Z_1),(X_2,Z_2)\in {\mathfrak{g}}_{\diamond }$. The adjoint representation ${\mathrm{Ad}}_{\diamond }$ of $G_{\diamond }$ is given by

$${\mathrm{Ad}}_{\diamond }\left((g,\alpha )\right)(X,Z)=(gX{g}^{-1},(Z-\alpha {}^{t}X){}^{t}g),$$

(84)

where $(g,\alpha )\in G_{\diamond }$ and $(X,Z)\in {\mathfrak{g}}_{\diamond }$. And the adjoint representation ${\mathrm{ad}}_{\diamond }$ of ${\mathfrak{g}}_{\diamond }$ on $\mathrm{End}\left({\mathfrak{g}}_{\diamond }\right)$ is given by

$${\mathrm{ad}}_{\diamond }\left((X,Z)\right)\left((X_1,Z_1)\right)={[(X,Z),(X_1,Z_1)]}_{\diamond }.$$

(85)

We easily see that the Killing form $B_{\diamond }$ of ${\mathfrak{g}}_{\diamond }$ is given by

$$B_{\diamond }((X_1,Z_1),(X_2,Z_2))=(m+4)\mathrm{Tr}\left(X_1{X}_2\right).$$

(86)

Therefore, the Killing form $B_{\diamond }$ is highly degenerate.

We let

$$K_{\diamond }=\left\{(k,0)\in G_{\diamond }|k\in SO(n,\mathbb{R})\right\}\cong SO(n,\mathbb{R})$$

be the compact subgroup of $G_{\diamond }$. Then, the Lie algebra ${\mathfrak{k}}_{\diamond }$ of $K_{\diamond }$ is

$${\mathfrak{k}}_{\diamond }=\{(X,0)\in {\mathfrak{g}}_{\diamond }|X+{}^{t}X=0, X\in {\mathbb{R}}^{(n,n)}, 0\in {\mathbb{R}}^{(m,n)}\}.$$

We let ${\mathfrak{p}}_{\diamond }$ be the subspace of ${\mathfrak{g}}_{\diamond }$ defined by

$${\mathfrak{p}}_{\diamond }=\left\{(X,Z)\in {\mathfrak{g}}_{\diamond }|X={}^{t}X\in {\mathbb{R}}^{(n,n)}, \mathrm{Tr}\left(X\right)=0,Z\in {\mathbb{R}}^{(m,n)}\right\}.$$

Then, we have the following relation:

$${[{\mathfrak{k}}_{\diamond },{\mathfrak{k}}_{\diamond }]}_{\diamond }\subset {\mathfrak{k}}_{\diamond }\hspace{1em}\mathrm{and}\hspace{1em}[{\mathfrak{k}}_{\diamond },{\mathfrak{p}}_{\diamond }]\subset {\mathfrak{p}}_{\diamond }.$$

(87)

In addition, we have

$${\mathfrak{g}}_{\diamond }={\mathfrak{k}}_{\diamond }\oplus {\mathfrak{p}}_{\diamond }\hspace{1em} (\mathrm{direct} \mathrm{sum}).$$

(88)

We note that the restriction of the Killing form $B_{\diamond }$ to ${\mathfrak{k}}_{\diamond }$ is negative definite and the restriction of $B_{\diamond }$ to the abelian subalgebra $\mathfrak{r}=\{(0,Z)\in {\mathfrak{g}}_{\diamond }\}$ is identically zero. Since $\mathfrak{r}$ is the radical of $B_{\diamond }$, $B_{\diamond }$ is degenerate (see Formula (86)).

An Iwasawa decomposition of the group $S{L}_{n,m}\left(\mathbb{R}\right)$ is given by

$$G_{\diamond }=N_{\diamond }{A}_{\diamond }{K}_{\diamond },$$

(89)

where

$$N_{\diamond }=\left\{\left(\left(\begin{array}{ccccc}1& \ast & \cdots & \cdots & \ast \\ 0& 1& \cdots & \cdots & \ast \\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& 1& \ast \\ 0& 0& 0& 0& 1\end{array}\right),a\right)\in G_{\diamond }|a\in {\mathbb{R}}^{(m,n)}\right\}$$

and

$$A_{\diamond }=\{(\mathrm{diag}(a_1,\cdots ,a_n),0)\in G_{\diamond }|a_i\in \mathbb{R},\prod _{k=1}^n{a}_k=1,\hspace{1em}1\le i\le n\}.$$

An Iwasawa decomposition of the Lie algebra ${\mathfrak{g}}_{\diamond }$ of $G_{\diamond }$ is given by

$${\mathfrak{g}}_{\diamond }={\mathfrak{n}}_{\diamond }+{\mathfrak{a}}_{\diamond }+{\mathfrak{k}}_{\diamond },$$

(90)

where

$${\mathfrak{n}}_{\diamond }=\left\{\left(\left(\begin{array}{ccccc}0& \ast & \cdots & \cdots & \ast \\ 0& 0& \cdots & \cdots & \ast \\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& 0& \ast \\ 0& 0& 0& 0& 0\end{array}\right),a\right)\in {\mathfrak{g}}_{\diamond }|a\in {\mathbb{R}}^{(m,n)}\right\}$$

and

$${\mathfrak{a}}_{\diamond }=\{(\mathrm{diag}(c_1,\cdots ,c_n),0)\in {\mathfrak{g}}_{\diamond }|c_i\in \mathbb{R},\sum _{k=1}^n{c}_k=0, 1\le i\le n\}.$$

In fact, ${\mathfrak{a}}_{\diamond }$ is the Lie algebra of $A_{\diamond }$ and ${\mathfrak{n}}_{\diamond }$ is the Lie algebra of $N_{\diamond }$.

Since ${\mathrm{Ad}}_{\diamond }\left(k\right){\mathfrak{p}}_{\diamond }\subset {\mathfrak{p}}_{\diamond }$ for any $k\in K_{\diamond },$ $K_{\diamond }$ acts on ${\mathfrak{p}}_{\diamond }$ via the adjoint representation of $K_{\diamond }$ on ${\mathfrak{p}}_{\diamond }$ by

$$k_{\diamond }·(X,Z)=\left(kX{}^{t}X,Z{}^{t}k\right),$$

(91)

where $k_{\diamond }=(k,0)\in K_{\diamond }$ with $k\in SO(n,\mathbb{R})$ and $(X,Z)\in {\mathfrak{p}}_{\diamond }.$

