Cells in Weyl group

By Kyo NISHIYAMA

Division of Mathematics, Faculty of Integrated Human Studies, Kyoto University, Kyoto 606, JAPAN

kyo@math.h.kyoto-u.ac.jp

Note for a talk at Strasbourg 1995/9/26

Revised for a talk at Tottori 1996/1/8

Revised after it

1996/1/12

Ver. 1.0 [00/11/25 12:04]

1 Introduction to cells

cells: \{ \begin{matrix}∗ & primitive ideals of U(𝔤) (𝔤 : semisimple Lie algebra) \\ ∗ & representation theory of finite Chevalley groups (unipotent representations) \\ ∗ & modular representation theory \\ & etc. \end{matrix}

Since I am not a specialist of finite Chevalley groups or modular representation theory, I will explain properties of cells through primitive ideals of U(𝔤).

Role in the theory of primitive ideals: 𝔤 : a semisimple Lie algebra /ℂ U(𝔤) : the enveloping algebra ⊃ ℨ : center 𝔤 ⊃ 𝔥 : CSA, W = W (𝔤, 𝔥) : Weyl group

ℨ →^{∼} U(𝔥)^{W} : Harish-Chandra isomorphism

For λ ∈ 𝔥^{∗}, ∃χ_{λ} : ℨ  ⟶  ℂ : algebra hom. ↘  ↗ evaluation at λ U(𝔥)^{W} = S(𝔥)^{W}

χ_{λ} : the central character corresponding to λ: Note that χ_{λ} = χ_{wλ} (w ∈ W).

L : an irreducible left U(𝔤)-module

Ann L = {X ∈ U(𝔤)| Xl = 0 (∀l ∈ L)} : primitive ideal^{1}

Ann L ∩ ℨ = ker χ_{λ} (∃λ ∈ 𝔥^{∗}) ⟺^{def.} Ann L ∈ Prim(λ) : primitive ideals with central character χ_{λ}

Prim U(𝔤) = ∐_{λ∈𝔥^{∗}/W} Prim(λ)

Example 1.1 Prim(ρ) ∋ Ann(trivial rep.)

(1) I ⊂ A : prime ideal ⇔ ( \begin{matrix}A/I ⊃ ∀J_{1}, J_{2} : ideals ≠ 0 \\ ⇒ J_{1} · J_{2} ≠ (0) \end{matrix} )

1There are several notions of primitivity in U(𝔤) (cf. Dixmier [Dixmier]).

I : maximal  I : completely prime

⇓  ⇓ I : primitive ⇒  I : prime  ⇒ I : semi-prime A ∋ 1 : ring (2) I ⊂ A : completely prime ⇔ A/I : integral domain (3) I ⊂ A : semi-prime ideal ⇔ ( \begin{matrix}A/I ⊃ J : nilpotent ideal \\ ⇒ J = (0) \end{matrix} )

By translation principle, Prim(ρ) ≃ Prim(ρ + λ) ∀λ ∈ P^{+}(Δ) Δ = Δ(𝔤, 𝔥) : roots P^{+}(Δ) : dominant integral weights

Parametrization of Prim(ρ)? w ∈ W

M_{w} = M(wρ − ρ) : Verma module with h.w. wρ − ρ

↓ : surjection

L_{w} = L(wρ − ρ) : its irreducible quotient

Theorem 1.2 (Duflo, 1977 [Duflo]) φ : W ∋ w ↦ Ann L_{w} ∈ Prim(ρ) is surjective.

Actually Duflo proved more. He proved the theorem for general central character χ_{λ}’s, which are possibly singular.

Classification of Prim(ρ) ⇔ determination of fibers of φ

⇔ {description of equivalence relation on W : w ∼ w′ ⟺^{def.} φ(w) = φ(w′)

This equivalence relation defines the left cells in W, i.e., Γ = Γ(w) = {w′| φ(w) = φ(w′)} is a left cell (Joseph and Vogan, 1980).^{2} Write ∼_{L} instead of ∼.

