On Lusztig’s Character Formula for Chevalley Groups of Type A_l

Abstract

For a Chevalley group $G$ over an algebraically closed field $K$ of characteristic $p>0$ with the irreducible root system $R,$ Lusztig’s character formula expresses the formal character of a simple $G$-module by the formal characters of the Weyl modules and the values of the Kazhdan–Lusztig polynomials at 1. It is known that, for a sufficiently large characteristic $p$ of the field $K,$ Lusztig’s character formula holds. The known lower bound of the characteristic $p$ is much larger than the Coxeter number $h$ of the root system $R.$ Observations show that for simple modules with restricted highest weights of small Chevalley groups such as those of types $A_1,A_2,A_3,B_2,B_3$, and $C_3$, Lusztig’s character formula holds for all $p\ge h$. For large Chevalley groups, no other examples are known. In this paper, for $G$ of type $A_l,$ we give some series of simple modules for which Lusztig’s character formula holds for all $p\ge h$. Using this result, we compute the cohomology of $G$ with coefficients in these simple modules. To prove the results, Jantzen’s filtration properties for Weyl modules and the properties of Kazhdan–Lusztig polynomials are used.

1. Introduction

Let $G$ be a Chevalley group over an algebraically closed field $K$ of characteristic $p>0.$ This was constructed from a finite-dimensional irreducible representation of the complex simple classical Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$ [1] (pp. 49–50). The Weyl module $V\left(\lambda \right)$ with the highest weight $\lambda$ over $G$ was obtained from the corresponding irreducible finite-dimensional module ${V\left(\lambda \right)}_{\mathbb{C}}$ over ${\mathfrak{g}}_{\mathbb{C}}$ [2] (pp. 291–292). The modules $V\left(\lambda \right)$ and ${V\left(\lambda \right)}_{\mathbb{C}}$ have the same formal characters, but $V\left(\lambda \right)$ need not be irreducible. A Chevalley group constructed over an algebraically closed field is an algebraic group. Therefore, the Weyl module can also be defined via the induced module of a semisimple algebraic group. Consider $G$ to be a semisimple algebraic group. Let $B$ be the Borel subgroup of $G$ generated by the positive root subgroups, let $U$ be the unipotent subgroup of $B,$ let $T$ be a maximal torus of $G,$ and let $\lambda$ be a weight of $G$, such that $\lambda :T\to K^{*}.$ Since $B=UT,$ the weight $\lambda$ can be extended to the character of $B$ by $\lambda \left(ut\right)=\lambda \left(t\right),$ where $u\in U,t\in T.$ Denote by $K_{\lambda }$ the corresponding one-dimensional $B$-module. If $H^0\left(\lambda \right)={\mathrm{i}\mathrm{n}\mathrm{d}}_B^G\left(K_{\lambda }\right)$ is an induced $G$-module, then $V\left(\lambda \right)={H^0\left(-w_0\lambda \right)}^{*},$ where $w_0$ is the maximal length element of the Weyl group for $R.$ Denote by $L\left(\lambda \right)$ an irreducible $G$-module with highest weight $\lambda .$ Let ${\mathrm{r}\mathrm{a}\mathrm{d}}_{G}V\left(\lambda \right)$ be the radical of $V\left(\lambda \right)$ and ${\mathrm{s}\mathrm{o}\mathrm{c}}_G{H}^0\left(\lambda \right)$ be the socle of $H^0\left(\lambda \right).$ Then, $L\left(\lambda \right)=V\left(\lambda \right)/{\mathrm{r}\mathrm{a}\mathrm{d}}_{G}V\left(\lambda \right)$ or $L\left(\lambda \right)={\mathrm{s}\mathrm{o}\mathrm{c}}_G{H}^0\left(\lambda \right).$ For formal characters of $V\left(\lambda \right)$ and ${V\left(\lambda \right)}_{\mathbb{C}}$, there is a Weyl’s character formula [1] (p. 139). In 1980, in the famous Lustig’s conjecture, a formula for calculating the formal character of $L\left(\lambda \right)$ was proposed [3]. This formula was later called Lustig’s character formula. It was proved that for a fixed root datum there exists a lower bound $b$ such that Lusztig’s character formula holds for all $p>b$ [4]. The current established bound [5] is quite large compared to the standard Coxeter number $h.$ For example, $p>l^{l^2}$ when $R=A_l$. In addition, there are Williamson’s counterexamples to the expected bounds in Lusztig’s conjecture [6,7]. There are also examples for which Lustig’s character formula is not valid in small characteristics $p<h$ [8] (p. 36). It is also known that, in the lower affine alcoves and for small groups, Lusztig’s character formula holds for all $p\ge h.$ This is confirmed by examples of small algebraic groups, such as ${SL}_2\left(K\right),$ ${SL}_3\left(K\right),$ ${Sp}_4\left(K\right),$ $G_2,$ ${SL}_4\left(K\right),$ ${Sp}_5\left(K\right),$ and ${SO}_7\left(K\right).$ It is also known that, for $p=5$ or $p=7$ and ${SL}_6\left(K\right),$ Lustig’s character formula also holds for all restricted weights [9] (p. 422). However, in the general case, it is not known which weights will satisfy Lusztig’s character formula, provided $p\ge h.$ So, the question of computing formal characters of simple modules for $G$ using Luisztig’s character formula remains largely open. In this paper, we investigate this question for the Chevalley group $G$ of type $A_l.$ The goal is to obtain examples of simple modules for which the Lustig’s character formula is valid for all $p\ge h.$ We prove that, for a series of simple modules whose highest weights are contained in affine alcoves along the walls of the fundamental Weyl chambers and in the alcoves adjacent to them, Lusztig’s character formula is valid for all $p\ge h.$

2. Materials and Methods

2.1. Preliminaries and Notation

Let $\mathbb{Z}$ be the set of integers, let $R$ be an irreducible root system of type $A_l$, let $R^{+}$ be the set of positive roots, let $∆=\{{\alpha }_1,\cdots ,{\alpha }_l\}$ be the set of simple roots, and let $h$ be the Coxeter number of $R$ and $E$ be the affine space with the base vector space generated by $R.$ Consider a Chevalley group $G$ over an algebraically closed field $K$ of characteristic $p>0.$ The affine Weyl group $W_p$ of $G$ is generated by affine reflections $s_{\alpha ,np},\alpha \in R,n\in \mathbb{Z},$ with an action given by

$$s_{\alpha ,np}·\lambda =\lambda -\left(〈\lambda +\rho ,{\alpha }^{\vee }〉-np\right)\alpha ,\lambda \in E,$$

where $\rho$ is the half-sum of positive roots. The affine hyperplane $H_{\alpha ,n}$ ($\alpha \in R^{+},$ $n\in \mathbb{Z}$) fixed by $s_{\alpha ,np}$ is the set

$$H_{\alpha ,n}=\left\{v\in E\right|〈v,{\alpha }^{\vee }〉=np\}.$$

Let $A$ be the set of connected components of $E\backslash \left({\cup }_{\alpha \in R^{+},n\in \mathbb{Z}}{H}_{\alpha ,n}\right).$ The elements of $A$ are called the affine alcoves. Let ${\tilde{\alpha }}_0\in R^{+}$ be the maximal root. Denote by $C_1$ the initial alcove in $A.$ It is obvious that $C_1$ is bounded by the affine hyperplanes $H_{{\tilde{\alpha }}_0,1},H_{{\alpha }_1,0},\cdots ,H_{{\alpha }_l,0}.$ Then, $A=\left\{w·C_1\right|w\in W_a\}.$

The affine Weyl group $W_p$ of the Chevalley group of type $A_l$ is the Coxeter group of type ${\tilde{A}}_l$ generated by the set of affine reflections

$$S_p=\left\{s_0=s_{{\tilde{\alpha }}_0,1},s_1=s_{{\alpha }_1,0},\cdots ,s_l=s_{{\alpha }_l,0}\right\}$$

with the following relations:

$${\left(s_i{s}_j\right)}^{m_{ij}},$$

(1)

where $i,j\in \{0,1,\cdots ,l\}$ and

$$m_{ij}=\left\{\begin{array}{l}1\mathrm{if}i=j,\\ 2\mathrm{if}\left|i-j\right|>1\mathrm{and}\left(i,j\right)\notin \left\{\left(l,0\right),\left(0,l\right)\right\},\\ 3\mathrm{if}\left|i-j\right|=1\mathrm{or}\left(i,j\right)\in \left\{\left(l,0\right),\left(0,l\right)\right\}.\end{array}\right.$$

Then, the subset $S=\left\{s_1,\cdots ,s_l\right\}$ of $S_p$ generates the Weyl group $W$ of $G.$

Each element $w\in W_p$ can be written as a product $w_1\cdots w_n$ of generators $w_i\in S_p.$ The length $l\left(w\right)$ of $w$ is smallest $n$ among such expressions of $w.$ In this case, the expression $w_1\cdots w_n$ of $w$ is called a reduced expression. We define the partial Chevalley–Bruhat order on $W_p,$ using the reduced expressions of its elements. Let $w_1\cdots w_n$ be a reduced expression of $w,$ then $u\le w$ if and only if there are indexes $1\le i_1<\cdots <i_m\le n$ such that $w_{i_1}\cdots w_{i_m}$ is a reduced subexpression of $u.$ For $u,w\in W_p$ with $u\le w,$ the Bruhat interval $[u,w]$ is defined by $\left[u,w\right]=\left\{x\in W_p\right|u\le x\le w\}.$

Denote by $P$ the weight lattice generated by the roots of $R.$ Let

$$P^{++}=\left\{\lambda \in P\right|〈\lambda ,{\alpha }^{\vee }〉\ge 0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\alpha \in ∆\}$$

be the set of dominant weights and $\tau \in C_1\cap P.$ Let $w_0$ be the unique maximal element of $W.$ Denote by ${\omega }_1,{\omega }_2,\cdots ,{\omega }_l$ the fundamental weights. We will need the following subsets$:$

$$\begin{array}{c}{W}_p^{+}=\left\{w\in W_p\right|w·\tau \in P^{++}\};\\ W_p^{-}=\left\{w_{0}w\right|w\in W_p\};\\ W_p^{res,+}=\left\{w\in W_p\right|0\le 〈w·\tau ,{\alpha }^{\vee }〉<p\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\alpha \in ∆\};\end{array}$$

The first of them is called the set of dominant elements of $W_p,$ the second the set of antidominant elements of $W_p,$ and the last the set of restricted elements of group $W_p.$

