Heisenberg Parabolic Subgroup of SO ∗ (10) and Invariant Differential Operators

: In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra so ∗ ( 10 ) . We use the maximal Heisenberg parabolic subalgebra P = M ⊕ A ⊕ N with M = su ( 3,1 ) ⊕ su ( 2 ) ∼ = so ∗ ( 6 ) ⊕ so ( 3 ) . We give the main and the reduced multiplets of indecomposable elementary representations. This includes the explicit parametrization of the intertwining differential operators between the ERS. Due to the recently established parabolic relations the multiplet classiﬁcation results are valid also for the algebras so ( p , q ) (with p + q = 10, p ≥ q ≥ 2) with maximal Heisenberg parabolic subalgebra: P (cid:48) = M (cid:48) ⊕ A (cid:48) ⊕ N (cid:48) , M (cid:48) = so ( p − 2, q − 2 ) ⊕ sl ( 2, IR ) , M (cid:48) C ∼ = M C .


Introduction
Invariant differential operators play a very important role in the description of physical symmetries. Recently, Refs. [1,2] we started the systematic explicit construction of invariant differential operators. We gave an explicit description of the building blocks, namely, the parabolic subgroups and subalgebras from which the necessary representations are induced. Thus, we have set the stage for a study of different non-compact groups.
In the present paper, we focus on the algebra so * (10). The algebras so * (2n) (for n ≥ 2) form a class of Lie algebras that have maximal Heisenberg parabolic subalgebras. The latter are given as: P = M ⊕ A ⊕ N , where M = so * (2n − 4) ⊕ so (3).
In order to avoid repetition, we refer to [1][2][3] for motivations and an extensive list of literature on the subject.

Preliminaries
Let G be a semisimple non-compact Lie group, and K a maximal compact subgroup of G. Then we have an Iwasawa decomposition G = KA 0 N 0 , where A 0 is abelian simply connected vector subgroup of G, N 0 is a nilpotent simply connected subgroup of G preserved by the action of A 0 . Further, let M 0 be the centralizer of A 0 in K. Then the subgroup P 0 = M 0 A 0 N 0 is a minimal parabolic subgroup of G. A parabolic subgroup P = MAN is any subgroup of G which contains a minimal parabolic subgroup.
The importance of the parabolic subgroups comes from the fact that the representations induced from them generate all (admissible) irreducible representations of G [4][5][6][7].
Let ν be a (non-unitary) character of A, ν ∈ A * , let µ fix an irreducible representation D µ of M on a vector space V µ .
When (3) holds then the Verma module with shifted weight V Λ−mβ (orṼ Λ−mβ for GVM and β non-compact) is embedded in the Verma module V Λ (orṼ Λ ). This embedding is realized by a singular vector v s determined by a polynomial P m,β (G − ) in the universal enveloping algebra (U(G − )) v 0 , G − is the subalgebra of G C generated by the negative root generators [14]. More explicitly, ref. [11], v s m,β = P m β v 0 (or v s m,β = P m β V µ v 0 for GVMs). Then there exists [11] an intertwining differential operator given explicitly by: where G − denotes the right action on the functions F , cf. (1).

The Non-Compact Lie
Algebra so * (10) 3.1. The General Case of so * (2n) The group G = SO * (2n) consists of all matrices in SO(2n, C) which commute with a real skew-symmetric matrix times the complex conjugation operator C: The Lie algebra G = so * (2n) is given by: The Cartan involution is given by: ΘX = −X † . Thus, K ∼ = u(n): Thus, G = so * (2n) has discrete series representations and highest/lowest weight representations. The complementary space P is given by: dim R P = n(n − 1). The split rank is r ≡ [n/2]. We need also the root system of G C = so(2n, C) . The positive roots are given standardly as: where i are standard orthonormal basis: i , j = δ ij . We shall need the scalar products of the roots: Note that the highest root is β 12 . The simple roots are: The compact roots w.r.t. the real form SO * (2n) are α ij -they form (by restriction) the root system of the semisimple part of K C , namely, K C s ∼ = su(n) C ∼ = sl(n, C), while the roots β ij are noncompact.
The minimal parabolics of SO * (2n) depend on whether n is even or odd and are: The subalgebras N ± 0 which form the root spaces of the root system (G, A 0 ) are of real dimension n(n − 1) − [n/2].
The maximal parabolic subalgebras have M-factors as follows [1]: The N ± factors in the maximal parabolic subalgebras have dimensions: dim (N ± j ) max = j(4n − 6j − 1). The case j = 1 is special. In this case, we have a maximal Heisenberg parabolic with M-factor: which we use in this paper.

The Case so * (10)
Further, we restrict to our case of study G = so * (10) with minimal parabolic: The Satake-Dynkin diagram of G is: where, by standard convention, the black dots represent the so(3) subalgebras of M 0 , and the left-right arrow represents the so(2) subalgebra of M 0 . We shall use the Heisenberg maximal parabolic (15) with M-subalgebra: The Satake-Dynkin diagram of M is a subdiagram of (17): where the single black dot represents the so(3) subalgebra, while the connected part of the diagram represents the su(3, 1) subalgebra. From the above follows that the M-compact roots of G C are (given in terms of the simple roots): By definition the above are the positive roots of M C , namely: su(2) C (20a), and su(3, 1) C = sl(4, C) (20b).
The positive M-noncompact roots of G C in terms of the simple roots are: where for convenience we use the notation γ ij ≡ α i,j+1 To characterize the Verma modules we shall use first the Dynkin labels: where ρ is half the sum of the positive roots of G C . Thus, we shall use: Note that when all m i ∈ IN then χ Λ characterizes the finite-dimensional irreps of G C and its real forms, in particular, so * (10). Furthermore, m 1 ∈ IN characterizes the finite-dimensional irreps of the su(2) subalgebra, while the set of positive integers {m 3 , m 4 , m 5 } characterizes the finite-dimensional irreps of su (3,1).
For the M-noncompact roots of G C we shall use also the Harish-Chandra parameters: and explicitly in terms of the Dynkin labels (compare (21)):