We put $K_{♮}=SO(n,\mathbb{R}).$ Action (91) induces the action of $K_{♮}$ on the polynomial algebra $\mathrm{Pol}\left({\mathfrak{p}}_{\diamond }\right)$ of ${\mathfrak{p}}_{\diamond }$ and the symmetric algebra $S\left({\mathfrak{p}}_{\diamond }\right)$. We denote by $\mathrm{Pol}{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}$ (resp. $S{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}$) the subalgebra of $\mathrm{Pol}\left({\mathfrak{p}}_{\diamond }\right)$ (resp. $S\left({\mathfrak{p}}_{\diamond }\right)$) consisting of all $K_{♮}$-invariants. The following inner product ${(,)}_{\diamond }$ on ${\mathfrak{p}}_{\diamond }$ defined by

$${\left((X_1,Z_1),(X_2,Z_2)\right)}_{\diamond }=\mathrm{Tr}\left(X_1{X}_2\right)+\mathrm{Tr}\left(Z_1{}^{t}Z_2\right),\hspace{1em}(X_1,Z_1),(X_2,Y_2)\in {\mathfrak{p}}_{\diamond }$$

gives an isomorphism as vector spaces

$${\mathfrak{p}}_{\diamond }\cong {\mathfrak{p}}_{\diamond }^{\ast },\hspace{1em}(X,Z)\mapsto f_{X,Z},\hspace{1em}(X,Z)\in {\mathfrak{p}}_{\diamond },$$

(92)

where ${\mathfrak{p}}_{\diamond }^{\ast }$ denotes the dual space of ${\mathfrak{p}}_{\diamond }$ and $f_{X,Z}$ is the linear functional on ${\mathfrak{p}}_{\diamond }$ defined by

$$f_{X,Z}\left((X_1,Z_1)\right)={\left((X,Z),(X_1,Z_1)\right)}_{\diamond },\hspace{1em}(X_1,Z_1)\in {\mathfrak{p}}_{\diamond }.$$

We let $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$ be the algebra of all differential operators on ${\mathfrak{P}}_{n,m}$ that are invariant under Action (81) of $G{L}_{n,m}\left(\mathbb{R}\right)$. It is known that there is a canonical linear bijection of $S{\left({\mathfrak{p}}_{\diamond }\right)}^K$ onto $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$. Identifying ${\mathfrak{p}}_{\diamond }$ with ${\mathfrak{p}}_{\diamond }^{\ast }$ by the above Isomorphism (92), we obtain a canonical linear bijection

$${\Psi }_{n,m}:\mathrm{Pol}{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}⟶\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$$

(93)

of $\mathrm{Pol}{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}$ onto $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$. The map ${\Psi }_{n,m}$ is described explicitly as follows. We put $N_{\diamond }=n(n+1)/2+mn-1$. We let $\left\{{\varsigma }_{\alpha }|1\le \alpha \le N_{\diamond }\right\}$ be a basis of ${\mathfrak{p}}_{\diamond }$. If $P\in \mathrm{Pol}{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}$; then,

$$\left({\Psi }_{n,m}\left(P\right)f\right)\left(g{K}_{\diamond }\right)={\left[P\left({\displaystyle \frac{\partial }{\partial t_{\alpha }}}\right)f\left(g·\exp \left(\sum _{\alpha =1}^{N_{\diamond }}{t}_{\alpha }{\varsigma }_{\alpha }\right)K_{\diamond }\right)\right]}_{\left(t_{\alpha }\right)=0},$$

(94)

where $f\in C^{\infty }\left({\mathfrak{P}}_{n,m}\right)$. We refer to [4] (pp. 280–289). In general, it is very hard to express ${\Psi }_{n,m}\left(P\right)$ explicitly for a polynomial $P\in \mathrm{Pol}{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}$.

We take a coordinate $(X,Z)$ in ${\mathfrak{p}}_{\diamond }$ such that

$$X=\left(\begin{array}{cccc}{x}_{11}& {\displaystyle \frac{1}{2}}{x}_{12}& \dots & {\displaystyle \frac{1}{2}}{x}_{1n}\\ {\displaystyle \frac{1}{2}}{x}_{12}& x_{22}& \dots & {\displaystyle \frac{1}{2}}{x}_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \frac{1}{2}}{x}_{1n}& {\displaystyle \frac{1}{2}}{x}_{2n}& \dots & x_{nn}\end{array}\right)\in {\mathfrak{p}}_{♭}\hspace{1em}\mathrm{and}\hspace{1em}Z=\left(z_{kl}\right)\in {\mathbb{R}}^{(m,n)},$$

where

$${\mathfrak{p}}_{♭}=\{X\in {\mathbb{R}}^{(n,n)}|X={}^{t}X,\hspace{1em}\mathrm{Tr}\left(X\right)=0\}.$$

We propose the following natural problems.

Problem 10. 

Find a complete list of explicit generators of $\mathrm{Pol}{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}$.

Problem 11. 

Find all the relations among a set of generators of $\mathrm{Pol}{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}$.

Problem 12. 

Find an easy or effective way to express the images of the above invariant polynomials under the Helgason map ${\Psi }_{n,m}$ explicitly.

Problem 13. 

Decompose $\mathrm{Pol}{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}$ into $SO(n,\mathbb{R})$-irreducibles.

Problem 14. 

Find a complete list of explicit generators of the algebra $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$ or construct explicit $S{L}_{n,m}\left(\mathbb{R}\right)$-invariant differential operators on ${\mathfrak{P}}_{n,m}.$

Problem 15. 

Find all the relations among a set of generators of $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$.

Problem 16. 

Is $\mathrm{Pol}{\left({\mathfrak{p}}_{\diamond }\right)}^{K_{♮}}$ finitely generated? Is $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$ finitely generated?

Problem 17. 

Find the center ${\mathfrak{Z}}_{n,m}$ of $\mathbb{D}\left({\mathfrak{P}}_{n,m}\right)$.

Problem 18. 

Decompose the Hilbert space $L^2(S{L}_{n,m}\left(\mathbb{Z}\right)\setminus S{L}_{n,m}\left(\mathbb{R}\right))$ into irreducible unitary representations of $S{L}_{n,m}\left(\mathbb{R}\right)$.

8. Invariant Differential Operators on ${\mathit{G}}^{\mathit{J}}/(\mathit{U}\left(\mathit{n}\right)\times \mathit{S}(\mathit{m},\mathbb{R}))$

The first part of this section is based on the author’s papers [38] and [30] (pp. 285–288).