Each rational $G$-module $V$ is a direct sum of $T$-modules: $V={\oplus }_{\lambda \in P}{V}_{\lambda },$ where $V_{\lambda }=\left\{v\in V\right|tv=\lambda \left(t\right)v\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}t\in T\}.$ A formal character of $V$ is an element $\left[V\right]$ of the grout ring $\mathbb{Z}\left[P\right]$ defined by

$$\left[V\right]=\sum _{\lambda \in P}\dim V_{\lambda }{e}^{\lambda },$$

where $e^{\lambda }$ is the basis element of $\mathbb{Z}\left[P\right]$ corresponding to $\lambda \in P.$ Weyl’s character formula yields the formal character of $\left[V\left(\lambda \right)\right]:$

$$\left[V\left(\lambda \right)\right]={\displaystyle \frac{\sum _{w\in W}{(-1)}^{l\left(w\right)}{e}^{w(\lambda +\rho )}}{{\sum }_{w\in w}{(-1)}^{l\left(w\right)}{e}^{w\left(\rho \right)}}},\lambda \in P^{++}.$$

Weyl’s character formula gives the following Weyl’s dimension formula

$$\dim V\left(\lambda \right)={\displaystyle \frac{{\prod }_{\alpha \in R^{+}}〈\lambda +\rho ,\alpha 〉}{{\prod }_{\alpha \in R^{+}}〈\rho ,\alpha 〉}},\lambda \in P^{++}.$$

2.2. Kazhdan–Lusztig Polynomials and Lusztig’s Conjecture

In this subsection, we give some properties of the Kazhdan–Lusztig polynomials and formulate Lusztig’s conjecture on the irreducible characters in positive characteristics. The Kazhdan–Lusztig polynomials are introduced in [10] (pp. 166, 170–171) as the coefficients in the transition matrix for expanding the basis in the Hecke algebra associated with the Coxeter group into the standard basis. A Kazhdan–Lusztig polynomial $P_{y,w}$ is a polynomial in $q$ of degree $\le {\displaystyle \frac{1}{2}}\left(l\left(w\right)-l\left(y\right)-1\right)$ if $y<w\in W_p$ and $P_{w,w}=1.$ For $w$ in $W_p,$ let $q_w=q^{l\left(w\right)}$ and ${\epsilon }_w={(-1)}^{l\left(w\right)}.$ Given $y,w\in W_p,$ we can say that $y\prec w$ if the following conditions hold: $y<w,$ ${\epsilon }_y=-{\epsilon }_w$ and $P_{y,w}$ is a polynomial in $q$ of degree exactly ${\displaystyle \frac{1}{2}}\left(l\left(w\right)-l\left(y\right)-1\right)$. In this case, the coefficient of the highest power of $q$ in $P_{y,w}$ is denoted $\mu \left(y,w\right).$ Suppose that $w=vs,$ where $s\in S_p$ and $l\left(w\right)=l\left(v\right)+1.$ If $y\le w,$ then

$$P_{y,w}=q^{1-c}{P}_{ys,v}+q^c{P}_{y,v}-\sum _{z,y\le z\prec v,zs<z}\mu \left(z,v\right)q^{{\displaystyle \frac{1}{2}}\left(l\left(v\right)-l\left(z\right)+1\right)}{P}_{y,z},$$

where $c=1$ is $ys<y,$ $c=0$ if $ys>y.$ We can say that $P_{y,w}=0$ when $y\nleqq w.$

We use the polynomial $R_{x,y}$ in $q$ defined for each $x,y\in W_p$ introduced also by Kazhdan and Lusztig in [10] (p. 169). We may compute $R_{x,y}$ inductively by the formulas:

$$\begin{array}{c}{R}_{x,y}=0\mathit{if}x\nleqq y,\\ R_{x,y}=1\mathit{if}x=y,\end{array}$$

$$R_{x,y}=\left\{\begin{array}{c}{R}_{sx,sy}\mathrm{if}sx<x\mathrm{and}sy<y,\\ R_{xs,ys}\mathrm{if}xs<x\mathrm{and}ys<y;\end{array}\right.$$

(2)

$$R_{x,y}=\left\{\begin{array}{c}\left(q-1\right)R_{sx,y}+q{R}_{sx,sy}\mathrm{if}sx>x\mathrm{and}sy<y,\\ \left(q-1\right)R_{xs,y}+q{R}_{xs,ys}\mathrm{if}xs>x\mathrm{and}ys<y.\end{array}\right.$$

(3)

Let

$$N_{y,w}=q_y{\sum }_{y\le z\le w}{R}_{y,z},$$

(4)

where $y,w\in W_p$ and $y\le w.$ The polynomials $R_{x,y}$ and $P_{y,w}$ are elements of the ring $\mathbb{Z}[q^{{\displaystyle \frac{1}{2}}},q^{-{\displaystyle \frac{1}{2}}}]$ of Laurent polynomials in $q^{{\displaystyle \frac{1}{2}}}.$ Let $a⟼\overline{a}$ be the involution of $\mathbb{Z}[q^{{\displaystyle \frac{1}{2}}},q^{-{\displaystyle \frac{1}{2}}}]$ given by $\overline{q^{{\displaystyle \frac{1}{2}}}}=q^{-{\displaystyle \frac{1}{2}}}.$

We will need the following properties of the polynomials $R_{x,y}$ and $P_{y,w}$ [10] (pp. 169, 172):

Lemma 1.

The following hold:

(a) 

${\overline{R}}_{x,y}={\epsilon }_x{\epsilon }_y{q}_x{R}_{x,y};$

(b) 

$R_{x,y}={(q-1)}^{l\left(y\right)-l\left(x\right)}$ for all $x\le y$ such that $l\left(x\right)\ge l\left(y\right)-2;$

(c) 

for all $x\le w,$ ${\epsilon }_x{\epsilon }_w{q}_w^{{\displaystyle \frac{1}{2}}}{q}_x^{-1}{\overline{P}}_{x,w}=\sum _{x\le y\le w}{\epsilon }_y{\epsilon }_w{q}_w^{-{\displaystyle \frac{1}{2}}}{q}_y{q}_x^{-1}{\overline{R}}_{x,y}{P}_{y,w};$

(d) 

given $y<w$ in $W_p$the following two conditions are equivalent:

$$P_{y^{\prime },w}=1\mathrm{for}\mathrm{all}y\le y^{\prime }\le w\mathrm{and}{N}_{y^{\prime },w}=q_w\mathrm{for}\mathrm{all}y\le y^{\prime }\le w;$$

Lusztig’s conjecture. Let $w\in W_p^{res,+}$ and $\tau \in C_1\cap P.$ Then,

$$\left[L\left(w·\tau \right)\right]={\sum }_{y\in W_p^{res,+},y\le w}{\left(-1\right)}^{l\left(w\right)-l\left(y\right)}{P}_{w_{0}y,w_{o}w}\left(1\right)\left[V\right(y·\tau \left)\right],$$

(5)

 where $P_{w_{0}y,w_{o}w}\left(1\right)$ is the value of $P_{w_{0}y,w_{o}w}$ when $q=1.$

Formula (5) is called Lusztig’s character formula.

For $R$-polynomials, several combinatorial formulas and interpretations are known [11,12,13]. The properties of the Kazhdan–Lusztig $P$-polynomials are studied in [14,15,16,17,18].

2.3. Jantzen Filtration and Jantzen’s Sum Formula

Consider the Weyl module $V\left(\lambda \right)$ with the highest weigh $\lambda \in P^{++}.$ There is a filtration of submodules

$$V\left(\lambda \right)={V\left(\lambda \right)}^0\supset {V\left(\lambda \right)}^1\supset {V\left(\lambda \right)}^2\supset \cdots$$

(6)

such that

$$V\left(\lambda \right)/{V\left(\lambda \right)}^1\cong L\left(\lambda \right)$$

(7)

and

$${\sum }_{j>0}\left[{V\left(\lambda \right)}^j\right]={\sum }_{\alpha \in R^{+}}{\sum }_{0<np<〈\lambda +\rho ,{\alpha }^{\vee }〉}{\nu }_p\left(np\right)\left[V\right(s_{\alpha ,n}·\lambda \left)\right],$$

(8)

where ${\nu }_p\left(m\right)=\max \left\{i\in \mathbb{N}\right|p^i\left|n\right\}$ [19] (pp. 282–283). The filtration (6) is called Jantzen’s filtration and the Formula (8) is called Jantzen’s sum formula. Let $V$ be a $G$-module. We define a composition coefficient $[V:L(\mu \left)\right]$ for $\mu \in P^{++}$ such that

$$\left[V\right]=\sum _{\lambda \in P^{++}}\left[V:L\left(\mu \right)\right]\left[L\right(\mu \left)\right].$$

If $\left[V:L\left(\mu \right)\right]\ne 0$ we can say that $L\left(\mu \right)$ is a composition factor of $V.$ For the Weyl modules $V\left(\lambda \right),$ composition factors were studied in [2,20,21,22,23,24,25,26,27,28,29,30,31,32].

2.4. Methods

Let $\left[y,w\right]$ be a Bruhat interval in $W_p.$ Denote by $f_{y,w}^k$ a number of $l\left(y\right)+k$ length elements of $\left[y,w\right],$ where $k\in \left\{0,1,\cdots ,l\left(w\right)-l\left(y\right)\right\}.$

To prove Lusztig’s character formula for a simple module with the highest weight $w·\tau ,$ where $w\in Y\cup Z,$ $\tau \in C_1\cap P,$ we will calculate the formal characters of this simple module in two ways:

(A)

A calculation using Lusztig’s character formula itself;

(B)

A calculation using Janzen’s sum formula for Janzen’s filtration of Weyl modules.

For $L(w·\tau )$, the same result of the calculations (A) and (B) guarantees the validity of Lustig’s character formula.

According to (5), the first way (A) is reduced to calculation of the Kazhdan–Lustig polynomials $P_{w_{0}y,w_{o}w}$ for all $y,w\in \left[1,w\right],$ where $1$ is the identity element of $W_p.$ The affine Weyl group $W_p$ is a Coxeter group; therefore, to calculate the Kazhdan–Lustig polynomials, we will use the properties of Coxeter groups such as the Subword Property and the Exchange Condition. We will also use the properties of the Kazhdan–Luisztig polynomials in Section 2.2. So, we realize (A) by the following steps:

(A1)

Calculation of $f_{w_{0}y,w_{o}w}^k$ for all $y\in [1,w]$ and for all $k\in \left\{0,1,\cdots ,l\left(w\right)-l\left(y\right)\right\};$

(A2)

Calculation of $f_{x,z}^k$ for all $x,z\in W_0\left(w\right),$ where

$$W_0\left(w\right)=\left\{w_{0}vy|y\in \left[1,w\right],1\le v\le s_{0}y\mathrm{a}\mathrm{n}\mathrm{d}l\left(vy\right)=l\left(v\right)+l(y)\right\};$$

(A3)

Calculation of the Kazhdan–Luisztig polynomial $R_{x,z}$ for all $x,z\in W_0\left(w\right);$

(A4)

Calculation of the Kazhdan–Luisztig polynomial $P_{w_{0}y,w_{o}w}$ for all $y\in \left[1,w\right];$

(A5)

Calculation of $\left[L\left(w·\tau \right)\right].$

The steps (A1) and (A2) are realized by using the Subword Property [33] (p. 34) and the Exchange Condition [34] (p. 94) of Coxeter groups.