Example 1.3 type A_{n−1}. W ≃ 𝔖_{n} : symmetric group of order n

𝔖_{n} ∋ w ⟼ (T_{w}^{L}, T_{w}^{R}) : pair of standard tableaux of the same shape

Robinson-Schensted

w∼_{L}w′ ⇔ T_{w}^{L} = T_{w′}^{L} (Barbasch-Vogan, 1982 [BVI, p.171]).^{3}

𝔥^{∗} ∋ μ ↦ rk(U(𝔤)/ Ann L(μ − ρ)) =: p(μ) rk : Goldie rank^{4}. P′(Δ) : regular integral weights, P (Δ)^{++} = {μ ∈ P′(Δ)| μ : dominant}

Theorem 1.4 (Joseph, 1980 [JI, Cor. 5.12]) For w ∈ W, p(wμ) is a polynomial on P (Δ)^{++}. Denote this polynomial by p_{w}(μ) : Joseph’s Goldie rank polynomial.

Remark. Using the notion of “coherent family”, this theorem is more comprehensive. p_{w}(μ) coincides with the character polynomial (up to scalar multiple) of the coherent family containing L_{w}. (Joseph, 1980 [JII, Theorem 5.1]. Cf. King, 1981 [King].) p_{w}(μ) : a harmonic polynomial on 𝔥^{∗}, generates an irreducible W -module.

2This is the definition of left cell Γ here, but historically, the left cells are defined by Lusztig by using purely algebraic structures of Coxeter groups. The equivalence of two definitions are established by Joseph and Vogan. See Theorem 1.6.

3If ∼_{R} is similarly defined by using the right annihilators, then w∼_{R}w′ ⇔ T_{w}^{R} = T_{w′}^{R}.

For two-sided cells, w ∼_{LR}w′ ⇔ T_{w}^{L} and T_{w′}^{L} have the same shape ⇔ T_{w}^{R} and T_{w′}^{R} have the same shape .

4A : a prime ring (i.e., (0) is a primitive ideal ⇔ ∃ faithful irreducible representation)

Then Fract(A) ≃ Mat(n × n, D) for some division ring D. Define rk A := n.

Γ : left cell ∋ w, w′ ⇒ p_{w}(μ) = p_{w′}(μ) (by definition they take the same value on ρ + they are proportional. See [JI, Cor.5.12]) ⟹ Write p_{Γ}(μ) = p_{w}(μ) (w ∈ Γ).

∀σ ∈ W^{∨}

C_{σ} := ∐{Γ|ℂW p_{Γ} ≃ σ} ⊂ W

C = C_{σ} : two-sided cell in W (if C_{σ} ≠ ∅^{5}) equivalence relation w∼_{LR}w′ (⇔ w, w′ ∈ ∃C)

σ ∈ W^{∨} : special representation^{6} ⇔^{def.} C_{σ} ≠ ∅

dim σ = #{Γ|Γ ⊂ C_{σ} : left cell } ⇒ #Prim(ρ) = ∑_{σ∈W^{∨} : special} dim σ

Example 1.5 type A_{n−1}. ∀σ ∈ 𝔖_{n}^{∨} : special ↔ Y_{σ} : Young diagram

C_{σ} = {w ↔ (T_{w}^{L}, T_{w}^{R})| T_{w}^{L} has the shape Y_{σ}}
∪
Γ = {(T_{w}^{L}, T_{w}^{R})} : for the fixed standard tableau T_{w}^{L}

⟹ #{Γ| Γ ⊂ C_{σ}} = #{ standard tableau of the shape Y_{σ}} = dim σ

#Prim(ρ) = ∑_{σ∈𝔖_{n}^{∨}} dim σ = #{ involutions in 𝔖_{n}}
#{Γ ⊂ C} = #{ involutions ∈ C} ([Duflo, Proposition 9], cf. [Borho, §5.9])

Remark. The above left (or two-sided) cells can be defined by using only combinatoric properties of a Coxeter system (by Kazhdan-Lusztig [KL]). So they are defined for general Coxeter groups, not only for Weyl groups. See [BVII, Remark after Cor.2.16].

Char L_{w} = ∑_{w′∈W} a_{w,w′} Char M_{w′} (a_{w,w′} ∈ ℤ)

a_{w,w′} is explicitly determined (Kazhdan-Lusztig conjecture). But we do not need the explicit information.

a(w) := ∑_{w′∈W} a_{w,w′}w′ ∈ ℂW
D(w) := {w′| a(w′) ∈ [ℂW a(w)]} = {w′| Ann L_{w′} ⊃ Ann L_{w}}

The second equality is due to Vogan [Vogan].

5This is not empty iff σ is a special representation. See below.

6Special representations are originally introduced by Lusztig ([LI, LII]). However, two different notions coincide ([BVII, Theorem 2.29]).

Here, [S] means a-basal subspace^{7} in ℂW spanned by S.