For two positive integers m and n, we consider the Heisenberg group

$$H_{\mathbb{R}}^{(n,m)}=\left\{(\lambda ,\mu ;\kappa )\right|\lambda ,\mu \in {\mathbb{R}}^{(m,n)},\kappa \in {\mathbb{R}}^{(m,m)},\kappa +\mu {}^t\lambda \mathrm{symmetric}\}$$

endowed with the following multiplication:

$$(\lambda ,\mu ;\kappa )\circ ({\lambda }^{\prime },{\mu }^{\prime };{\kappa }^{\prime })=(\lambda +{\lambda }^{\prime },\mu +{\mu }^{\prime };\kappa +{\kappa }^{\prime }+\lambda {}^t\mu ^{\prime }-\mu {}^t\lambda ^{\prime })$$

with $(\lambda ,\mu ;\kappa ),({\lambda }^{\prime },{\mu }^{\prime };{\kappa }^{\prime })\in H_{\mathbb{R}}^{(n,m)}.$ We define the Jacobi group $G^J$ of degree n and index m that is the semidirect product of $Sp(2n,\mathbb{R})$ and $H_{\mathbb{R}}^{(n,m)}$

$$G^J=Sp(2n,\mathbb{R})⋉H_{\mathbb{R}}^{(n,m)}$$

endowed with the following multiplication law:

$$\left(M,(\lambda ,\mu ;\kappa )\right)·\left(M^{\prime },({\lambda }^{\prime },{\mu }^{\prime };{\kappa }^{\prime })\right)=\left(M{M}^{\prime },(\tilde{\lambda }+{\lambda }^{\prime },\tilde{\mu }+{\mu }^{\prime };\kappa +{\kappa }^{\prime }+\tilde{\lambda }{}^t\mu ^{\prime }-\tilde{\mu }{}^t\mu ^{\prime })\right)$$

(95)

with $M,M^{\prime }\in Sp(2n,\mathbb{R}),(\lambda ,\mu ;\kappa ),({\lambda }^{\prime },{\mu }^{\prime };{\kappa }^{\prime })\in H_{\mathbb{R}}^{(n,m)}$, and $(\tilde{\lambda },\tilde{\mu })=(\lambda ,\mu )M^{\prime }$. Then, $G^J$ acts on ${\mathbb{H}}_n\times {\mathbb{C}}^{(m,n)}$ transitively by

$$\left(M,(\lambda ,\mu ;\kappa )\right)·(\Omega ,Z)=\left(M·\Omega ,(Z+\lambda \Omega +\mu ){(C\Omega +D)}^{-1}\right),$$

(96)

where $M=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in Sp(2n,\mathbb{R}),(\lambda ,\mu ;\kappa )\in H_{\mathbb{R}}^{(n,m)}$, and $(\Omega ,Z)\in {\mathbb{H}}_n\times {\mathbb{C}}^{(m,n)}.$ We note that the Jacobi group $G^J$ is $\mathsf{not}$ a reductive Lie group and the homogeneous space ${\mathbb{H}}_n\times {\mathbb{C}}^{(m,n)}$ is not a symmetric space. From now on, for brevity, we write ${\mathbb{H}}_{n,m}={\mathbb{H}}_n\times {\mathbb{C}}^{(m,n)}$. The homogeneous space ${\mathbb{H}}_{n,m}$ is called the $\mathsf{Siegel}–\mathsf{Jacobi} \mathsf{space}$ of degree n and index m.

For a coordinate $(\Omega ,Z)\in {\mathbb{H}}_{n,m}$ with $\Omega =\left({\omega }_{\mu \nu }\right)$ and $Z=\left(z_{kl}\right)$, we write $\Omega =X+iY$ with $X=\left(x_{ij}\right),Y=\left(y_{ij}\right)$ real. We put $d\Omega =\left(d{\omega }_{ij}\right)$ and $d\overline{\Omega }=\left(d{\overline{\omega }}_{ij}\right)$ and set

$${\displaystyle \frac{\partial }{\partial \Omega }}=\left({\displaystyle \frac{1+{\delta }_{ij}}{2}}{\displaystyle \frac{\partial }{\partial {\omega }_{ij}}}\right)\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}{\displaystyle \frac{\partial }{\partial \overline{\Omega }}}=\left({\displaystyle \frac{1+{\delta }_{ij}}{2}}{\displaystyle \frac{\partial }{\partial {\overline{\omega }}_{ij}}}\right).$$

We write

$$\begin{array}{ccc}\hfill Z& =& U+iV,\hspace{1em}U=\left(u_{kl}\right),\hspace{1em}V=\left(v_{kl}\right)\mathrm{real},\hfill \\ \hfill dZ& =& \left(d{z}_{kl}\right),\hspace{1em}d\overline{Z}=\left(d{\overline{z}}_{kl}\right),\hfill \end{array}$$

$${\displaystyle \frac{\partial }{\partial Z}}=\left(\begin{array}{ccc}{\displaystyle \frac{\partial }{\partial z_{11}}}& \dots & {\displaystyle \frac{\partial }{\partial z_{m1}}}\\ ⋮& \ddots & ⋮\\ {\displaystyle \frac{\partial }{\partial z_{1n}}}& \dots & {\displaystyle \frac{\partial }{\partial z_{mn}}}\end{array}\right),\hspace{1em}{\displaystyle \frac{\partial }{\partial \overline{Z}}}=\left(\begin{array}{ccc}{\displaystyle \frac{\partial }{\partial {\overline{z}}_{11}}}& \dots & {\displaystyle \frac{\partial }{\partial {\overline{z}}_{m1}}}\\ ⋮& \ddots & ⋮\\ {\displaystyle \frac{\partial }{\partial {\overline{z}}_{1n}}}& \dots & {\displaystyle \frac{\partial }{\partial {\overline{z}}_{mn}}}\end{array}\right).$$

The author proved the following theorems in [39].

Theorem 13. 

For any two positive real numbers A and B,

$$\begin{array}{ccc}\hfill d{s}_{n,m;A,B}^2& =& A·\mathrm{Tr}\left(Y^{-1}d\Omega Y^{-1}d\overline{\Omega }\right)\hfill \\ & & +B\{\mathrm{Tr}\left(Y^{-1}{}^{t}VV{Y}^{-1}d\Omega Y^{-1}d\overline{\Omega }\right)+\mathrm{Tr}\left(Y^{-1}{}^t\left(dZ\right)d\overline{Z}\right)\hfill \\ & & -\mathrm{Tr}\left(V{Y}^{-1}d\Omega Y^{-1}{}^t\left(d\overline{Z}\right)\right)-\mathrm{Tr}\left(V{Y}^{-1}d\overline{\Omega }{Y}^{-1}{}^t\left(dZ\right)\right)\}\hfill \end{array}$$

is a Riemannian metric on ${\mathbb{H}}_{n,m}$ which is invariant under Action (96) of $G^J.$ In fact, $d{s}_{n,m;A,B}^2$ is a Kähler metric of ${\mathbb{H}}_{n,m}.$

Proof. 

See Theorem 1.1 in [39]. □

Theorem 14. 