In the next step (A3), we use the induction on $l\left(x\right)-l\left(z\right),$ the Coxeter group Subword Property, and properties (2) and (3) of the Kazhdan–Lusztig polynomial $R_{x,y}$. The induction is based on the following:

Lemma 2.

Let $s\in S_p,$ and let $z\in W_p$ satisfy $zs<z.$ Suppose that $x<z.$

(a)

If $xs<s,$ then $xs<zs.$

(b)

If $xs>s,$ then $xs\le z$ and $x<zs.$

This lemma is a right-hand version of Lemma 7.4 in [34] (p. 151), and can easily be proved similarly.

In the step (A4), the results of steps (A1)–(A3), the properties of the Kazhdan–Lustig polynomials from Lemma 1, and the Formula (4) are used.

In the last step (A5) of the first way of calculation, we calculate the formal characters of $L\left(w·\tau \right)$ using the results of step (A4) and Lusztig’s character formula, Formula (5).

To realize (B), we can use the computation algorithm for formal characters of Weyl modules in terms of the formal characters of simple modules, as used in [2]. In our case, first we:

(B1)

Express the sum ${\sum }_{j>0}\left[{V\left(y·\tau \right)}^j\right]$ in terms of the formal characters of simple modules with highest weights in {$x·\tau |x\in \left[1,y\right]\}$ for all $y\le w,$ using the Formulas (6)–(8);

(B2)

Determine the multiplicities $\left[V(y·\tau ):L\left(x·\tau \right)\right],x\in \left[1,y\right]$ of the composition factors $L\left(x·\tau \right)$ and $x\in \left[1,y\right]$ of $V(y·\tau )$ for all $y\le w,$ using the Jantzen’s translation principle [2] (p. 297) and the results of step (B1);

(B3)

Express the formal character of $L(w·\tau )$ in terms of the formal characters of Weyl modules $V\left(x·\tau \right)$ and $x\in \left[1,w\right],$ using the results of step (B2).

For specific examples of the application of (B1)–(B2) to calculate the formal character of Weyl modules, see [2] (pp. 284–286).

The validity of Lustig’s character formula for $L(w·\tau )$ yields the validity of the following dimension formula for all $y\le w$ [35] (p. 523):

$${\sum }_n\dim {Ext}_G^n\left(L\right(y·\tau ),L(w·\tau \left)\right)t^n={\sum }_{z·\tau \in W_p^{+}}{t}^{l\left(y\right)+l\left(w\right)-2l\left(z\right)}{\overline{P}}_{w_{0}z,w_{0}y}{\overline{P}}_{w_{0}z,w_{0}w},$$

where $P_{a,b}$ is the Kazhdan–Lusztig polynomial associated with $a,b\in W_p$ in $t^2=q.$ In particular, for $y=1$ and $\tau =0,$ we obtain

$${\sum }_n\dim H^n(G,L(w·\tau \left)\right)t^n=t^{l\left(w\right)}{\overline{P}}_{w_0,w_{0}w}.$$

(9)

So, using (9) and the result of step (A3), we can compute the cohomology of $G$ with the coefficients in $L\left(w·0\right).$

In positive characteristics, cohomology with coefficients in simple modules was studied in [36,37,38,39,40,41,42,43,44,45].

3. Results

3.1. On Lustig’s Character Formula for Simple Modules—In This Subsection We Prove the Following

Theorem 1.

Let $G$ be a Chevalley group of type $A_l$ over an algebraically closed field $K$ of characteristic $p\ge h$ and let $W_p$ be the affine Weyl group of $G.$ Consider the following subsets of $W_p:$

$$\begin{array}{c}Y=\left\{y_{-1}=1,y_i=\left\{s_0{s}_1\cdots s_i\right|i=0,1,\cdots ,l-2\right\},\\ Z=\left\{z_{-1}=1,z_0=s_0{s}_l,z_i=\left\{s_0{s}_l{s}_1\cdots s_i\right|i=1,2,\cdots ,l-2\right\}.\end{array}$$

For all  $w\in Y\cup Z\subset W_p$ and for all $\tau \in C_1\cap P$, Lusztig’s character formula, Formula (5), holds.

Proof of Theorem 1.

The proof is divided into two parts. In the first part, we realize the steps (A1)–(A4). Similarly, in the second part, steps (B1)–(B3) are realized. □

• First part. Step (A1) is the proof of the following:

Lemma 3.

Let $w_0,$$y_i$, $z_i$ be as above. If $y,w\in Y\cup Z$ and $y\le w,$ then 

$$f_{w_{0}y,w_{0}w}^k=C_{l\left(w_{0}w\right)-l\left(w_{0}y\right)}^k,$$

except for the cases $y=1$ and $w\in \left\{z_i\right|i=1,2,\cdots ,l-2\}.$ In the latter case,

$$f_{w_0,w_0{z}_i}^k=\left\{\begin{array}{c}1\hspace{1em}\hspace{1em}\hspace{1em}\mathit{if}k=0,\\ 2C_{i+1}^1\hspace{1em}\mathit{if}k=1,\\ C_{i+1}^k+C_{i+2}^{k}otherwise.\end{array}\right.$$

Proof. 

Let $y=1$ and $w\in \left\{y_i\right|i=0,1,\cdots ,l-2\}.$ So, we consider the Bruhat interval $\left[w_0,w_0{y}_i\right].$ If $x\in W_p$ and $s_{i_1}{s}_{i_2}\cdots s_{i_q}$ is a reduced expression of $x$, then using the Subword Property [33] (p. 34), the inequality $w_0\le x$ holds if and only if there exists a reduced expression

$$w_0=s_{j_1}{s}_{j_2}\cdots s_{j_k},1\le j_1<j_2<\cdots <j_k\le i_q.$$

Let $x=w_{0}u$ and $w_0\le x.$ If $l\left(u\right)=0,$ then it is obvious that $x=w_0=s_{j_1}{s}_{j_2}\cdots s_{j_k}.$

If $l\left(u\right)=1,$ then $u\in S_p.$ Suppose that $u\in S,$ then $x\in W,$ so $l\left(x\right)=l\left(w_0\right)-1$ (see, [1], p. 16), which contradicts $w_0\le x$. Then, $u\in S_p\backslash S,$ so $u=s_0.$ Hence, $x=w_0{s}_0.$

In the case where $l\left(u\right)=2,$ we obtain that $u\in \{s_0{s}_1,s_0{s}_l\}.$ Indeed, let $u=s_{i_1}{s}_{i_2},$ where $s_{i_1}{s}_{i_2}\in S_p.$ If $s_{i_1}{s}_{i_2}\in S,$ then $l\left(x\right)=l\left(w_0\right)-2$, which contradicts $w_0\le x$. So, we need to check the following remaining possibilities: $s_{i_1}{s}_{i_2}=s{s}_0$ and $s_{i_1}{s}_{i_2}=s_{0}s$ with $s\in S.$ In the first case, $l\left(w_{0}s\right)<l\left(w_0\right),$ since $w_{0}s\in W.$ Then, using the Exchange Property [34] (p. 94), $w_{0}s=s_{j_1}{s}_{j_2}\cdots \widehat{s_j}{\cdots s}_{j_k}$ for some $j_1\le j\le j_k,$ which contradicts the Subword Property. In the second case, we will assume that $s_{0}s\ne s{s}_0,$ otherwise we will return to the first case. Then, the defining relation (1) gives $s\in \{s_1,s_l\}.$ Hence, $x=w_0{s}_0{s}_1,w_0{s}_0{s}_l.$

Continuing this process, we obtain $u=y{s}_0{y}^{\prime },$ where $y$ and $y^{\prime }$ satisfy the following conditions:

(1)

$y,y^{\prime }\in W$ and $y\in \left[1,y^{\prime }\right]$;

(2)

$l\left(u\right)=l\left(y\right)+l\left(s_0\right)+l\left(y^{\prime }\right)=l\left(y\right)+l\left(y^{\prime }\right)+1.$

Using the Subword Property, $x=w_{0}y{s}_0{y}^{\prime }\le w_{0}w=w_0{y}_i,$ $i\in \left\{1,2,\cdots ,l-2\right\},$ if and only if the conditions (1), (2), and

(3)

$y^{\prime }\le y_i,$

hold, since all reduced expressions of $y_i$ begins with $s_0$ and $l\left(w_{0}y\right)=l\left(w_0\right)-l\left(y\right)$ [1] (p. 16).

The conditions (1)–(3) provide that

(4)

$s_0{y}^{\prime }=y_j$ for some $j\le i.$

Hence, we can write the elements of the Bruhat interval $[w_0,w_0{y}_i]$ in the form $x=w_{0}y{y}_j.$ If $l\left(w_{0}y{y}_j\right)=l\left(w_0\right)+k,$ then, by (2) and (4), $l\left(y_j\right)-l\left(y\right)=k,$ since $l\left(w_{0}y\right)=l\left(w_0\right)-l\left(y\right)$ [1] (p. 16). Since $l\left(y_j\right)=j+1,$ from $l\left(y_j\right)-l\left(y\right)=k$ and the conditions (1) and (4) it follows that $j\ge k-1,$ So, $w_{0}y{y}_j$ is an element of length $l\left(w_0\right)+k$ of the Bruhat interval $\left[w_0,w_0{y}_i\right]$ if and only if

(a1)

$j\in \left\{k-1,k,\cdots ,i\right\};$

(a2)

$l\left(y_j\right)-l\left(y\right)=k;$

(a3)

$y\in \left[1,s_0{y}_j\right];$

(a4)

$l\left(y{y}_j\right)=l\left(y\right)+l\left(y_j\right).$

Now, we calculate $f_{w_0,w_0{y}_i}^k$ for the Bruhat interval $\left[w_0,w_0{y}_i\right].$ Consider elements of the form $w_{0}y{y}_j.$

If $j=k-1$, then, by the condition (a2), $l\left(y\right)=0,$ hence $y=1.$ It is easy to see that $y=1$ also satisfies the conditions (a3) an (a4). Therefore, $w_0{y}_{k-1}$ is an $l\left(w_0\right)+k$ length element in $\left[w_0,w_0{y}_i\right].$

If $j=k+t,$ then, by the condition (a1), $0\le t\le i-k,$ and by the condition (a2), $l\left(y\right)=t+1.$ Then, the conditions (a3) and (a4) are satisfied only by the following elements of length $l\left(w_0\right)+k:$

$$w_0{s}_{i_1}{s}_{i_2}\cdots s_{i_{t+1}}{y}_{k+t},i_1<i_2<\cdots <i_{t+1},i_1,i_2,\cdots ,i_{t+1}\in \left\{1,2,\cdots ,k+t\right\}.$$