Theorem 1.6 (Joseph [Joseph], Vogan [Vogan])

w′∼_{L}w ⟺ [ℂW a(w′)] = [ℂW a(w)]

For Γ : a left cell (w ∈ Γ), we put^{8}

σ_{Γ} := ∑_{w′∈D(w)}[ℂW a(w′)]/ ∑_{w′′∈D(w),w′′∉Γ}[ℂW a(w′′)] : left cell representation, not irreducible in general

Similarly put

C(w) = {w′| a(w′) ∈ [ℂW a(w)W]}

Theorem 1.7 ([BVII, Cor.2.16])

w∼_{LR}w′ ⟺ [ℂW a(w′)W] = [ℂW a(w)W]

For C : a two-sided cell (w ∈ C), we put^{9}

σ_{C} := ∑_{w′∈C(w)}[ℂW a(w′)W]/ ∑_{w′′∈C(w),w′′∉C}[ℂW a(w′′)W] : two-sided cell representation

As a W × W -module,

σ_{C} = \begin{matrix}∑ & ⊕ \\ τ \end{matrix} τ ⊗ τ (multiplicity free)

F_{C} := {τ ∈ W^{∨}| [σ_{C} : τ ⊗ τ] = 1} ⊂ W^{∨} : a family associated to C

F = F_{C} is also called two-sided cell^{10} in W^{∨}.

W^{∨} = ∐_{C : two-sided cell} F_{C}

∀F_{C} contains exactly one special representation, i.e., {special representations ∈ W^{∨}} gives a complete system of representatives for families.

7A subspace in ℂW is called a-basal if it has basis consisting of a(w)’s. See Joseph II 298p. Note that {a(w)| w ∈ W } forms a basis of ℂW. In the original version of this note, I misunderstood the definition of D(w); Tanisaki kindly pointed it out.

8The first summation in the right hand side of the next formula is surplus.

9Here, the summation is also surplus.

10Original definition of family is very complicated one ([Orange Book, §4.2]). However it coincides with the definition presented here (Barbasch-Vogan [BVII, Theorem 2.29], cf. [Orange Book, Theorem 5.25]).

2 Invariant for cells

We want to define an invariant for families {F}. The present observations are experimental ones^{11}.

Gyoja(1983) [Gyoja] : σ ∈ H_{q}(W)^{∨}, H_{q}(W) : an Iwahori-Hecke algebra^{12}

L(t, σ) = L(t, q, σ) := ∑_{w∈W} σ(w)t^{l(w)} (l(w) : length function)

L(t, σ) : {∃ functional equation distribution of zeros and poles^{13} are similar to zeta functions^{14}(congruence zeta functions on algebraic varieties).

⟹^{Iwahori’s suggestion} Take l′(w) instead of l(w);

l′(w) := \binom{minimal of m, where w = r_{1}r_{2} ⋯ r_{m},}{r_{i} : reflection (not necessarily simple)}

(Cf. for type A, #Prim(ρ) = #{involutions})

Remark. W ↷ V : natural representation, V (w) := {v ∈ V | wv = v} (w ∈ W). Then l′(w) = codim V (w).

Gyoja-N.-Shimura : σ ∈ W^{∨}, W : not an Iwahori-Hecke algebra, but a finite Weyl group.

c(σ, t) := ∑_{w∈W}(trace σ(w))t^{l′(w)}

Does c(σ, t) classify cells? : (c(σ, t) is computable!) For types A_{l}, B_{l}(= C_{l}), G_{2} : yes! i.e.,

σ_{1} ∼_{LR}σ_{2}^{ι} (ι ∈ Aut(W, S)) ⇔ c(σ_{1}, t) = c(σ_{2}, t)

(σ_{1} and σ_{2}^{ι} belong to the same family)

For Weyl groups of the other types, some deviation occurs.

11M. Sato: Every mathematician should be an experimentalist.

12For a long time, H_{q}(W) is simply called Hecke algebra. Recently, it becomes to be called Iwahori-Hecke algebra. For this, see Introduction of [Orange Book].

14In case of affine Weyl groups, poles appear. However, in this note, Weyl groups always remain finite.

14Functional equation:

L(t, σ) = σ(T_{w_{0}})t^{l(w_{0})}L((−qt)^{−1}, σ̂), w_{0} : the longest element, σ̂(T_{w}) = (−q)^{l(w)} (T_{w^{−1}})^{−1} .

zeros:

t = ζq^{−i/2m}, ζ : root of unity, i, m ∈ ℤ s.t. 1 ≤ m ≤ l(w_{0}), 0 ≤ i/2m ≤ 1.