The Laplace operator ${\Delta }_{m,m;A,B}$ of the $G^J$-invariant metric $d{s}_{n,m;A,B}^2$ is given by

$${\Delta }_{n,m;A,B}={\displaystyle \frac{4}{A}}{\mathbb{M}}_1+{\displaystyle \frac{4}{B}}{\mathbb{M}}_2,$$

(97)

where

$$\begin{array}{ccc}\hfill {\mathbb{M}}_1& =& \mathrm{Tr}\left(Y{}^t\left(Y{\displaystyle \frac{\partial }{\partial \overline{\Omega }}}\right){\displaystyle \frac{\partial }{\partial \Omega }}\right)+\mathrm{Tr}\left(V{Y}^{-1}{}^{t}V{}^t\left(Y{\displaystyle \frac{\partial }{\partial \overline{Z}}}\right){\displaystyle \frac{\partial }{\partial Z}}\right)\hfill \\ & & +\mathrm{Tr}\left(V{}^t\left(Y{\displaystyle \frac{\partial }{\partial \overline{\Omega }}}\right){\displaystyle \frac{\partial }{\partial Z}}\right)+\mathrm{Tr}\left({}^{t}V{}^t\left(Y{\displaystyle \frac{\partial }{\partial \overline{Z}}}\right){\displaystyle \frac{\partial }{\partial \Omega }}\right)\hfill \end{array}$$

and

$${\mathbb{M}}_2=\mathrm{Tr}\left(Y{\displaystyle \frac{\partial }{\partial Z}}{}^t\left({\displaystyle \frac{\partial }{\partial \overline{Z}}}\right)\right).$$

Furthermore, ${\mathbb{M}}_1$ and ${\mathbb{M}}_2$ are differential operators on ${\mathbb{H}}_{n,m}$ invariant under Action (96) of $G^J.$

Proof. 

See Theorem 1.2 in [39]. □

Remark 15. 

Erik Balslev [40] developed the spectral theory of ${\Delta }_{1,1;1,1}$ on $H_{1,1}$ for certain arithmetic subgroups of the Jacobi modular group to prove that the set of all eigenvalues of ${\Delta }_{1,1;1,1}$ satisfies the Weyl law.

Remark 16. 

Yang et al. [41] proved that the scalar curvature of $({\mathbb{H}}_{1,1},d{s}_{1,1;A,B}^2)$ is $-{\displaystyle \frac{3}{A}}$ and hence is independent of parameter B.

Remark 17. 

The scalar and Ricci curvatures of the Siegel–Jacobi space $({\mathbb{H}}_{1,m},d{s}_{1,m;A,B}^2)(m\ge 1)$ were completely computed by G. Khan and J. Zhang [42] (Proposition 8, pp. 825–826). Furthermore, Khan and Zhang proved that $({\mathbb{H}}_{1,m},d{s}_{1,m;A,B}^2)(m\ge 1)$ has non-negative orthogonal anti-bisectional curvature (cf. [42]) (Proposition 9, p. 826).

Remark 18. 

For an application of the invariant metric $d{s}_{n,m;A,B}^2$, we refer to [42,43,44].

Now, we investigate differential operators on the Siegel–Jacobi space ${\mathbb{H}}_{n,m}$ invariant under Action (96) of $G^J$. The stabilizer $K^J$ of $G^J$ at $(i{I}_n,0)$ is given by

$$K^J=\left\{\left(k,(0,0;\kappa )\right)|k\in K,\kappa ={}^t\kappa \in {\mathbb{R}}^{(m,m)}\right\},$$

where

$$K=\left\{\left(\begin{array}{cc}A& -B\\ B& A\end{array}\right)|A{}^{t}B+B{}^{t}B=I_n,\hspace{1em}A{}^{t}B=B{}^{t}A,\hspace{1em}A,B\in {\mathbb{R}}^{(n,n)}\right\}\cong U\left(n\right).$$

Therefore, ${\mathbb{H}}_{n,m}\cong G^J/K^J$ is a homogeneous space which is not symmetric. The Lie algebra ${\mathfrak{g}}^J$ of $G^J$ has a decomposition

$${\mathfrak{g}}^J={\mathfrak{k}}^J+{\mathfrak{p}}^J,$$

where

$${\mathfrak{g}}^J=\left\{\left(Z,(P,Q,R)\right)|Z\in ,P,Q\in {\mathbb{R}}^{(m,n)}, R={}^{t}R\in {\mathbb{R}}^{(m,m)}\right\},$$

$${\mathfrak{k}}^J=\left\{\left(X,(0,0,R)\right)|X\in \mathfrak{k},R={}^{t}R\in {\mathbb{R}}^{(m,m)}\right\},$$

$${\mathfrak{p}}^J=\left\{\left(Y,(P,Q,0)\right)|Y\in \mathfrak{m},P,Q\in {\mathbb{R}}^{(m,n)}\right\}.$$

Here,

$$\mathfrak{g}=\left\{\left(\begin{array}{cc}{X}_1& X_2\\ X_3& -{}^{t}X_1\end{array}\right)|X_1,X_2,X_3\in {\mathbb{R}}^{(n,n)},X_2={}^{t}X_2,X_3={}^{t}X_3\right\},$$

$$\mathfrak{k}=\left\{\left(\begin{array}{cc}X& -Y\\ Y& X\end{array}\right)\in {\mathbb{R}}^{(2n,2n)}|{}^{t}X+X=0,Y={}^{t}Y\right\}$$

and

$$\mathfrak{m}=\left\{\left(\begin{array}{cc}X& Y\\ Y& -X\end{array}\right)|X={}^{t}X,Y={}^{t}Y,X,Y\in {\mathbb{R}}^{(n,n)}\right\}.$$

We note that is the Lie algebra of $Sp(2n,\mathbb{R})$ and $\mathfrak{k}$ is the Lie algebra of $K\cong U\left(n\right)$. Thus, the tangent space of the homogeneous space ${\mathbb{H}}_{n,m}$ at $(i{I}_n,0)$ is identified with ${\mathfrak{p}}^J$.

If $\alpha =\left(\left(\begin{array}{cc}{X}_1& Y_1\\ Z_1& -{}^{t}X_1\end{array}\right),(P_1,Q_1,R_1)\right)$ and $\beta =\left(\left(\begin{array}{cc}{X}_2& Y_2\\ Z_2& -{}^{t}X_2\end{array}\right),(P_2,Q_2,R_2)\right)$ are elements of ${\mathfrak{g}}^J$, then the Lie bracket $[\alpha ,\beta ]$ of $\alpha$ and $\beta$ is given by

$$[\alpha ,\beta ]=\left(\left(\begin{array}{cc}{X}^{\ast }& Y^{\ast }\\ Z^{\ast }& -{}^{t}X^{\ast }\end{array}\right),(P^{\ast },Q^{\ast },R^{\ast })\right),$$