The number such elements is equal to $C_{k+t}^{t+1}.$ Therefore,

$$f_{w_0,w_0{y}_i}^k=1+\sum _{t=0}^{i-k}{C}_{k+t}^{t+1}=\sum _{t=-1}^{i-k}{C}_{k+t}^{t+1}=C_{i+1}^{i-k+1}=C_{i+1}^k.$$

Since $l\left(w_0{y}_i\right)=l\left(w_0\right)+i+1,$ we obtain that $f_{w_0,w_0{y}_i}^k=C_{l\left(w_0{y}_i\right)-l\left(w_0\right)}^k.$

If $y,w\in \left\{y_i\right|i=0,1,\cdots ,l-2\},$ then, using similar arguments as above, we see that the set of $l\left(w_0{y}_i\right)+k$ length elements of the Bruhat interval $[w_0{y}_i,w_0{y}_j]$ consists only of elements of the form $w_{0}y{y}_t$, satisfying the following conditions:

(b1)

$t\in \left\{i+k,i+k+1,\cdots ,j\right\};$

(b2)

$l\left(y_t\right)-l\left(y\right)=k+i+1;$

(b3)

$y\in \left[1,y_i^{-1}{y}_t\right];$

(b4)

$l\left(y{y}_t\right)=l\left(y\right)+l\left(y_t\right).$

If $t=i+k$, then, by the condition (b2), $l\left(y\right)=0,$ hence $y=1.$ It is easy to see that $y=1$ also satisfies the conditions (b3) and (b4). Therefore, $w_0{y}_{i+k}$ is an $l\left(w_0{y}_i\right)+k$ length element in $\left[w_0{y}_i,w_0{y}_j\right].$

If $t=i+k+m,$ then, by the condition (b1), we have $1\le m\le j-i-k.$ The condition (b2) yields $l\left(y\right)=m.$ Then, the conditions (b3) and (b4) are satisfied only by the following $l\left(w_0{y}_i\right)+k$ length elements:

$$w_0{s}_{a_1}{s}_{a_2}\cdots s_{a_m}{y}_{i+k+m},a_1<a_2<\cdots <a_m,a_1,a_2,\cdots ,a_m\in \left\{i+1,i+3,\cdots ,i+k+m\right\}.$$

The number of such elements is equal to $C_{k+m-1}^m.$ Therefore,

$$f_{w_0{y}_i,w_0{y}_j}^k=1+\sum _{m=1}^{j-i-k}{C}_{k+m-1}^m=\sum _{m=0}^{j-i-k}{C}_{k+m-1}^m=C_{j-i}^{j-i-k}=C_{j-i}^k.$$

Since $l\left(w_0{y}_j\right)-l\left(w_0{y}_i\right)=j-i,$ we obtain that $f_{w_0{y}_i,w_0{y}_j}^k=C_{l\left(w_0{y}_j\right)-l\left(w_0{y}_i\right)}^k.$

Next, we calculate $f_{w_0,w_0{z}_i}^k$ for the Bruhat interval $\left[w_0,w_0{z}_i\right].$ Let $k>1.$ Then, the set of $l\left(w_0\right)+k$ length elements of the Bruhat interval $\left[w_0,w_0{z}_i\right]$ consists only of elements of the form $w_{0}y{y}_j$, satisfying the conditions (a1)–(a4), and of the form $w_{0}x{z}_t$, satisfying the following conditions:

(c1)

$t\in \left\{k-2,k-1,\cdots ,i\right\};$

(c2)

$l\left(z_t\right)-l\left(x\right)=k;$

(c3)

$x\in \left[1,s_0{z}_t\right];$

(c4)

$l\left(x{z}_t\right)=l\left(x\right)+l\left(z_t\right).$

If $t=k-2$, then, by condition (c2), $l\left(x\right)=0.$ Hence, using the condition (c3), we have one $l\left(w_0\right)+k$ length element $w_0{z}_{k-2}$ in $\left[w_0,w_0{z}_i\right].$ If $t=k-1$, then $l\left(x\right)=1,$ and using the conditions (c3) and (c3), we have the following $k=C_k^1$ elements of length $l\left(w_0\right)+k$ in $\left[w_0,w_0{z}_i\right]:$

$$w_0{s}_{a_1}{z}_{k-1},a_1\in \left\{1,2,\cdots ,k-1,l\right\}.$$

If $t=k+m$, then, by (c1), $-1\le m\le i-k,$ and by (c2), $l\left(x\right)=m+2.$ Then, using the conditions (c3) and (c4), we have the following $C_{k+m+1}^{m+2}$ elements of length $l\left(w_0\right)+k$ in $\left[w_0,w_0{z}_i\right]:$

$$w_0{s}_{a_1}{s}_{a_2}\cdots s_{a_{m+2}}{z}_{k+m},a_1<a_2<\cdots <a_{m+2},a_1,a_2,\cdots ,a_m\in \left\{1,2,\cdots ,k+m,l\right\}.$$

Hence, the number of $l\left(w_0\right)+k$ length elements in $\left[w_0,w_0{z}_i\right]$, satisfying the conditions (c1)–(c4), is equal to

$$\sum _{m=-2}^{i-k}{C}_{k+m+1}^{m+2}=C_{i+2}^{i-k+2}=C_{i+2}^k.$$

We know that the number of elements of the form $w_{0}y{y}_j,$ satisfying the conditions (a1)–(a4), are equal to $C_{i+1}^k.$ Hence, if $k>1,$ then

$$f_{w_0,w_0{z}_j}^k=C_{i+1}^k+C_{i+2}^k.$$

If $k=0,$ then the statement is obvious. Let $k=1.$ Then, replacing the condition (c1) with the condition $t\in \left\{k-1,k,\cdots ,i\right\},$ we obtain

$$f_{w_0,w_0{z}_j}^1=C_{i+1}^1+C_{i+2}^1-1=2C_{i+1}^1.$$

Thus,

$$f_{w_0,w_0{z}_i}^k=\left\{\begin{array}{c}1\hspace{1em}\hspace{1em}\hspace{1em}\mathit{if}k=0,\\ 2C_{i+1}^1\hspace{1em}\mathit{if}k=1,\\ C_{i+1}^k+C_{i+2}^{k}otherwise.\end{array}\right.$$

Consider now the Bruhat interval $\left[w_0{y}_i,w_0{z}_j\right].$ The set of $l\left(w_0{y}_i\right)+k$ length elements of this interval consists only of elements of the form $w_{0}y{y}_t$, satisfying the conditions (b1)–(b4), and of the form $w_{0}x{z}_m$, satisfying the following conditions:

(d1)

$m\in \left\{i+k-1,i+k,\cdots ,j\right\};$

(d2)

$l\left(z_m\right)-l\left(x\right)=i+k+1;$

(d3)

$x\in \left[1,y_i^{-1}{z}_m\right];$

(d4)

$l\left(x{z}_m\right)=l\left(x\right)+l\left(z_m\right).$

The number of elements $w_{0}y{y}_t$ satisfying the conditions (b1)–(b4), as we already know from the above calculations, is equal to $C_{j-i}^k.$ It is easy to calculate that the number of elements $w_{0}x{z}_m$ satisfying the conditions (d1)–(d4) is equal to $C_{j-i}^{k-1}.$ Then,

$$f_{w_0{y}_i,w_0{z}_j}^k=C_{j-i}^k+C_{j-i}^{k-1}=C_{j-i+1}^k.$$

The last equality yields $f_{w_0{y}_i,w_0{z}_j}^k=C_{l\left(w_0{z}_j\right)-l\left(w_0{y}_i\right)}^k,$ since

$$l\left(w_0{z}_j\right)-l\left(w_0{y}_i\right)=l\left(w_0\right)+j+1-\left(l\left(w_0\right)+i\right)=j-i+1.$$

Finally, the set of $l\left(w_0{y}_i\right)+k$ length elements of the Bruhat interval $\left[w_0{z}_i,w_0{z}_j\right]$ consists only of elements of the form $w_{0}x{z}_t$ satisfying the following conditions:

(e1)

$t\in \left\{i+k,i+k+1,\cdots ,j\right\};$

(e2)

$l\left(z_t\right)-l\left(x\right)=i+k+2;$

(e3)

$x\in \left[1,z_i^{-1}{z}_t\right];$

(e4)

$l\left(x{z}_t\right)=l\left(x\right)+l\left(z_t\right).$

The number of such elements is equal to $C_{j-i}^k.$ Since $l\left(w_0{z}_j\right)-l\left(w_0{z}_i\right)=j-i,$ we obtain that $f_{w_0{z}_i,w_0{z}_j}^k=C_{l\left(w_0{z}_j\right)-l\left(w_0{z}_i\right)}^k.$

Step (A2) From the proof of Lemma 4 we see that the elements of the Bruhat intervals $\left[w_{0}y,w_{0}w\right]$ with $y,w\in Y\cup Z$ belong to the set

$$W_0(Y\cup Z)=\left\{w_{0}vx|x\in Y\cup Z,1\le v\le s_{0}x\mathrm{a}\mathrm{n}\mathrm{d}l\left(vx\right)=l\left(v\right)+l(x)\right\}.$$

Note that for all $w\in Y\cup Z$, the element $w_{0}w$ is also contained in $W_0\left(Y\cup Z\right).$

Further, to compute the Kazhdan–Lusztig polynomials, we need the numbers $f_{y,w}^k$ for the Bruhat intervals $\left[y,w\right],$ where $y,w\in W_0(Y\cup Z).$ □

Lemma 4.

Let $y,w\in W_0(Y\cup Z)$. Suppose that $y\ne w_0$ and $w\notin \left\{z_i\right|i=1,2,\cdots ,l-2\}.$ If $y\le w,$ then $f_{y,w}^k=C_{l\left(w\right)-l\left(y\right)}^k.$

Proof. 

