CUSPIDAL MODULES AS SUMMANDS OF A GELFAND-GRAEV MODULE

I. TULUNAY

Abstract. Let G = GL_{n}(q) be the general linear group over a finite field 𝔽_{q} with q elements. We call a Gel’fand-Graev module to be the module which affords the Gel’fand-Graev character defined in 1.1. It is known that every cuspidal module of G is isomorphic to a (unique) direct summand of a Gel’fand-Graev module. In this article, we investigate a certain endomorphism so that each irreducible cuspidal module is contained in a certain eigenspace corresponding to the cuspidal character. Furthermore, we determine the eigenvalue of that endomorphism by using character theory of finite general linear group.

1. PRELIMINARIES

Let G = GL_{n}(q), the general linear group of degree n over a finite field 𝔽_{q}. Let B be the group of upper triangular matrices (a Borel subgroup) and H the diagonal matrices (torus). For each i ∈ {1, 2, …, n−1}, define s_{i} be the permutation matrix corresponding to the transposition (i i + 1). Let S = {s_{i} | 1 ⩽ i ⩽ n − 1}. Then W = ⟨S⟩ is the Weyl group and isomorphic to the symmetric group on {1, 2, …, n}. We shall frequently identify elements of W with the corresponding permutations.

Identify the root system of W by

Φ = { α_{i j} | i ≠ j , 1 ⩽ i, j ⩽ n }.

where α_{i j} is the i-th minus the j-th ortonormal basis vectors of an n - dimensional Euclidean space.

The subgroup U = { (a_{ij}) ∈ B | a_{ii} = 1 } (upper unitriangular matrices) is a maximal unipotent subgroup and Sylow p-subgroup of G. We have B = U ⋊ H. For each α = α_{ij} ∈ Φ, define

x_{α}(t) = x_{ij}(t) = I_{n} + te_{ij}

for any t ∈ 𝔽_{q}, where I_{n} is the identity matrix. The group X_{α} = X_{ij} = { x_{ij}(t) | t ∈ 𝔽_{q} } is the root subgroup corresponding to α. For any w ∈ W,

(1)  wx_{ij}(t)w^{−1} = x_{w(i),w(j)}(t).

Date: October 2002.

1991 Mathematics Subject Classification. [2000] Primary 20C33.

This material in this paper forms a part of the author’s doctoral thesis at the University of Sydney. I thank my supervisor R. B. Howlett for his invaluable assistance.

Furthermore, U = ∏_{1⩽i<j⩽n} X_{ij}.

Througout this paper, let θ : 𝔽_{q}^{+} ⟶ ℂ^{∗} be a fixed nontrivial homomorphism.

Definition 1.1. Define a linear character Ψ : U ⟶ ℂ^{∗} by

Ψ((u_{ij})) = θ (n−1 ; ∑_{i=1}u_{i i+1})

The linear character Ψ is in general position in the sense that the restriction of Ψ to each fundamental root subgroup X_{i i+1} is nontrivial. The induced character Ψ^{G} is called the Gel’fand-Graev character of G.

Define an idempotent e in ℂG by

(2)  e := \frac{1}{|U|} ∑_{u∈U} Ψ(u^{−1})u.

Then the left ideal ℂGe of ℂG affords the Gel’fand-Graev representation.

Let M be an irreducible cuspidal module of G. Then M has a unique (up to scalars) vector b such that

u.b = Ψ(u)b

for all u ∈ U. Thus ⟨b⟩ is the unique ℂU-submodule of Res \begin{matrix}G \\ U \end{matrix} M affording Ψ (See [7]). Since eb = b, the left ideal ℂGe of ℂG (which affords the Gel’fand-Graev representation) has a (unique) direct summand isomorphic to M. Identify M with this summand, that is M = ℂGb. Then, up to a scalar, b is an idempotent that commutes with e.

2. MAIN RESULT

Hereafter let 𝔽_{q^{n}}, (n ⩾ 1) be an extension of degree n of 𝔽_{q}. Consider a multiplicative character χ of 𝔽_{q^{n}} satisfying the following condition: Condition C: For all d ∈ ℤ^{+} such that n is divisible by d and d ≠ n, χ^{q^{d}−1} is not identically equal to unity. Such a multiplicative character χ defines an irreducible cuspidal character ξ_{χ} of G (See [8]). If M is an irreducible cuspidal module affording ξ_{χ} then the elements zI of the center Z of G act on M via the restriction of χ to 𝔽_{q}^{∗}; that is

(zI)m = χ(z)m

for all z ∈ 𝔽_{q}^{∗} and m ∈ M. From now on fix such a multiplicative character χ of 𝔽_{q^{n}}.