Main Multiplets of SO * (10)
The main multiplets are in 1-to-1 correspondence with the finite-dimensional irreps of so * (10), i.e., they are labeled by the five positive Dynkin labels m i ∈ IN.
We take χ 0 = χ HC . It has one embedded Verma module with HW Λ a = Λ 0 − m 2 γ 2 . The number of ERs/GVMs in the main multiplet is 40. We give the whole multiplet as follows: We shall label the signature of the ERs of G also as follows: where the first entry n = m 1 labels the finite-dimensional irreps of su (2), the second entry labels the characters of A , the last three entries of χ are labels of the finite-dimensional (nonunitary) irreps of M = su(3, 1) when all n j > 0 or limits of the latter when some n j = 0. Note that m 15,23 = m 1 + 2m 2 + 2m 3 + m 4 + m 5 is the Harish-Chandra parameter for the highest root β 12 . These labeling signatures may be given in the following pair-wise manner: The ERs in the multiplet are related also by intertwining integral operators introduced in [15,16]. These operators are defined for any ER, the general action being: The main multiplets are given explicitly in Figure 1. The pairs χ ± are symmetric w.r.t. to the dashed line in the middle of the figure-this represents the Weyl symmetry realized by the Knapp-Stein operators (29): G KS : C χ ∓ ←→ C χ ± . Some comments are in order.
Matters are arranged so that in every multiplet only the ER with signature χ − 0 contains a finite-dimensional nonunitary subrepresentation in a finite-dimensional subspace E . The latter corresponds to the finite-dimensional irrep of so * (10) with signature {m 1 , . . . , m 5 }.
The subspace E is annihilated by the operator G + , and is the image of the operator G − . The subspace E is annihilated also by the intertwining differential operator acting from χ − 0 to χ − a . When all m i = 1 then dim E = 1, and in that case E is also the trivial onedimensional UIR of the whole algebra G. Furthermore in that case the conformal weight is zero: In the conjugate ER χ + 0 there is a unitary discrete series subrepresentation in an infinitedimensional subspace D. It is annihilated by the operator G − , and is the image of the operator G + .

Fig. 1.
Main multiplets for SO * (10) using induction from maximal Heisenberg parabolic Thus, for so * (10) the ER with signature χ + 0 contains both a holomorphic discrete series representation and a conjugate anti-holomorphic discrete series representation. The direct sum of the holomorphic and the antiholomorphic representation spaces form the invariant subspace D mentioned above. Note that the corresponding lowest weight GVM is infinitesimally equivalent only to the holomorphic discrete series, while the conjugate highest weight GVM is infinitesimally equivalent to the anti-holomorphic discrete series.
In Figure 1 we use the notation: Λ ± = Λ(χ ± ). Each intertwining differential operator is represented by an arrow accompanied either by a symbol i jk encoding the root γ jk and the number m γ jk which is involved in the BGG criterion, or a symbol i jk encoding the root β jk and the number m β jk from BGG.
Finally, we remind that according to [3] the above considerations for the intertwining differential operators are applicable also for the algebras so(p, q) (with p + q = 10, p ≥ q ≥ 2) with maximal Heisenberg parabolic subalgebras: P = M ⊕ A ⊕ N , M = so(p − 2, q − 2) ⊕ sl(2, I R).

Main Reduced Multiplets
Intertwining differential operators occur not only in the main multiplets, but also in their reductions. There are five main reduced multiplets M k , k = 1, 2, 3, 4, 5, which may be obtained by setting the parameter m k = 0.
The main reduced multiplet M 1 contains 27 GVMs (ERs), see Figure 2. Their signatures are given as follows: Note that some of the inducing representations, namely, χ ± 0 , χ ± g , χ ± k , χ ± l , χ ± p , χ ± q , χ ± r , are limits of M representations, while the rest are finite-dimensional IRs (as in the main multiplets).
The main reduced multiplet M 2 contains 27 GVMs (ERs), see Figure 3, with signatures given as follows: Main reduced multiplets for SO * (10) of type M 1 Figure 2. Main reduced multiplets for SO * (10) of type M 1 . Fig. 2b.
Main reduced multiplets for SO * (10) of type M 2 Main reduced multiplets for SO * (10) of type M 3

Next Reduced Multiplets
There are intertwining differential operators also in the next reduced multiplets. We start with cases M ij (i < j) when m i = m j = 0.
The reduced multiplet M 12 contains 15 GVMs (ERs) with signatures given as follows: Here we note only the ER χ s which is induced from finite-dimensional M-irrep, and where the subrepresentation is a singlet.
The reduced multiplet M 13 contains 18 GVMs (ERs): Here we note the ERs χ ± h , χ ± m induced from finite-dimensional M-irreps, which doublets are related by KS operators, yet for the pair Λ ± m the KS operator G + is actually the differential operator D m 3 α 12 .
The reduced multiplet M 14 contains 18 GVMs (ERs): Here we note the ERs χ ± h , χ ± m induced from finite-dimensional M-irreps, and forming a sub-multiplet as follows: Here we note the ERs χ ± i , χ ± m induced from finite-dimensional M-irreps, and forming a sub-multiplet as follows: Here we note only the ER χ q which is induced from finite-dimensional M-irrep, and where the subrepresentation is a singlet.