If $y=w_0,$ $w\in \left\{z_i\right|i=1,2,\cdots ,l-2\}$ and $k\ne 0,$ then by Lemma 3,

$$f_{y,w}^k\ne C_{l\left(w\right)-l\left(y\right)}^k.$$

Therefore, we suppose that $y\ne w_0$ and $w\notin \left\{z_i\right|i=1,2,\cdots ,l-2\}.$

Let $y\le w$ and $y=w_{0}vx,w=w_0{v}^{\prime }u,$where $x,u\in Y\cup Z.$ Then, the Subword Property yields

$$v^{\prime }\le v,$$

otherwise we would obtain $w_0{v}^{\prime }<w_{0}v$, which contradicts $y\le w.$ From the inequalities $v^{\prime }\le v,$ $v\le s_{0}x$ and $v^{\prime }\le s_{0}u$ it follows that

$$v,v^{\prime }\in \left[1,s_{0}x\right]\mathrm{and}x\le u.$$

Hence, the set of $l\left(w_{0}vx\right)+k$ length elements of the Bruhat interval $[w_{0}vx,w_0{v}^{\prime }u]$consists only of elements of the following two forms:

(1)

$w_0{x}^{\prime }x\in W_0(Y\cup Z),$ where $x^{\prime }$ satisfies the following conditions:

(a)

$l\left(x^{\prime }\right)=l\left(v\right)-k;$

(b)

$l\left(x^{\prime }x\right)=l\left(x^{\prime }\right)+l\left(x\right);$

(c)

$x^{\prime }\le v.$

(2)

$w_0{x}^{″}z\in W_0(Y\cup Z),$ where $x^{″}$ and $z$ satisfy the following conditions:

(a)

$l\left(z\right)-l\left(x^{″}\right)=l\left(x\right)-l\left(v\right)+k$;

(b)

$l\left(x^{″}z\right)=l\left(x^{″}\right)+l\left(z\right)$;

(c)

$x\le z\le u$;

(d)

$v^{\prime }\le x^{″}\le x.$

The number of elements in form (1) is equal to $C_{l\left(v\right)}^k,$ and for the elements in form (2) it is equal to $C_{l\left(v\right)}^{k-1}+C_{l\left(v\right)+1}^{k-1}+\cdots +C_{l\left(u\right)-l\left(x\right)+l\left(v\right)-l\left(v^{\prime }\right)-1}^{k-1}$. Then,

$$f_{y,w}^k=f_{w_{0}vx,w_0{v}^{\prime }u}^k=C_{l\left(v\right)}^k+C_{l\left(v\right)}^{k-1}+C_{l\left(v\right)+1}^{k-1}+\cdots +C_{l\left(u\right)-l\left(x\right)+l\left(v\right)-l\left(v^{\prime }\right)-1}^{k-1}=C_{l\left(u\right)-l\left(x\right)+l\left(v\right)-l\left(v^{\prime }\right)}^k.$$

Since $l\left(w\right)-l\left(y\right)=l\left(w_0{v}^{\prime }u\right)-l\left(w_{0}vx\right)=l\left(v\right)-l\left(v^{\prime }\right)+l\left(u\right)-l\left(x\right),$ then

$$f_{y,w}^k=C_{l\left(u\right)-l\left(x\right)+l\left(v\right)-l\left(v^{\prime }\right)}^k=C_{l\left(w\right)-l\left(y\right)}^k.$$

□

Step (A3). Calculation of the Kazhdan–Luisztig R-polynomials. We use the following:

Lemma 5.

Let $y,w\in W_0\left(Y\cup Z\right)$ and $y\le w.$ Then, $R_{y,w}=({q-1)}^{l\left(w\right)-l\left(y\right)}.$

Proof. 

As in Lemma 4, we can write the elements $y$ and $w$ in the forms

$$\begin{array}{c}y=w_{0}vx,\mathrm{where}1\le v\le s_{0}x\mathrm{a}\mathrm{n}\mathrm{d}l\left(vx\right)=l\left(v\right)+l\left(x\right),\\ w=w_0{v}^{\prime }u,\mathrm{where}1\le v^{\prime }\le s_{0}u\mathrm{a}\mathrm{n}\mathrm{d}l\left(v^{\prime }u\right)=l\left(v^{\prime }\right)+l\left(u\right),\end{array}$$

respectively.

Now, we use induction on $l\left(w\right)-l\left(y\right).$ If $l\left(w\right)-l\left(y\right)\le 2,$ then according to statement (b) of Lemma 1, $R_{y,w}=({q-1)}^{l\left(w\right)-l\left(y\right)}.$ Now, let $l\left(w\right)-l\left(y\right)>2.$ We fix $s\in S$ such that $ws<w.$ For $ys$ there are two possible cases: $ys<y$ and $ys>y.$

If $ys<y$, then, by statement (a) of Lemma 2, $ys<ws$ since $y<w.$ Then, using (2) and the induction hypothesis, we obtain

$$R_{y,w}=R_{ys,ws}={(q-1)}^{l\left(ws\right)-l\left(ys\right)}={(q-1)}^{l\left(w\right)-l\left(y\right)}.$$

If $y<ys,$according to statement (b) of Lemma 2, $ys<w$ and $y\le ws$ since $y<w.$ So, we can use the induction hypothesis and Formula (3). Using (3), we obtain

$$R_{y,w}=\left(q-1\right)R_{y,ws}+q{R}_{ys,ws}.$$

(10)

Now, we prove that $R_{ys,ws}=0$. As above, let $y=w_{0}vx.$ If $y<ys,$ then by the Subword Property, $x<xs$ and $xs\ne sx.$ Indeed, if $xs\le x,$ then by the Exchange Property, $l\left(ys\right)=l\left(y\right)-1$, which contradicts $y<ys.$ Similarly, if $xs=sx,$ then $ys=w_{0}vsx<y.$ Hence, there is only one case when $xs=u$ and $us<u.$

Suppose that $ys\le ws.$ Then, by the Subword Property,

$$v^{\prime }\le v,v,v^{\prime }\in \left[1,s_{0}x\right]\mathrm{and}xs\le us.$$

The condition $xs\le us$ contradicts the conditions $xs=u$ and $us<u.$ Then, $ys\nleqq ws.$ Therefore, $R_{ys,ws}=0.$

Thus, according to (10), $R_{y,w}=\left(q-1\right)R_{y,ws}.$ By the induction hypothesis,

$$R_{y,ws}={(q-1)}^{l\left(ws\right)-l\left(y\right)}.$$

Then,

$$R_{y,w}=\left(q-1\right){(q-1)}^{l\left(ws\right)-l\left(y\right)}={(q-1)}^{l\left(w\right)-l\left(y\right)}.$$

□

Step (A4). Calculation of the Kazhdan–Luisztig $P$-polynomials. Using the previous results and the statements (a), (c), and (d) of Lemma 1, we can completely calculate the Kazhdan–Lusztig polynomial $P_{w_{0}y,w_{0}w}$ for all $y,w\in Y\cup Z$ with $y\le w.$

Proposition 1.

Let $y,w\in Y\cup Z$ and $y\le w.$ Then, $R_{w_{0}y,w_{0}w}=1,$ except for the case where $y=1$ and $w\in \left\{z_i\right|i=1,2,\cdots ,l-2\}.$ In the latter case, $R_{w_{0}y,w_{0}w}=q+1.$

Proof. 

Let $y\ne 1$ and $w\notin \left\{z_i\right|i=1,2,\cdots ,l-2\}.$ Using (4) and Lemmas 4–6, for all $y^{\prime }\in [w_{0}y,w_{0}w],$ we obtain

$$N_{y^{\prime },w_{0}w}=q^{l\left(y^{\prime }\right)}\sum _{k=0}^{l\left(w_{0}w\right)-l\left(y^{\prime }\right)}{C}_{l\left(w_{0}w\right)-l\left(y^{\prime }\right)}^k{(q-1)}^k=q^{l\left(y^{\prime }\right)}{q}^{l\left(w_{0}w\right)-l\left(y^{\prime }\right)}=q^{l\left(w_{0}w\right)}.$$

Then, according to statement (d) of Lemma 1, for all $y^{\prime }\in [w_{0}y,w_{0}w],$ $P_{y^{\prime },w_{0}w}=1.$ In particular, $P_{w_{0}y,w_{0}w}=1$ for all $y,w\in Y\cup Z$ with $y\ne 1$ and $w\notin \left\{z_i\right|i=1,2,\cdots ,l-2\}.$

Now we prove that $P_{w_{0}y,w_{0}w}=q+1$ if $y=1$ and $w\in \left\{z_i\right|i=1,2,\cdots ,l-2\}.$ From statements (a) and (c) of Lemma 1, for all $x\le w$, it follows that

$$q^{l\left(w\right)-l\left(x\right)}{\overline{P}}_{x,w}={\sum }_{x\le y\le w}{R}_{x,y}{P}_{y,w,}.$$

(11)

We write (11) for $x=w_0$ and $w=w_0{z}_i,$ where $i\in \left\{1,2,\cdots ,l-2\right\}:$

$$q^{l\left(w_0{z}_i\right)-l\left(w_0\right)}{\overline{P}}_{w_0,w_0{z}_i}={\sum }_{w_0\le y\le w_0{z}_i}{R}_{w_0,y}{P}_{y,w_0{z}_i,}$$

(12)

By Formula (4) and Lemmas 3–5, for all $y^{\prime }\in \left[y,w_0{z}_i\right],$

$$N_{y^{\prime },w_0{z}_i}=q^{l\left(w_0{z}_i\right)},$$

if $y\in [w_0,w_0{z}_i]\backslash \left\{w_0\right\}$ and $i\in \left\{1,2,\cdots ,l-2\right\}.$ Then, by statement (d) of Lemma 1, $P_{y,w_0{z}_i}=1$ for all $y\in [w_0,w_0{z}_i]\backslash \left\{w_0\right\}$ and for all $i\in \left\{1,2,\cdots ,l-2\right\}.$ Then, using Lemmas 3 and 4 for the right-hand side of (12), we obtain

$$q^{i+2}{\overline{P}}_{w_0,w_0{z}_i}=P_{w_0,w_0{z}_i,}+2C_{i+1}^1+\sum _{k=2}^{i+2}(C_{i+1}^k+C_{i+2}^k){(q-1)}^k,$$

where $i\in \left\{1,2,\cdots ,l-2\right\}.$ Simplify the right-hand side of the last equality:

$$q^{i+2}{\overline{P}}_{w_0,w_0{z}_i}=P_{w_0,w_0{z}_i,}-1-q+q^{i+1}+q^{1+2}.$$

The last equality yields $P_{w_0,w_0{z}_i}=q+1$ for all $i\in \left\{1,2,\cdots ,l-2\right\}.$ □

Step (A5). Calculation of the formal characters of simple modules. We prove the following:

Proposition 2.

The following character formulas hold:

(a) 

$\left[L\left(\tau \right)\right]=\left[V\left(\tau \right)\right]$ and for all $i\in \left\{0,1,\cdots ,l-2\right\},$ $\left[L\left(y_i·\tau \right)\right]={\sum }_{j=-1}^i{\left(-1\right)}^{i-j}\left[V\left(y_j·\tau \right)\right];$

(b) 

$\left[L\left(z_0·\tau \right)\right]=\left[V\left(\tau \right)\right]-\left[V\left(y_0·\tau \right)\right]+\left[V\left(z_0·\tau \right)\right];$

(c) 

for all $i\in \left\{1,2,\cdots ,l-2\right\},$

$$\left[L\left(z_i·\tau \right)\right]={\left(-1\right)}^{i+2}·2\left[V\left(\tau \right)\right]+\sum _{j=0}^i{\left(-1\right)}^{i-j+1}\left[V\left(y_j·\tau \right)\right]+\sum _{j=0}^i{\left(-1\right)}^{i-j}\left[V\left(z_j·\tau \right)\right].$$

Proof. 