Want to avoid the deviation. W ↷ V : natural representation, {m_{i}} : exponents, H(V) = ⊕_{n}H^{n}(V) : harmonic polynomials σ ∈ W^{∨}, χ = Char σ

τ̃(q, y)(w) := \frac{det(1 + yw|V)}{det(1 − qw|V)},
τ̃(σ; q, y) := ⟨χ, τ̃(q, y)⟩_{W}
= \frac{1}{#W} ∑_{w∈W} χ(w^{−1})τ̃(q, y)(w)
= ∑_{i,j}⟨S^{i}(V) ⊗ ∧^{j}(V), χ⟩_{W} q^{i}y^{j}

= \frac{dim σ}{∏_{i=1}^{l}(1 − q^{m_{i}+1})} ∑_{n,m≥0} [χ : H^{n}(V) ⊗ ⋀^{m}(V)] q^{n}y^{m},

Proposition 2.1 l := dim V : rank of W

\frac{#W}{dim σ} lim_{q→1} τ̃(σ; q, −1 + t(1 − q)) = t^{l}c(σ; t^{−1})

⟹ τ̃(σ; q, y) refines the property of c(σ, t).

From now on, W is assumed to be irreducible.

Observation 2.2 τ̃(σ; q, y) has many factors of the same type (1 + q^{c}y) (c ∈ ℤ).

Example 2.3 (Or a proposition)

#(family) = 1 ⇒ τ̃(σ; q, y) = f(q) ∏_{i=1}^{rank}(1 + q^{c_{i}}y)

τ̃(trivial; q, y) = ∏_{i=1}^{l} \frac{1 + yq^{m_{i}}}{1 − q^{m_{i}+1}}

Example 2.4 (See §3) type A_{l−1} : W ≃ 𝔖_{l}

τ̃(χ^{λ}; q, y) = q^{n(λ)} ∏_{x∈λ} \frac{1 + q^{c(x)}y}{1 − q^{h(x)}} λ : partition and tableau

notations: (cf. Macdonald’s book [M])

n(λ) = ∑(i − 1)λ_{i}, x = (i, j) ∈ λ ∈ ℤ^{2}, c(x) = j − i, h(x) = hook length

\begin{matrix}λ : \\ c(x) : & 0 1 2 3 4 ; −1 0 1 2 ; −2 −1 ; −3 , & h(x) : & 8 6 4 3 1 6 4 2 1 3 1 1 \end{matrix}

We put τ_{1}(σ; q, y) = ∏_{i=1}^{κ(σ)}(1+q^{c_{i}}y) : the largest factor of τ̃(σ; q, y) including only the factors

(1 + q^{c}y) (c ∈ ℤ).

Theorem 2.5 If

(1) W is of type A_{l}, B_{l}(= C_{l}), G_{2} or

(2) W is of type D_{l}, F_{4}, E_{l} (l = 6, 7, 8) and #(family) ≤ 3, then

σ_{1} ∼_{LR}σ_{2}^{ι} (ι ∈ Aut(W, S)) ⇔ τ_{1}(σ_{1}; q, y) = τ_{1}(σ_{2}; q, y)

i.e., σ_{1} and σ_{2}^{ι} belong to the same family.

Remark. Even for the case #(family) ≥ 5, the above theorem is almost true. We can attach linear factors like τ_{1}(σ; q, y) for each σ ∈ W^{∨}, but it is not the largest linear factor (i.e., we must choose “special” linear factors).

Remark. If #(family) = 3,

τ̃(σ_{1}; q, y) + τ̃(σ_{2}; q, y) = f(q) ∏_{i}(1 + q^{d_{i}}y) (d_{i} ∈ ℤ)

for any two representations σ_{1}, σ_{2} in the family. Similar phenomena occur for the case of type D_{l} (and #(family) ≥ 5). The set of (computable) integers

(c_{1}, c_{2}, ⋯, c_{κ(σ)})

determines a family (or two-sided cell) in W^{∨} (for W in the theorem). If we overcome the deviation explained above (i.e., the case of #(family) ≥ 5), the computable invariants (c_{i}) really classifies families (or two-sided cells) in W^{∨} (even for nonirreducible W ’s).

Problem 2.6 (1) Find a method to choose special linear factors for exceptions of the above theorem.