(98)

where

$$\begin{array}{ccc}\hfill X^{\ast }& =& X_1{X}_2-X_2{X}_1+Y_1{Z}_2-Y_2{Z}_1,\hfill \\ \hfill Y^{\ast }& =& X_1{Y}_2-X_2{Y}_1+Y_2{}^{t}X_1-Y_1{}^{t}X_2,\hfill \\ \hfill Z^{\ast }& =& Z_1{X}_2-Z_2{X}_1+{}^{t}X_2{Z}_1-{}^{t}X_1{Z}_2,\hfill \\ \hfill P^{\ast }& =& P_1{X}_2-P_2{X}_1+Q_1{Z}_2-Q_2{Z}_1,\hfill \\ \hfill Q^{\ast }& =& P_1{Y}_2-P_2{Y}_1+Q_2{}^{t}X_1-Q_1{}^{t}X_2,\hfill \\ \hfill R^{\ast }& =& P_1{}^{t}Q_2-P_2{}^{t}Q_1+Q_2{}^{t}P_1-Q_1{}^{t}P_2.\hfill \end{array}$$

Lemma 3. 

$$[{\mathfrak{k}}^J,{\mathfrak{k}}^J]\subset {\mathfrak{k}}^J,\hspace{1em}[{\mathfrak{k}}^J,{\mathfrak{p}}^J]\subset {\mathfrak{p}}^J.$$

Proof. 

The proof follows immediately from Formula (98). □

Lemma 4. 

Let

$$k^J=\left(\left(\begin{array}{cc}A& -B\\ B& A\end{array}\right),(0,0,\kappa )\right)\in K^J$$

with $\left(\begin{array}{cc}A& -B\\ B& A\end{array}\right)\in K,\kappa ={}^t\kappa \in \mathbb{R}(m,m)$ and

$$\alpha =\left(\left(\begin{array}{cc}X& Y\\ Y& -X\end{array}\right),(P,Q,0)\right)\in {\mathfrak{p}}^J$$

with $X={}^{t}X,Y={}^{t}Y\in {\mathbb{R}}^{(n,n)},P,Q\in {\mathbb{R}}^{(m,n)}.$ Then, the adjoint action of $K^J$ on ${\mathfrak{p}}^J$ is given by

$$\mathrm{Ad}\left(k^J\right)\alpha =\left(\left(\begin{array}{cc}{X}_{\ast }& Y_{\ast }\\ Y_{\ast }& -X_{\ast }\end{array}\right),(P_{\ast },Q_{\ast },0)\right),$$

(99)

where

$$\begin{array}{ccc}\hfill X_{\ast }& =& AX{}^{t}A-\left(BX{}^{t}B+BY{}^{t}A+AY{}^{t}B\right),\hfill \end{array}$$

(100)

$$\begin{array}{ccc}\hfill Y_{\ast }& =& \left(AX{}^{t}B+AY{}^{t}A+BX{}^{t}A\right)-BY{}^{t}B,\hfill \end{array}$$

(101)

$$\begin{array}{ccc}\hfill P_{\ast }& =& P{}^{t}A-Q{}^{t}B,\hfill \end{array}$$

(102)

$$\begin{array}{ccc}\hfill Q_{\ast }& =& P{}^{t}B+Q{}^{t}A.\hfill \end{array}$$

(103)

Proof. 

We leave the proof to the reader. □

We recall that $T_n$ denotes the vector space of all $n\times n$ symmetric complex matrices. For brevity, we put $T_{n,m}:=T_n\times {\mathbb{C}}^{(m,n)}.$ We define the real linear isomorphism $\Phi :{\mathfrak{p}}^J⟶T_{n,m}$ by

$$\Phi \left(\left(\begin{array}{cc}X& Y\\ Y& -X\end{array}\right),(P,Q,0)\right)=\left(X+iY,P+iQ\right),$$

(104)

where $\left(\begin{array}{cc}X& Y\\ Y& -X\end{array}\right)\in \mathfrak{m}$ and $P,Q\in {\mathbb{R}}^{(m,n)}.$

We let $S(m,\mathbb{R})$ denote the additive group consisting of all $m\times m$ real symmetric matrices. Now, we define the isomorphism $\theta :K^J⟶U\left(n\right)\times S(m,\mathbb{R})$ by

$$\theta (h,(0,0,\kappa \left)\right)=\left(\delta \right(h),\kappa ),\hspace{1em}h\in K,\kappa \in S(m,\mathbb{R}),$$

(105)

where $\delta :K⟶U\left(n\right)$ is the map defined by Formula (49). Identifying ${\mathbb{R}}^{(m,n)}\times {\mathbb{R}}^{(m,n)}$ with ${\mathbb{C}}^{(m,n)}$, we can identify ${\mathfrak{p}}^J$ with $T_{n,m}=T_n\times {\mathbb{C}}^{(m,n)}$.

Theorem 15. 

The adjoint representation of $K^J$ on ${\mathfrak{p}}^J$ is compatible with the natural action of $U\left(n\right)\times S(m,\mathbb{R})$ on $T_{n,m}$ defined by

$$(h,\kappa )·(\omega ,z):=(h\omega {}^{t}h,z{}^{t}h),\hspace{1em}\hspace{1em}h\in U\left(n\right),\kappa \in S(m,\mathbb{R}),(\omega ,z)\in T_{n,m}$$

(106)

through maps Φ and θ. Precisely, if $k^J\in K^J$ and $\alpha \in {\mathfrak{p}}^J$, then we have the following equality:

$$\Phi \left(\mathrm{Ad}\left(k^J\right)\alpha \right)=\theta \left(k^J\right)·\Phi \left(\alpha \right).$$

(107)

Here, we regard the complex vector space $T_{n,m}$ as a real vector space.

Proof. 

The proof can be found in [30] (pp. 286–287). □

We now study algebra $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$ of all differential operators on ${\mathbb{H}}_{n,m}$ invariant under the natural Action (96) of $G^J$. Action (106) induces the action of $U\left(n\right)$ on the polynomial algebra ${\mathrm{Pol}}_{n,m}:=\mathrm{Pol}\left(T_{n,m}\right).$ We denote by ${\mathrm{Pol}}_{n,m}^{U\left(n\right)}$ the subalgebra of ${\mathrm{Pol}}_{n,m}$ consisting of all $U\left(n\right)$-invariants. Similarly, Action (99) of K induces the action of K on the polynomial algebra $\mathrm{Pol}\left({\mathfrak{p}}^J\right)$. We see that through the identification of ${\mathfrak{p}}^J$ with $T_{n,m}$, the algebra $\mathrm{Pol}\left({\mathfrak{p}}^J\right)$ is isomorphic to ${\mathrm{Pol}}_{n,m}.$ The following $U\left(n\right)$-invariant inner product ${(,)}_J$ of the complex vector space $T_{n,m}$ defined by

$${\left((\omega ,z),({\omega }^{\prime },z^{\prime })\right)}_J=\mathrm{Tr}\left(\omega \overline{{\omega }^{\prime }}\right)+\mathrm{Tr}\left(z^t\overline{z^{\prime }}\right),\hspace{1em}(\omega ,z),({\omega }^{\prime },z^{\prime })\in T_{n,m}$$

gives a canonical isomorphism

$$T_{n,m}\cong T_{n,m}^{\ast },\hspace{1em}(\omega ,z)\mapsto f_{\omega ,z},\hspace{1em}(\omega ,z)\in T_{n,m},$$

where $f_{\omega ,z}$ is the linear functional on $T_{n,m}$ defined by

$$f_{\omega ,z}\left(({\omega }^{\prime },z^{\prime })\right)={\left(({\omega }^{\prime },z^{\prime }),(\omega ,z)\right)}_J,\hspace{1em}({\omega }^{\prime },z^{\prime })\in T_{n,m}.$$