(a) By (5), $\left[L\left(\tau \right)\right]=\left[L\left(1·\tau \right)\right]=P_{w_0,w_o}\left(1\right)\left[V\left(1·\tau \right)\right].$ By Proposition 1, $P_{w_0,w_o}\left(1\right)=1,$ so $\left[L\left(\tau \right)\right]=\left[V\left(1·\tau \right)\right]=\left[V\left(\tau \right)\right].$ Similarly, by (5), for all $i\in \left\{0,1,\cdots ,l-2\right\},$

$$\left[L\left(y_i·\tau \right)\right]=\sum _{j=-1}^i{\left(-1\right)}^{l\left(y_i\right)-l\left(y_j\right)}{P}_{w_0{y}_j,w_o{y}_i}\left(1\right)\left[V\right(y_j·\tau \left)\right].$$

By Proposition 1, $P_{w_0{y}_j,w_o{y}_i}=1$ for all $j\le i.$ Then, for all $i\in \left\{0,1,\cdots ,l-2\right\},$

$$\left[L\left(y_i·\tau \right)\right]=\sum _{j=-1}^i{\left(-1\right)}^{i-j}\left[V\left(y_j·\tau \right)\right],$$

since $l\left(y_i\right)-l\left(y_j\right)=i+1-\left(j+1\right)=i-j.$

(b) By (5),

$$\left[L\left(z_0·\tau \right)\right]=P_{w_0,w_o{z}_0}\left(1\right)\left[V\left(\tau \right)\right]-P_{w_0{y}_0,w_o{z}_0}\left(1\right)\left[V\left(y_0·\tau \right)\right]+P_{w_0{z}_0,w_o{z}_0}\left(1\right)\left[V\left(z_0·\tau \right)\right].$$

According to Proposition 1, $P_{w_0,w_o{z}_0}\left(1\right)=P_{w_0{y}_0,w_o{z}_0}\left(1\right)=P_{w_0{z}_0,w_o{z}_0}\left(1\right)=1.$ Then,

$$\left[L\left(z_0·\tau \right)\right]=\left[V\left(\tau \right)\right]-\left[V\left(y_0·\tau \right)\right]+\left[V\left(z_0·\tau \right)\right].$$

(c) By (5), for all $i\in \left\{1,2,\cdots ,l-2\right\},$

$$\begin{array}{c}\left[L\left(z_i·\tau \right)\right]={\left(-1\right)}^{i+2}·P_{w_0,w_o{z}_i}\left(1\right)\left[V\left(\tau \right)\right]+{\displaystyle \sum _{j=0}^i}{\left(-1\right)}^{i-j+1}{P}_{w_0{y}_j,w_o{z}_i}\left(1\right)\left[V\left(y_j·\tau \right)\right]\\ +{\displaystyle \sum _{j=0}^i}{\left(-1\right)}^{i-j}{P}_{w_0{z}_j,w_o{z}_1}\left[V\left(z_j·\tau \right)\right].\end{array}$$

By Proposition 1, for all $i\in \left\{1,2,\cdots ,l-2\right\},$ $P_{w_0,w_o{z}_i}\left(1\right)=2,$ and $P_{w_{0}z,w_o{z}_1}\left(1\right)=1$ if

$$z\in \left\{y_j,z_j|j=0,1,\cdots ,l-2\right\}.$$

Then, for all $\in \left\{1,2,\cdots ,l-2\right\},$

$$\left[L\left(z_i·\tau \right)\right]={\left(-1\right)}^{i+2}·2\left[V\left(\tau \right)\right]+{\sum }_{j=0}^i{\left(-1\right)}^{i-j+1}\left[V\left(y_j·\tau \right)\right]+{\sum }_{j=0}^i{\left(-1\right)}^{i-j}\left[V\left(z_j·\tau \right)\right].$$

□

• Second part. First, we prove some preliminary lemmas.

Lemma 6.

Let $\tau ={\sum }_{i=1}^l{a}_i{\omega }_i\in C_1\cap P,$ where ${\omega }_1,{\omega }_2,\cdots ,{\omega }_l$ are the fundamental weights and $a_1,a_2,\cdots ,a_l\in \mathbb{Z}.$ Then,

(a) 

$y_0·\tau =\left(p-{\sum }_{j=2}^l{a}_j-l\right){\omega }_1+{\sum }_{j=2}^{l-1}{a}_j{\omega }_j+\left(p-{\sum }_{j=1}^{l-1}{a}_j-l\right){\omega }_l;$

(b) 

for all$i\in \left\{1,2,\cdots ,l-2\right\},$

$$y_i·\tau =\left(p-\sum _{j=1}^l{a}_j-l-1\right){\omega }_1+a_1{\omega }_2+\cdots +a_{i-1}{\omega }_i+(a_i+a_{i+1}+1){\omega }_{i+1}+\sum _{j=i+2}^{l-1}{a}_j{\omega }_j+\left(p-\sum _{j=i+1}^{l-1}{a}_j-l+i\right){\omega }_l;$$

(c) 

$$z_0·\tau =\left(p-\sum _{j=2}^{l-1}{a}_j-l+1\right){\omega }_1+\sum _{j=2}^{l-2}{a}_j{\omega }_j+\left(a_{l-1}+a_l+1\right){\omega }_{l-1}+\left(p-\sum _{j=1}^l{a}_j-l-1\right){\omega }_l;$$

(d) 

for all$i\in \left\{1,2,\cdots ,l-2\right\},$

$$\begin{array}{c}{z}_i·\tau =\left(p-{\sum }_{j=1}^{l-1}{a}_j-l\right){\omega }_1+a_1{\omega }_2+\cdots +a_{i-1}{\omega }_i+\left(a_i+a_{i+1}+1\right){\omega }_{i+1}+{\sum }_{j=i+2}^{l-2}{a}_j{\omega }_j+\\ \left(a_{l-1}+a_l+1\right){\omega }_{l-1}+\left(p-{\displaystyle \sum _{j=i+1}^l}{a}_j-l+i-1\right){\omega }_l.\end{array}$$

Proof. 

The set of positive roots of $G$ can be seen as the set

$$R^{+}=\left\{{\alpha }_i+\cdots +{\alpha }_j={\epsilon }_i-{\epsilon }_{j+1}|i\le j=1,2,\cdots ,l\right\},$$

where ${\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_{l+1}$ are the orthonormal basis of ${\mathbb{R}}^{l+1}.$ Then,

$$〈\tau +\rho ,{\alpha }^{\vee }〉=\left\{\begin{array}{l}{a}_i+1,\mathit{if}\alpha ={\alpha }_i,i=1,2,\cdots ,l;\\ a_i+\cdots +a_i+j-i+1,\mathit{if}\alpha ={\alpha }_i+\cdots {+\alpha }_i,i<j.\end{array}\right.$$

(13)

(a) By (13), $y_0·\tau =\tau -\left(〈\tau +\rho ,{{\tilde{\alpha }}_0}^{\vee }〉-p\right){\tilde{\alpha }}_0=\tau +\left(p-\sum _{j=1}^l{a}_j-l-1\right)\left({\omega }_1+{\omega }_l\right)=$

$$\left(p-\sum _{j=2}^l{a}_j-l\right){\omega }_1+\sum _{j=2}^{l-1}{a}_j{\omega }_j+\left(p-\sum _{j=1}^{l-1}{a}_j-l\right){\omega }_l.$$

(b) We use induction on $i.$ According to (13),

$$s_1·\tau =\left(-a_1-1\right){\omega }_1+\left(a_1+a_2+1\right){\omega }_2+\sum _{j=3}^l{a}_j{\omega }_j.$$

Then, $y_1·\tau =s_1·\tau -\left(〈s_1·\tau +\rho ,{{\tilde{\alpha }}_0}^{\vee }〉-p\right){\tilde{\alpha }}_0=s_1·\tau ++\left(p-{\sum }_{j=2}^l{a}_j-l+1\right)\left({\omega }_1+{\omega }_l\right)=$

$$\left(p-\sum _{j=1}^l{a}_j-l-1\right){\omega }_1+\left(a_1+a_2+2\right){\omega }_2+\sum _{j=3}^{l-1}{a}_j{\omega }_j+\left(p-\sum _{j=2}^{l-1}{a}_i-l+1\right){\omega }_l.$$

Therefore, the statement is true for $i=1.$ Suppose the statement is true for all $i<t,$ where $t\le l-2.$ Using (13), we have

$$s_t·\tau =\sum _{j=1}^{t-2}{a}_j{\omega }_j+\left(-a_t-2\right){\omega }_t+\left(a_t+a_{t+1}+1\right){\omega }_{t+1}+\sum _{j=t+2}^l{a}_j{\omega }_j.$$

By the induction hypothesis,

$$y_{t-1}·\tau =\left(p-\sum _{j=1}^l{a}_j-l-1\right){\omega }_1+a_1{\omega }_2+\cdots +a_{t-2}{\omega }_{t-1}+\left(a_{t-1}+a_t+1\right){\omega }_t+\sum _{j=t+1}^{l-1}{a}_j{\omega }_j+\left(p-\sum _{j=t}^{l-1}{a}_j-l+t-1\right){\omega }_l.$$

Then, $y_t·\tau =y_{t-1}·\left(s_t·\tau \right)=$

$$\left(p-\sum _{j=1}^l{a}_j-l-1\right){\omega }_1+a_1{\omega }_2+\cdots +a_{t-1}{\omega }_t+\left(a_t+a_{t+1}+1\right){\omega }_{t+1}+\sum _{j=t+2}^{l-1}{a}_j{\omega }_j+\left(p-\sum _{j=i+1}^{l-1}{a}_j-l+t\right){\omega }_l.$$

Hence, the statement is true for $i\in \left\{1,2,\cdots ,l-2\right\}.$

The statements (c) and (d) can easily be obtained in the same way as the previous statements.

Now let

$${\beta }_i=\sum _{j=i}^l{\alpha }_j,{{\beta }^{\prime }}_i=\sum _{j=1}^{l+1-i}{\alpha }_j.$$

Note that ${\beta }_i={{\beta }^{\prime }}_i={\tilde{\alpha }}_0.$ Then, $s_{{\beta }_1,p}={s^{\prime }}_{{\beta }_1,p}=s_0.$ □

Lemma 7.

The following hold:

(a) 

for all$i\in \left\{2,3,\cdots ,l-1\right\},$ $s_{{\beta }_i,p}=s_0{s}_1{s}_2\cdots s_{i-2}{s}_{i-1}{s}_{i-2}\cdots s_1{s}_0;$

(b) 

for all$i\in \left\{2,3,\cdots ,l-1\right\},$ $s_{{{\beta }^{\prime }}_i,p}=s_0{s}_l{s}_{l-1}\cdots s_{l-i+3}{s}_{l-i+2}{s}_{l-i+3}\cdots s_l{s}_0.$

Proof. 