(2) The above consideration is an experimental one. Clarify the meaning of the invariants (c_{i}) from the view points of the representation theory of Weyl groups or Iwahori-Hecke algebras, theory of primitive ideals, finite Chevalley groups, modular representation theory . . ..

3 Calculation of τ̃ for type A

This section is almost borrowed from Macdonald’s book: [M].

First, let us introduce some general theory from [M, §I.2 and §I.3]. In the following, t is an indeterminate and h_{r}, e_{r}, p_{r} are coefficients;

H(t) = ∑_{r≥0} h_{r}t^{r} (h_{0} = 1)
E(t) := 1/H(−t) = ∑_{r≥0} e_{r}t^{r}

P (t) := \frac{d}{dt} log H(t) = H′(t)/H(t) = ∑_{r≥0} p_{r}t^{r}

For a partition λ = (λ_{1}, λ_{2}, ⋯), put

h_{λ} = ∏_{i≥1} h_{i}^{λ_{i}}, e_{λ} = ∏_{i≥1} e_{i}^{λ_{i}}, p_{λ} = ∏_{i≥1} p_{i}^{λ_{i}}.

Theorem 3.1

h_{r} = ∑_{|λ|=r} \frac{1}{z_{λ}} p_{λ}, e_{r} = ∑_{|λ|=r} \frac{ε_{λ}}{z_{λ}} p_{λ}

Here, we used the following notation;

z_{λ} := ∏_{i≥1} i^{m_{i}}m_{i}! (λ = (1^{m_{1}} · 2^{m_{2}} · 3^{m_{3}} ⋯) i.e., m_{i} = #{j| λ_{j} = i})

= \frac{#(conjugacy class in 𝔖_{n} corresponding to λ)}{n!} (n = |λ|)
ε_{λ} := (−1)^{|λ|−l(λ)}

Theorem 3.2 Put

s_{λ} := det(h_{λ_{i}−i+j})_{1≤i,j≤n} (n ≥ l(λ))
= ∑_{w∈𝔖_{n}} ε(w)h_{λ+δ−wδ}
= det(e_{λ′_{i}−i+j})_{1≤i,j≤m} (m ≥ l(λ′))

Now put n := |λ|(≥ l(λ)) and take the character χ^{λ} of 𝔖_{n} corresponding to λ. Then we have

s_{λ} = ∑_{μ} \frac{1}{z_{μ}} χ^{λ}(μ)p_{μ},

where μ runs over all the partitions of n (i.e., conjugacy classes of 𝔖_{n}).

Example 3.3 ([M, §I.2 and §I.3]) Let {x_{i}} be the set of indeterminates.

H(t) := ∏_{i=1}^{n} \frac{1}{1 − x_{i}t}

⇒ \{ \begin{matrix}h_{r} & : complete symmetric function of degree r \\ e_{r} & : elementary symmetric function of degree r \\ p_{r} & = ∑_{i=1}^{n} x_{i}^{r} \end{matrix}

s_{λ} = \frac{a_{λ+δ}}{a_{δ}} : Schur function
= \frac{∑_{w∈𝔖_{n}} ε(w)w(x^{λ+δ})}{∑_{w∈𝔖_{n}} ε(w)w(x^{δ})} (δ = (n − 1, n − 2, ⋯, 0))

Example 3.4 ([M, §I.2 Ex.3 and §I.3 Ex.1]) Let q be an indeterminates.

H(t) := ∏_{i=0}^{n−1} \frac{1}{1 − q^{i}t}

⇒ h_{r} = [\begin{matrix}n + r − 1 \\ r \end{matrix}] , e_{r} = q^{r(r−1)/2} [\begin{matrix}n \\ r \end{matrix}] , p_{r} = \frac{1 − q^{nr}}{1 − q^{r}}

[\begin{matrix}n \\ r \end{matrix}] := \frac{∏_{j=1}^{r}(1 − q^{n−j+1})}{∏_{i=1}^{r}(1 − q^{i})} : q-binomial coefficient

s_{λ} = q^{n(λ)} ∏_{x∈λ} \frac{1 − q^{n+c(x)}}{1 − q^{h(x)}} \{ \begin{matrix}c(x) & : content \\ h(x) & : hook length \\ n(λ) & = ∑_{i≥1}(i − 1)λ_{i} \end{matrix} (See Example 2.4)

Example 3.5 ([M, §I.2 Ex.4 and §I.3 Ex.2]) Let n → ∞ in Example 3.4.