According to Helgason [4] (p. 287), one obtains a canonical linear bijection of $S{\left(T_{n,m}\right)}^{U\left(n\right)}$ onto $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$. Here, $S\left(T_{n,m}\right)$ denotes the symmetric algebra of $T_{n,m}$ and $S{\left(T_{n,m}\right)}^{U\left(n\right)}$ denotes the subalgebra of all $U\left(n\right)$-invariants in $S\left(T_{n,m}\right)$. Identifying $T_{n,m}$ with $T_{n,m}^{\ast }$ by the above isomorphism, one obtains a natural linear bijection

$${\Theta }_{n,m}:{\mathrm{Pol}}_{n,m}^{U\left(n\right)}⟶\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$$

of ${\mathrm{Pol}}_{n,m}^{U\left(n\right)}$ onto $\mathbb{D}\left({\mathbb{H}}_{n,m}\right).$ The map ${\Theta }_{n,m}$ is described explicitly as follows. We put $N_{\ast }=n(n+1)+2mn$. We let $\left\{{\eta }_{\alpha }|1\le \alpha \le N_{\ast }\right\}$ be a basis of ${\mathfrak{p}}^J$. If $P\in \mathrm{Pol}{\left({\mathfrak{p}}^J\right)}^K={\mathrm{Pol}}_{n,m}^{U\left(n\right)}$, then

$$\left({\Theta }_{n,m}\left(P\right)f\right)\left(g{K}^J\right)={\left[P\left({\displaystyle \frac{\partial }{\partial t_{\alpha }}}\right)f\left(g·\exp \left(\sum _{\alpha =1}^{N_{\ast }}{t}_{\alpha }{\eta }_{\alpha }\right)K^J\right)\right]}_{\left(t_{\alpha }\right)=0},$$

(108)

where $g\in G^J$ and $f\in C^{\infty }\left({\mathbb{H}}_{n,m}\right)$. In general, it is hard to express ${\Theta }_{n,m}\left(P\right)$ explicitly for a polynomial $P\in {\mathrm{Pol}}_{n,m}^{U\left(n\right)}$.

We propose the following natural problems.

Problem 19. 

Find a complete list of explicit generators of ${Pol}_{n,m}^{U\left(n\right)}$.

Problem 20. 

Find all the relations among a set of generators of ${Pol}_{n,m}^{U\left(n\right)}$.

Problem 21. 

Find an easy or effective way to express the images of the above invariant polynomials or generators of ${Pol}_{n,m}^{U\left(n\right)}$ under the Helgason map ${\Theta }_{n,m}$ explicitly.

Problem 22. 

Decompose ${Pol}_{n,m}^{U\left(n\right)}$ into $U\left(n\right)$-irreducibles.

Problem 23. 

Find a complete list of explicit generators of the algebra $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$ or construct explicit $G^J$-invariant differential operators on ${\mathbb{H}}_{n,m}.$

Problem 24. 

Find all the relations among a set of generators of $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$.

Problem 25. 

Is ${Pol}_{n,m}^{U\left(n\right)}$ finitely generated? Is $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$ finitely generated?

Problem 26. 

Are there canonical ways to find generators of ${Pol}_{n,m}^{U\left(n\right)}$?

Problem 27. 

Find the center ${\mathfrak{C}}_{n,m}$ of $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$.

We give answers to Problems 19 and 25. We put ${\phi }^{\left(2k\right)}=Tr\left({\left(w\overline{w}\right)}^k\right)$. Moreover, for $1\le a,b\le m$ and $k\ge 0$, we put

$$\begin{array}{cccc}\hfill {\psi }_{ba}^{(0,2k,0)}& ={\left(\overline{z}{\left(w\overline{w}\right)}^k{}^{t}z\right)}_{ba},\hfill & \hfill \hspace{1em}\hspace{1em}{\psi }_{ba}^{(1,2k,0)}& ={\left(z\overline{w}{\left(w\overline{w}\right)}^k{}^{t}z\right)}_{ba},\hfill \\ \hfill {\psi }_{ba}^{(0,2k,1)}& ={\left(\overline{z}{\left(w\overline{w}\right)}^{k}w{}^t\overline{z}\right)}_{ba},\hfill & \hfill \hspace{1em}\hspace{1em}{\psi }_{ba}^{(1,2k,1)}& ={\left(z\overline{w}{\left(w\overline{w}\right)}^{k}w{}^t\overline{z}\right)}_{ba}.\hfill \end{array}$$

Then, we have the following relations:

$${\phi }^{\left(2k\right)}={\overline{\phi }}^{\left(2k\right)},{\psi }_{ab}^{(1,2k,1)}={\psi }_{ba}^{(0,2k+2,0)},{\psi }_{ab}^{(1,2k,0)}={\psi }_{ba}^{(1,2k,0)}={\overline{\psi }}_{ab}^{(0,2k,1)}={\overline{\psi }}_{ba}^{(0,2k,1)}.$$

(109)

Minoru Itoh [36] proved the following theorems:

Theorem 16. 

The algebra ${Pol}_{n,m}^{U\left(n\right)}$ is generated by the following polynomials:

$${\phi }^{(2k+2)},\hspace{1em}\hspace{1em}Re{\psi }_{ab}^{(0,2k,0)},\hspace{1em}\hspace{1em}Im{\psi }_{cd}^{(0,2k,0)},\hspace{1em}\hspace{1em}Re{\psi }_{ab}^{(1,2k,0)},\hspace{1em}\hspace{1em}Im{\psi }_{ab}^{(1,2k,0)}.$$

Here, the indices run as follows:

$$0\le k\le n-1,\hspace{1em}\hspace{1em}1\le a\le b\le m,\hspace{1em}\hspace{1em}1\le c<d\le m.$$

This is seen from the following theorem by using (109):

Theorem 17. 

The algebra ${Pol}_{n,m}^{U\left(n\right)}$ is generated by ${\phi }^{(2k+2)}$, ${\psi }_{ba}^{(0,2k,0)}$, ${\psi }_{ba}^{(0,2k,1)}$, and ${\psi }_{ba}^{(1,2k,0)}$. Here, the indices run as follows:

$$0\le k\le n-1,\hspace{1em}1\le a,b\le m.$$

Proof. 