(a) Using (13) and statements (a) and (b) of Lemma 6, we obtain $s_{{\beta }_i,p}·\left(y_{i-1}·\tau \right)=y_{i-2}·\tau$ for all $i\in \left\{2,3,\cdots ,l-1\right\}.$ Then, for all $i\in \left\{2,3,\cdots ,l-1\right\},$

$$s_{{\beta }_i,p}=y_{i-2}{y}_{i-1}^{-1}=s_0{s}_1{s}_2\cdots s_{i-2}{s}_{i-1}{s}_{i-2}\cdots s_1{s}_0.$$

□

The proof of statement (b) is similar to the proof of the previous statement (a).

• Steps (B1) and (B2) are the proofs for the following:

Proposition 3.

The following character formulas hold:

(a) 

$\left[V\left(\tau \right)\right]=\left[L\left(\tau \right)\right]$ and for all $i\in \left\{0,1,\cdots ,l-2\right\},$ $\left[V\left(y_i·\tau \right)\right]=\left[L\left(y_i·\tau \right)\right]+\left[V\left(y_{i-1}·\tau \right)\right];$

(b) 

$\left[V\left(z_0·\tau \right)\right]=\left[L\left(z_0·\tau \right)\right]+\left[L\left(y_0·\tau \right)\right];$

(c) 

$\left[V\left(z_1·\tau \right)\right]=\left[L\left(z_1·\tau \right)\right]+\left[L\left(z_0·\tau \right)\right]+\left[L\left(y_1·\tau \right)\right]+\left[V\left(y_0·\tau \right)\right]+\left[L\left(\tau \right)\right];$

(d) 

for all $i\in \left\{2,3,\cdots ,l-2\right\},$

$$\left[V\left(z_i·\tau \right)\right]=\left[L\left(z_i·\tau \right)\right]+\left[L\left(z_{i-1}·\tau \right)\right]+\left[L\left(y_i·\tau \right)\right]+\left[V\left(y_{i-1}\left)\right]·\tau \right)\right].$$

Proof. 

(a) The first equality is evident. We use, for the other equalities, induction on $i.$ According to (13) and statement (a) of Lemma 6, $〈y_0·\tau +\rho ,{\alpha }^{\vee }〉<p,$ except in the case when $\alpha ={\tilde{\alpha }}_0.$ In the last case, $〈y_0·\tau +\rho ,{{\tilde{\alpha }}_0}^{\vee }〉=2p-{\sum }_{j=1}^l{a}_j-l.$ Using (8), we obtain

$$\sum _{j>0}\left[{V(y_0·\tau )}^j\right]=\left[V\left(s_0·\left(y_0·\tau \right)\right)\right]=\left[V\left(s_0{y}_0)·\tau \right)\right]=\left[V\left(s_0{s}_0)·\tau \right)\right]=\left[V\left(\tau \right)\right]=\left[L\left(\tau \right)\right].$$

Then, by (6), ${V(y_0·\tau )}^j=L\left(\tau \right).$ Therefore, by (7), $V(y_0·\tau )/L\left(\tau \right)=L(y_0·\tau ).$ This implies that $\left[V\left(y_0·\tau \right)\right]=\left[L\left(y_0·\tau \right)\right]+\left[L\left(\tau \right)\right].$ Thus, the statement is true for $i=0.$

Suppose the statement is true for all $i<t,$ where $t\le l-2.$ According to (13) and statement (b) of Lemma 6, $〈y_t·\tau +\rho ,{\alpha }^{\vee }〉<p,$ except in the following cases:

$$〈y_t·\tau +\rho ,{{\tilde{\alpha }}_0}^{\vee }〉=2p-{\sum }_{j=t+1}^l{a}_j-l,$$

and

$$〈y_t·\tau +\rho ,{{\beta }_i}^{\vee }〉=p+{\sum }_{j=i-1}^t{a}_j+t-i+2,i=2,3,\cdots ,t+1.$$

Using (8), (13), and statement (b) of Lemma 6, we obtain

$$\sum _{j>0}\left[{V(y_t·\tau )}^j\right]=\sum _{\alpha \in \{{\beta }_1,{\beta }_2,\cdots ,{\beta }_{t+1}\}}\left[V\left(s_{\alpha ,p}·\left(y_t·\tau \right)\right)\right]=\sum _{\alpha \in \{{\beta }_1,{\beta }_2,\cdots ,{\beta }_{t+1}\}}\left[V\left(\left(s_{\alpha ,p}{y}_t\right)·\tau \right)\right].$$

Then, by statement (a) of Lemma 7,

$$\begin{array}{c}{\displaystyle \sum _{j>0}}\left[{V(y_t·\tau )}^j\right]={\displaystyle \sum _{j=1}^{t+1}}\left[V\right((s_0{s}_1{s}_2\cdots s_{i-2}{s}_{i-1}{s}_{i-2}\cdots s_1{s}_0)(s_0{s}_1{s}_2\cdots s_{i-2})·\tau )]=\\ {\displaystyle \sum _{j=1}^{t+1}}\left[V\right((s_i{s}_{i+1}\cdots s_t{y}_{t-2})·\tau )]={\displaystyle \sum _{j=1}^{t+1}}{\left(-1\right)}^{l\left(s_i{s}_{i+1}\cdots s_t\right)}\left[V\right(y_{t-2}·\tau )].\end{array}$$

By the induction hypothesis, $\left[V\left(y_i·\tau \right)\right]=\left[L\left(y_i·\tau \right)\right]+\left[L\left(y_{i-1}·\tau \right)\right]$ for all $0\le i<t.$ Then,

$$\sum _{j>0}\left[{V(y_t·\tau )}^j\right]={\left(-1\right)}^t\left[L\left(\tau \right)\right]+\sum _{j=2}^{t+1}{\left(-1\right)}^{t+1-j}\left(\left[L\left(y_{j-2}·\tau \right)\right]+\left[L\left(y_{j-3}·\tau \right)\right]\right)=\left[L\left(y_{t-1}·\tau \right)\right].$$

This implies that $\left[V\left(y_t·\tau \right)\right]=\left[L\left(y_t·\tau \right)\right]+\left[L\left(y_{t-1}·\tau \right)\right].$ Therefore, the statement holds for all $i=0,1,\cdots ,l-2.$

(b) By (13) and statement (c) of Lemma 6, $〈z_0·\tau +\rho ,{\alpha }^{\vee }〉<p,$ except in the following cases: $〈z_0·\tau +\rho ,{{\tilde{\alpha }}_0}^{\vee }〉=2p-{\sum }_{j=1}^{l-1}{a}_j-l+1$ and $〈z_0·\tau +\rho ,{{{\beta }^{\prime }}_2}^{\vee }〉=p+a_l+1.$ Using (8), we obtain

$$\sum _{j>0}\left[{V(z_0·\tau )}^j\right]=\left[V\left(s_0·\left(z_0·\tau \right)\right)\right]+\left[V\left(s_{{{\beta }^{\prime }}_2,p}·\left(z_1·\tau \right)\right)\right].$$

Then, by Lemma 7, ${\sum }_{j>0}\left[{V(z_0·\tau )}^j\right]=-\left[V\left(\tau \right)\right]+\left[V\left(y_0·\tau \right)\right].$ Using the statements (a), we obtain ${\sum }_{j>0}\left[{V(z_0·\tau )}^j\right]=\left[L\left(y_0·\tau \right)\right].$ Hence, by (7), $\left[V\left(z_0·\tau \right)\right]=\left[L\left(z_0·\tau \right)\right]+\left[L\left(y_0·\tau \right)\right].$

(c) By (8), (13), and statement (d) of Lemma 6,

$$\sum _{j>0}\left[{V(z_1·\tau )}^j\right]=\left[V\left(s_{{\beta }_1,p}·\left(z_1·\tau \right)\right)\right]+\left[V\left(s_{{\beta }_2,p}·\left(z_1·\tau \right)\right)\right]+\left[V\left(s_{{{\beta }^{\prime }}_2,p}·\left(z_1·\tau \right)\right)\right].$$

Then, by Lemma 7, ${\sum }_{j>0}\left[{V(z_1·\tau )}^j\right]=\left[V\left(\tau \right)\right]+\left[V\left(y_1·\tau \right)\right]+\left[V\left(z_0·\tau \right)\right].$ Using statements (a) and (b) of this lemma, we obtain

$$\sum _{j>0}\left[{V(z_1·\tau )}^j\right]=\left[L\left(z_0·\tau \right)\right]+\left[L\left(y_1·\tau \right)\right]+2\left[L\left(y_0·\tau \right)\right]+\left[L\left(\tau \right)\right].$$

From this, it immediately follows that $\left[V\left(z_1·\tau \right):L\left(\mu \right)\right]=1$ for $\mu =\tau ,y_1·\tau ,z_0·\tau .$ Therefore, the simple modules $\left[L\left(\tau \right)\right],L\left(y_1·\tau \right),$ and $L\left(z_0·\tau \right)$ are the composition factors for ${V(z_1·\tau )}^1/{V(z_1·\tau )}^2.$ However, for $\left[V\left(z_1·\tau \right):L\left(y_0·\tau \right)\right]$, there are two possibilities. This composition coefficient could be equal to 1 or 2. However, according to the translation principle [2] (p. 297) and statement (a), $\left[V\left(z_1·\tau \right):L\left(y_0·\tau \right)\right]=\left[V\left(y_1·\tau \right):L\left(y_0·\tau \right)\right]=1.$ We thus obtain

$$\left[V\left(z_1·\tau \right)\right]=\left[L\left(z_1·\tau \right)\right]+\left[L\left(z_0·\tau \right)\right]+\left[L\left(y_1·\tau \right)\right]+\left[L\left(y_0·\tau \right)\right]+\left[L\left(\tau \right)\right].$$

(d) We again use induction on $i.$ According to (8), (13), and statement (d) of Lemma 6,

$$\sum _{j>0}\left[{V(z_2·\tau )}^j\right]=\left[V\left(s_{{\beta }_1,p}·\left(z_1·\tau \right)\right)\right]+\left[V\left(s_{{\beta }_2,p}·\left(z_1·\tau \right)\right)\right]+\left[V\left(s_{{\beta }_3,p}·\left(z_1·\tau \right)\right)\right]+\left[V\left(s_{{{\beta }^{\prime }}_2,p}·\left(z_1·\tau \right)\right)\right].$$

Then, by Lemma 7,

$$\sum _{j>0}\left[{V(z_2·\tau )}^j\right]=-\left[V\left(\tau \right)\right]+\left[V\left(y_2·\tau \right)\right]-\left[V\left(z_0·\tau \right)\right]+\left[V\left(z_1·\tau \right)\right].$$

Using the statements (a) and (b), we obtain

$$\sum _{j>0}\left[{V(z_2·\tau )}^j\right]=2\left[L\left(y_1·\tau \right)\right]+\left[L\left(y_2·\tau \right)\right]+\left[L\left(z_1·\tau \right)\right].$$

This implies that $\left[V\left(z_2·\tau \right):L\left(\mu \right)\right]=1$ for $\mu =z_1·\tau ,y_2·\tau .$ Therefore, the simple modules $L\left(z_1·\tau \right),$ and $L\left(y_2·\tau \right)$ are the composition factors for ${V(z_2·\tau )}^1/{V(z_2·\tau )}^2.$ However, for $\left[V\left(z_2·\tau \right):L\left(y_1·\tau \right)\right]$, there are two possibilities. This composition coefficient could be equal to 1 or 2. However, according to the translation principle [2] (p. 297) and statement (a), $\left[V\left(z_2·\tau \right):L\left(y_1·\tau \right)\right]=\left[V\left(y_z·\tau \right):L\left(y_0·\tau \right)\right]=1.$ So, we obtain

$$\left[V\left(z_2·\tau \right)\right]=\left[L\left(z_2·\tau \right)\right]+\left[L\left(z_1·\tau \right)\right]+\left[L\left(y_2·\tau \right)\right]+\left[L\left(y_1·\tau \right)\right].$$

Therefore, for $i=2$, the statement is established.