H(t) := ∏_{i≥0} \frac{1}{1 − q^{i}t}

⇒ h_{r} = ∏_{i=1}^{r} \frac{1}{1 − q^{i}} , e_{r} = q^{r(r−1)/2}h_{r}, p_{r} = \frac{1}{1 − q^{r}}

s_{λ} = q^{n(λ)} ∏_{x∈λ} \frac{1}{1 − q^{h(x)}}

Example 3.6 ([M, §I.2 Ex.5 and §I.3 Ex.3] and [Andrews, Chap.II]) Let a and b be indeterminates.

H(t) := ∏_{i≥0} \frac{1 − bq^{i}t}{1 − aq^{i}t}

⇒ h_{r} = ∏_{i=1}^{r} \frac{a − bq^{i−1}}{1 − q^{i}} , e_{r} = ∏_{i=1}^{r} \frac{aq^{i−1} − b}{1 − q^{i}}, p_{r} = \frac{a^{r} − b^{r}}{1 − q^{r}}

s_{λ} = q^{n(λ)} ∏_{x∈λ} \frac{a − bq^{c(x)}}{1 − q^{h(x)}}

Proof for the last statement on s_{λ}: Substitute t → t/a and we have s_{λ}/a^{|λ|} instead of s_{λ}. So we can assume a = 1. Next, note that the both hand sides of the equation to be proved are polynomials in b. If b = q^{n} (n = 1, 2, ⋯), it reduces to Example 3.5, hence the equation is valid for infinitely many value of b. We are done.

In the above Example 3.6, substitute a = 1 and b = −y. For σ = (1, 2, ⋯, r) ∈ 𝔖_{r} : cyclic permutation, we have

det(1 − qσ|ℂ^{r}) = 1 − q^{r}.

⟹ p_{r} = \frac{1 − (−y)^{r}}{1 − q^{r}} = \frac{det(1 + yσ|ℂ^{r})}{det(1 − qσ|ℂ^{r})}

⟹ For w ∈ 𝔖_{n} with cyclic type μ;

p_{μ} = \frac{det(1 + yw|ℂ^{n})}{det(1 − qw|ℂ^{n})} (= τ̃(q, y)(w))

s_{λ} = q^{n(λ)} ∏_{x∈λ} \frac{1 + yq^{c(x)}}{1 − q^{h(x)}} (by Example 3.6)
= ∑_{μ} \frac{1}{z_{μ}} χ^{λ}(μ)p_{μ} (by Theorem 3.2)

= \frac{1}{n!} ∑_{w∈𝔖_{n}} χ^{λ}(w)p_{μ(w)} (μ(w) : cyclic type of w)
= \frac{1}{n!} ∑_{w∈𝔖_{n}} χ^{λ}(w)τ̃(q, y)(w) (by the above)
= ⟨χ, τ̃(q, y)⟩_{𝔖_{n}} = τ̃(χ; q, y)

4 Comments and references

Mathiew : H := ℂ^{l}\{z_{i} = z_{j}} cohomology H^{i}(H) : dim H^{i}(H) = #{w ∈ 𝔖_{l}| l′(w) = i} Is there any relation? (Cf. Brieskorn [Brieskorn] and Orlik [Orlik].)

Schiffmann : Is there a formula for κ(σ)?

τ_{1}(σ; q, y) = ∏_{i=1}^{κ(σ)}(1 + q^{c_{i}}y)

partial answer:

type A_{l−1} :  κ(σ) = l
type E_{l} & #(family) = 3 : κ(σ) = l − 2
etc.

Tanisaki : How about for Hecke algebra H_{q}(W)?

L′(σ; t) := ∑_{w∈W} σ(T_{w})t^{l′(w)}

Oshima : l′(w) can be expressed as a linear sum of characters. Is there a closed formula indicating this fact? partial answer:

c(χ; t) := ∑_{w∈W} χ(w)t^{l′(w)} ⇒ \frac{d}{dt}c(χ; t)|_{t=1} = #W · ⟨χ, l′⟩_{W}

Since l′(w) = ∑_{χ}⟨χ, l′⟩_{W} χ(w), we have

l′(w) = \frac{1}{#W} ∑_{χ} ( \frac{d}{dt}c(χ; t)|_{t=1}) χ(w) = \frac{d}{dt} (\frac{1}{#W} ∑_{χ} c(χ; t)χ(w))|_{t=1} .

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