See Theorem 1.1 in [36]. □

We consider the case $n=m=1.$ For a coordinate $(w,\xi )$ in $T_{1,1}=\mathbb{C}\times \mathbb{C}$, we write $w=r+is,\xi =\zeta +i\eta \in \mathbb{C},r,s,\zeta ,\eta$ real. The author of [45] proved that algebra ${\mathrm{Pol}}_{1,1}^{U\left(1\right)}$ is generated by

$$\begin{array}{ccc}& & q(w,\xi )={\displaystyle \frac{1}{4}}w\overline{w}={\displaystyle \frac{1}{4}}\left(r^2+s^2\right),\hfill \\ & & \alpha (w,\xi )=\xi \overline{\xi }={\zeta }^2+{\eta }^2,\hfill \\ & & \varphi (w,\xi )={\displaystyle \frac{1}{2}}\mathrm{Re}\left({\xi }^2\overline{w}\right)={\displaystyle \frac{1}{2}}r\left({\zeta }^2-{\eta }^2\right)+s\zeta \eta ,\hfill \\ & & \psi (w,\xi )={\displaystyle \frac{1}{2}}\mathrm{Im}\left({\xi }^2\overline{w}\right)={\displaystyle \frac{1}{2}}s\left({\eta }^2-{\zeta }^2\right)+r\zeta \eta .\hfill \end{array}$$

In [45], using Formula (108), the author calculated explicitly the images

$$D_1={\Theta }_{1,1}\left(q\right),\hspace{1em}{D}_2={\Theta }_{1,1}\left(\alpha \right),\hspace{1em}{D}_3={\Theta }_{1,1}\left(\varphi \right)\hspace{1em}\mathrm{and}\hspace{1em}{D}_4={\Theta }_{1,1}\left(\psi \right)$$

of $q,\xi ,\varphi$, and $\psi$ under the Helgason map ${\Theta }_{1,1}$. We can show that algebra $\mathbb{D}\left({\mathbb{H}}_{1,1}\right)$ is generated by the following differential operators:

$$\begin{array}{ccc}\hfill D_1& =& y^2\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}\right)+v^2\left({\displaystyle \frac{{\partial }^2}{\partial u^2}}+{\displaystyle \frac{{\partial }^2}{\partial v^2}}\right)\hfill \\ \hfill & & +2yv\left({\displaystyle \frac{{\partial }^2}{\partial x\partial u}}+{\displaystyle \frac{{\partial }^2}{\partial y\partial v}}\right),\hfill \\ \hfill D_2& =& y\left({\displaystyle \frac{{\partial }^2}{\partial u^2}}+{\displaystyle \frac{{\partial }^2}{\partial v^2}}\right),\hfill \\ \hfill D_3& =& y^2{\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{{\partial }^2}{\partial u^2}}-{\displaystyle \frac{{\partial }^2}{\partial v^2}}\right)-2y^2{\displaystyle \frac{{\partial }^3}{\partial x\partial u\partial v}}\hfill \\ \hfill & & -\left(v{\displaystyle \frac{\partial }{\partial v}}+1\right)D_2\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill D_4& =& y^2{\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{{\partial }^2}{\partial v^2}}-{\displaystyle \frac{{\partial }^2}{\partial u^2}}\right)-2y^2{\displaystyle \frac{{\partial }^3}{\partial y\partial u\partial v}}\hfill \\ \hfill & & -v{\displaystyle \frac{\partial }{\partial u}}{D}_2,\hfill \end{array}$$

where $\tau =x+iy$ and $z=u+iv$ with real variables $x,y,u,v.$ Moreover, we have

$$\begin{array}{ccc}\hfill D_1{D}_2-D_2{D}_1& =& 2y^2{\displaystyle \frac{\partial }{\partial y}}\left({\displaystyle \frac{{\partial }^2}{\partial u^2}}-{\displaystyle \frac{{\partial }^2}{\partial v^2}}\right)\hfill \\ \hfill & & -4y^2{\displaystyle \frac{{\partial }^3}{\partial x\partial u\partial v}}-2\left(v{\displaystyle \frac{\partial }{\partial v}}{D}_2+D_2\right).\hfill \end{array}$$

In particular, algebra $\mathbb{D}\left({\mathbb{H}}_{1,1}\right)$ is not commutative. We refer to [45] for more detail.

Hiroyuki Ochiai [46] proved the following results.

Lemma 5. 

We have the following relation:

$${\varphi }^2+{\psi }^2=q{\alpha }^2.$$

This relation exhausts all the relations among the generators $q,\alpha ,\varphi$, and ψ of ${\mathrm{Pol}}_{1,1}^{U\left(1\right)}$.

Proof. 

This follows from a direct computation. □

Theorem 18. 

We have the following relations:

(a)	$[D_1,D_2]=2D_3$,
(b)	$[D_1,D_3]=2D_1{D}_2-2D_3$,
(c)	$[D_2,D_3]=-D_2^2$,
(d)	$[D_4,D_1]=0$,
(e)	$[D_4,D_2]=0$,
(f)	$[D_4,D_3]=0$,
(g)	$D_3^2+D_4^2=D_2{D}_1{D}_2$.

These seven relations exhaust all the relations among generators $D_1,D_2,D_3$, and $D_4$ of $\mathbb{D}\left({\mathbb{H}}_{1,1}\right)$.

Proof. 

The proof can be found in [46]. □

Finally, we see that for the case when $n=m=1$, the above eight problems are completely solved.

Remark 19. 

According to Theorem 18, we see that $D_4$ is a generator of the center of $\mathbb{D}\left({\mathbb{H}}_{1,1}\right)$. We observe that the Lapalcian

$${\Delta }_{1,1;A,B}={\displaystyle \frac{4}{A}}{D}_1+{\displaystyle \frac{4}{B}}{D}_2 \hspace{1em}(\mathrm{see} \mathrm{Formula} (97))$$

of $({\mathbb{H}}_{1,1},d{s}_{1,1;A,B}^2)$ does not belong to the center of $\mathbb{D}\left({\mathbb{H}}_{1,1}\right)$.

Remark 20. 

When $n=1$ and m is an arbitrary integer, Conley and Raum [47] found the $2m^2+m+1$ explicit generators of $\mathbb{D}\left({\mathbb{H}}_{1,m}\right)$ and the explicit one generator of the center of $\mathbb{D}\left({\mathbb{H}}_{1,m}\right)$. They also found the generators of the center of the universal enveloping algebra $\mathfrak{U}\left({\mathfrak{g}}^J\right)$ of the Jacobi Lie algebra ${\mathfrak{g}}^J$. The number of generators of the center of $\mathfrak{U}\left({\mathfrak{g}}^J\right)$ is $1+{\displaystyle \frac{m(m+1)}{2}}.$

We set ${\Gamma }_n^{♭}:=Sp(2n,\mathbb{Z})$ and ${\Gamma }^J:={\Gamma }_n^{♭}⋉H_{\mathbb{Z}}^{(n,m)}$ (see notations).