Now, suppose that the statement is true for all $i<t,$ where $t\le l-2.$ Using (8), (13), and statement (d) of Lemma 6, we obtain

$$\sum _{j>0}\left[{V(z_t·\tau )}^j\right]=\sum _{\alpha \in \{{\beta }_1,{\beta }_2,\cdots ,{\beta }_{t+1}\}}\left[V\left(s_{\alpha ,p}·\left(y_t·\tau \right)\right)\right]+\left[V\left(s_{{{\beta }^{\prime }}_2,p}·\left(z_t·\tau \right)\right)\right].$$

Then, by Lemma 7,

$$\sum _{j>0}\left[{V(z_t·\tau )}^j\right]={\left(-1\right)}^{t+1}\left[L\left(\tau \right)\right]+\sum _{j=2}^{t+1}{\left(-1\right)}^{t+1-j}\left[L\left(z_{j-2}·\tau \right)\right]+\left[L\left(y_t·\tau \right)\right].$$

By statements (a)–(c) and the induction hypothesis,

$$\begin{array}{c}\left[V\left(y_t·\tau \right)\right]=\left[L\left(y_t·\tau \right)\right]+\left[L\left(y_{t-1}·\tau \right)\right],\\ \left[V\left(z_1·\tau \right)\right]=\left[L\left(z_1·\tau \right)\right]+\left[L\left(z_0·\tau \right)\right]+\left[L\left(y_1·\tau \right)\right]+\left[L\left(y_0·\tau \right)\right]+\left[L\left(\tau \right)\right],\\ \left[V\left(z_i·\tau \right)\right]=\left[L\left(z_i·\tau \right)\right]+\left[L\left(z_{i-1}·\tau \right)\right]+\left[L\left(y_i·\tau \right)\right]+\left[L\left(y_{i-1}·\tau \right)\right],2\le i<t.\end{array}$$

Then, we obtain $\sum _{j>0}\left[{V(z_t·\tau )}^j\right]=2\left[L\left(y_{t-1}·\tau \right)\right]+\left[L\left(y_t·\tau \right)\right]+\left[L\left(z_{t-1}·\tau \right)\right].$ This implies that $\left[V\left(z_t·\tau \right):L\left(\mu \right)\right]=1$ for $\mu =y_t·\tau ,z_{t-1}·\tau .$ Therefore, the simple modules $L\left(z_t·\tau \right)$ and $L\left(z_{t-1}·\tau \right)$ are the composition factors for ${V(z_t·\tau )}^1/{V(z_t·\tau )}^2.$ However, for the composition coefficient $\left[V\left(z_t·\tau \right):L\left(y_{t-1}·\tau \right)\right]$ there are two possibilities. This composition coefficient could be equal to 1 or 2. However, according to the translation principle [2] (p. 297) and the statement (a), $\left[V\left(z_t·\tau \right):L\left(y_{t-1}·\tau \right)\right]=\left[V\left(z_{t-1}·\tau \right):L\left(y_{t-1}·\tau \right)\right]=1.$ So, we obtain

$$\left[V\left(z_t·\tau \right)\right]=\left[L\left(z_t·\tau \right)\right]+\left[L\left(z_{t-1}·\tau \right)\right]+\left[L\left(y_t·\tau \right)\right]+\left[L\left(y_{t-1}·\tau \right)\right].$$

□

• Steps (B3). Using Proposition 3, we obtain

  (i) 

  $\left[L\left(\tau \right)\right]=\left[V\left(\tau \right)\right]$ and for all $i\in \left\{0,1,\cdots ,l-2\right\},$ $\left[L\left(y_i·\tau \right)\right]=\sum _{j=-1}^i{\left(-1\right)}^{i-j}\left[V\left(y_j·\tau \right)\right];$

  (ii) 

  $\left[L\left(z_0·\tau \right)\right]=\left[V\left(\tau \right)\right]-\left[V\left(y_0·\tau \right)\right]+\left[V\left(z_0·\tau \right)\right];$

  (iii) 

  for all $i\in \left\{1,2,\cdots ,l-2\right\},$

$$\left[L\left(z_i·\tau \right)\right]={\left(-1\right)}^{i+2}·2\left[V\left(\tau \right)\right]+\sum _{j=0}^i{\left(-1\right)}^{i-j+1}\left[V\left(y_j·\tau \right)\right]+\sum _{j=0}^i{\left(-1\right)}^{i-j}\left[V\left(z_j·\tau \right)\right].$$

The obtained formal characters of simple modules (i), (ii), and (iii) completely coincide with the formal characters of the statements (a), (b), and (c) of Proposition 2, respectively. Hence, for all $w\in Y\cup Z\subset W_p$ and for all $\tau \in C_1\cap P,$ Lusztig’s character formula, Formula (5), holds. The proof of Theorem 1 is complete.

3.2. On Cohomology of Simple Modules

In this subsection, we calculate the cohomology of the simple modules $L(w·0)$ for all $w\in Y\cup Z\subset W_p,$ using Theorem 1, Proposition 1, and (9).

Proposition 4.

Let $G$ be a Chevalley group of type $A_l$ over an algebraically closed field $K$ of characteristic $p\ge h.$ Then,

(a) 

for all $i=0,1,\cdots ,l-2,$

$${dim}_K{H}^n\left(G,L\left(y_i·0\right)\right)=\left\{\begin{array}{l}1ifn=i+1,\\ 0otherwise;\end{array}\right.$$

(b) 

following hold:

$${dim}_K{H}^n\left(G,L\left(z_0·0\right)\right)=\left\{\begin{array}{c}1\mathrm{if}n=2,\\ 0otherwise;\end{array}\right.$$

(c) 

for all $i=1,2,\cdots ,l-2,$

$${dim}_K{H}^n\left(G,L\left(z_i·0\right)\right)=\left\{\begin{array}{l}1ifn=i,i+2,\\ 0otherwise.\end{array}\right.$$

Proof. 

According to Theorem 1, for all simple modules $L\left(w·0\right),$ $w\in Y\cup Z\subset W_p,$ Lusztig’s character formula holds. Therefore, to calculate the dimensions of their cohomology, we can use the Formula (9). Let $P_{w_0,w_{0}w}=1,$ then by (9),

$${dim}_K{H}^n\left(G,L\left(w·0\right)\right)=\left\{\begin{array}{l}1\mathrm{if}n=l\left(w\right),\\ 0\mathrm{otherwise}.\end{array}\right.$$

(14)

Similarly, if $P_{w_0,w_{0}w}=1+q=1+t^2,$ then by (9),

$${dim}_K{H}^n\left(G,L\left(w·0\right)\right)=\left\{\begin{array}{l}1\mathrm{if}n=l\left(w\right),l\left(w\right)-2\\ 0\mathrm{otherwise}.\end{array}\right.$$

(15)

According to Proposition 1, for all $i=0,1,\cdots ,l-2,$ $P_{w_0,w_0{y}_i}=1.$ According to the definition of $y_i,$ $l\left(y_i\right)=i+1$ for all $i=0,1,\cdots ,l-2.$ Then, using (14), we obtain statement (a).

By Proposition 1, $P_{w_0,w_0{z}_0}=1.$ Then, (14) yields statement (b), since $l\left(z_0\right)=2.$

By Proposition 1, for all $i=1,2,\cdots ,l-2,$ $P_{w_0,w_0{z}_i}=1+q=1+t^2.$ So, using (15), we obtain statement (c), since $l\left(z_i\right)=i+2$ for all $i=1,2,\cdots ,l-2.$ □

4. Discussion

We have proven that, in the case of a Chevalley group of type $A_l$, Lusztig’s character formula is valid for all dominant elements $w\in Y\cup Z$ for all $p\ge h,$ where

$$\begin{array}{c}Y=\left\{y_{-1}=1,y_i=\left\{s_0{s}_1\cdots s_i\right|i=0,1,\cdots ,l-2\right\},\\ Z=\left\{z_{-1}=1,z_0=s_0{s}_l,z_i=\left\{s_0{s}_l{s}_1\cdots s_i\right|i=1,2,\cdots ,l-2\right\}.\end{array}$$

Using this result, we computed the cohomology of $G$ with coefficients in the simple modules $L\left(w·0\right),w\in Y\cup Z$ for all $p\ge h.$ It is easy to see that a similar result also holds for all dominant elements $w^{\prime }\in Y^{\prime }\cup Z^{\prime }$ for all $p\ge h,$ where

$$\begin{array}{c}{Y}^{\prime }=\left\{{y^{\prime }}_{-1}=1,{y^{\prime }}_i=s_0,{y^{\prime }}_i=\left\{s_0{s}_l{s}_{l-1}\cdots s_{l-i+1}\right|i=1,2,\cdots ,l-2\right\},\\ Z^{\prime }=\left\{z_{-1}=1,{z^{\prime }}_0=s_0{s}_l,z_i=\left\{s_0{s}_1{s}_l{s}_{l-1}\cdots s_{l-i+1}\right|i=1,2,\cdots ,l-2\right\}.\end{array}$$

Proposition 4, obtained using Theorem 1, agrees with Corollary 4.1 in [32] (p. 3872). Note that, in the case of the group $G$ of type $A_l,$ our result extends the lower bound on characteristic $p$ of statement (c) to the Coxeter number $h.$

It can be assumed that a similar result holds for other Chevalley groups for restricted elements $w\in W_p^{res,+}$ when the reduced expressions of all dominant elements of the Bruhat interval $[1,w]$ do not contain doubling generators. It may be that Lusztig’s character formula also holds for some other alcoves in the restricted region for all $p\ge h.$ These assumptions need further investigation.