Definition 11. 

Function $f:{\mathbb{H}}_{n,m}⟶\mathbb{C}$ is called a $\mathsf{Maass}–\mathsf{Jacobi} \mathsf{function}$ for ${\Gamma }^J$ if it satisfies the following conditions (MJ1)–(MJ4):

(MJ1)

f is real analytic;

(MJ2)

$f({\gamma }^J·(\Omega ,Z))=f(\Omega ,Z)\mathrm{for} \mathrm{all} {\gamma }^J\in {\Gamma }^J\mathrm{and}(\Omega ,Z)\in {\mathbb{H}}_{n,m};$

(MJ3)

f is an eigenfunction of each diffrential operator in the center ${\mathfrak{C}}_{n,m}$ of $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$;

(MJ4)

f has a polynomial growth, i.e., there exist a constant $C>0$ such that

$$\left|f\right(X+iY,Z\left)\right|\le C\left|p\right(Y\left)\right|\hspace{1em}\mathrm{as} \det \left(Y\right)⟶\infty ,$$

where $p\left(Y\right)$ is a polynomial in $Y=\left(y_{ij}\right).$

We give another notion of Maass–Jacobi functions in the following way.

Definition 12. 

We let $\mathbb{D}{\left({\mathbb{H}}_{n,m}\right)}^{\ast }$ be the commutative subalgebra of $\mathbb{D}\left({\mathbb{H}}_{n,m}\right)$ containing the Laplace operator ${\Delta }_{n,m;A,B}$ of the Siegel–Jacobi space $({\mathbb{H}}_{n,m},d{s}_{n,m;A,B}^2)$. Function $f:{\mathbb{H}}_{n,m}⟶\mathbb{C}$ is called a $\mathsf{Maass}\mathsf{–}\mathsf{Jacobi} \mathsf{function}$ for ${\Gamma }^J$ with respect to $\mathbb{D}{\left({\mathbb{H}}_{n,m}\right)}^{\ast }$ if it satisfies the following conditions $\left(\mathrm{MJ}1\right),\left(\mathrm{MJ}2\right)$ and $\left(\mathrm{MJ}4\right)$ together with ${\left(\mathrm{MJ}3\right)}^{\ast }$:

${\left(\mathrm{MJ}3\right)}^{\ast }$ f is an eigenfunction of each diffrential operator in $\mathbb{D}{\left({\mathbb{H}}_{n,m}\right)}^{\ast }$.

Function $f:{\mathbb{H}}_{n,m}⟶\mathbb{C}$ is called a $\mathsf{weak} \mathsf{Maass}\mathsf{–}\mathsf{Jacobi} \mathsf{function}$ for ${\Gamma }^J$ if it satisfies the following conditions, $\left(\mathrm{MJ}1\right),\left(\mathrm{MJ}2\right)$, and $\left(\mathrm{MJ}4\right)$, together with ${\left(\mathrm{MJ}3\right)}^{\diamond }$;

${\left(\mathrm{MJ}3\right)}^{\diamond }$ f is an eigenfunction of the Laplace operator ${\Delta }_{n,m;A,B}$ of $({\mathbb{H}}_{n,m},d{s}_{n,m;A,B}^2)$.

We denote by ${\mathfrak{W}}_{n,m}$ the complex vector space of all ${\Gamma }^J$-invariant real analytic functions on ${\mathbb{H}}_{n,m}.$ We define formally the following inner product:

$${\langle \phi ,\psi \rangle }_{n,m}:={\int }_{{\Gamma }^J\setminus {\mathbb{H}}_{n,m}}\phi (\Omega ,Z)\overline{\psi (\Omega ,Z)}d{v}_{n,m}(\Omega ,Z).$$

Here,

$$d{v}_{n,m}(\Omega ,Z):=det{\left(Y\right)}^{-(n+m+1)}\left[dX\right]\left[dY\right]\left[dU\right]\left[dV\right]$$

is a $G^J$-invariant volume element on ${\mathbb{H}}_{n,m}$, where for a coordinate $(\Omega ,Z)\in {\mathbb{H}}_{n,m}$ with $\Omega =X+iY,X=\left(x_{ij}\right),Y=\left(y_{ij}\right),Z=U+iV,U=\left(u_{ij}\right),V=\left(v_{ij}\right),X,Y,U,V$ real,

$$\left[dX\right]=\underset{i\le j}{\bigwedge }d{x}_{ij},\hspace{1em}\left[dY\right]=\underset{i\le j}{\bigwedge }d{y}_{ij},\hspace{1em}\left[dU\right]=\underset{k,l}{\bigwedge }d{u}_{kl},\hspace{1em}\left[dV\right]=\underset{k,l}{\bigwedge }d{v}_{kl}.$$

We let

$${\mathcal{H}}_{n,m}:=L^2({\Gamma }^J\setminus {\mathbb{H}}_{n,m})=\{\phi \in {\mathfrak{W}}_{n,m}|{\langle \phi ,\phi \rangle }_{n,m}<\infty \}$$

be the Hilbert space with the hermitian inner product ${\langle ,\rangle }_{n,m}$.

Problem 28. 

Develop the theory of harmonic analysis on ${\mathcal{H}}_{n,m}$ with respect to $\mathbb{D}{\left({\mathbb{H}}_{n,m}\right)}^{\ast }$. In particular, develop the spectral theory of the Laplace operator ${\Delta }_{n,m;A,B}$ on ${\mathcal{H}}_{n,m}$.

Problem 29. 

Let $L^2({\Gamma }^J\setminus G^J)$ denote the Hilbert space of all real analytic ${\Gamma }^J$-invariant functions on $G^J$ such that

$${\int }_{{\Gamma }^J\setminus G^J}{\left|f\left(x\right)\right|}^{2}d\xi \left(x\right)<\infty ,$$

where $d\xi \left(x\right)$ is a Haar measure on $G^J$. So we have the hermitian inner product on $L^2({\Gamma }^J\setminus G^J)$ defined by

$${\langle f,g\rangle }_J:={\int }_{{\Gamma }^J\setminus G^J}f\left(x\right)\overline{g\left(x\right)}d\xi \left(x\right)\hspace{1em}\hspace{1em}(f,g\in L^2({\Gamma }^J\setminus G^J)).$$

Decompose the Hilbert space $L^2({\Gamma }^J\setminus G^J)$ into irreducible unitary representations of $G^J$.