Let

Λ(χ) := \frac{(−1)^{n+1}}{q^{n−1}} ∑_{\begin{matrix}λ∈𝔽_{q^{n}} \\ N(λ)=(−1)^{n+1} \end{matrix}} θ(Tr(λ))χ(λ^{−1}),

where N and Tr are the usual norm and trace maps of 𝔽_{q^{n}}.

Define a ℂG-endomorphism f of ℂGe by

(3)  f(α) = α(es_{n−1}s_{n−2} … s_{1}e)

for all α ∈ ℂGe (right multiplication by es_{n−1}s_{n−2} … s_{1}e).

The main result of this paper is as follows.

Theorem 2.1. With above notation, we have

es_{n−1}s_{n−2} … s_{1}b = Λ(χ)b.

Furthermore an irreducible cuspidal module ℂGb corresponding the multiplicative character χ of 𝔽_{q^{n}} satisfying Condition C given above is contained in the Λ(χ)– eigenspace of the endomorphism f defined as in (3) of ℂGe.

The rest of the paper is devoted to proving this theorem. The proof is done in two stages. The first stage is to prove the following lemma.

Lemma 2.2.

es_{n−1}s_{n−2} … s_{1}b = \frac{κ(1)}{q^{n−2}} b

where κ(1) is a certain constant in complex number field.

The second stage is to prove, by using the character theory of G, that

\frac{κ(1)}{q^{n−2}} = Λ(χ).

3. PROOF OF LEMMA 2.2

For each root α ∈ Φ, we define

\overline{X}_{α} := \frac{1}{q} ∑_{a∈𝔽_{q}} x_{α}(a),
X̃_{α} := \frac{1}{q} ∑_{a∈𝔽_{q}} θ(−a)x_{α}(a).

Thus \overline{X}_{α} and X̃_{α} are respectively the idempotents of the trivial character and θ. Furthermore,

(4)  e = ∏_{\begin{matrix}1⩽i⩽n−2 \\ 3⩽j⩽n \\ i+1<j \end{matrix}} \overline{X}_{ij} ∏_{1⩽k⩽n−1} X̃_{k k+1}.

The following results are trivial and so we omit the proofs.

Lemma 3.1. For each α ∈ Φ,

(i) X̃_{α}X̃_{α} = X̃_{α} and \overline{X}_{α}\overline{X}_{α} = \overline{X}_{α}. (ii) \overline{X}_{α}X̃_{α} = X̃_{α}\overline{X}_{α} = 0.

(iii)

\overline{X}_{α}b = \begin{cases}b & if α ∈ Φ^{+} \ Π, \\ 0 & if α ∈ Π, \end{cases}

X̃_{α}b = \begin{cases}0 & if α ∈ Φ^{+} \ Π, \\ b & if α ∈ Π. \end{cases}

Lemma 3.2.

es_{n−1}s_{n−2} … s_{1}b = X̃_{n−1 n} ∏_{1⩽i⩽n−2} \overline{X}_{in}s_{n−1}s_{n−2} … s_{1}b.

Proof. Observe that for all i, j, k ∈ {1, 2, …, n} with 1 ⩽ i ⩽ n − 3, 3 ⩽ j ⩽ n − 1, i + 1 < j, 1 ⩽ k ⩽ n − 2,

s_{1}s_{2} … s_{n−1}(i j) = (i + 1 j + 1)
s_{1}s_{2} … s_{n−1}(k k + 1) = (k + 1 k + 2).

Hence, by (4), (1) and Lemma 3.1(iii),

es_{n−1}s_{n−2} … s_{1}b = ∏_{\begin{matrix}1⩽i⩽n−2 \\ 3⩽j⩽n \\ i+1<j \end{matrix}} \overline{X}_{ij} ∏_{1⩽k⩽n−1} X̃_{k k+1}s_{n−1}s_{n−2} … s_{1}b

= (X̃_{n−1 n} ∏_{1⩽i⩽n−2} \overline{X}_{in}s_{n−1}s_{n−2} … s_{1}) ∏_{\begin{matrix}1⩽i⩽n−3 \\ 3⩽j⩽n−1 \\ i+1<j \end{matrix}} \overline{X}_{i+1 j+1} ∏_{1⩽k⩽n−2} X̃_{k+1 k+2}b

= X̃_{n−1 n} ∏_{1⩽i⩽n−2} \overline{X}_{in}s_{n−1}s_{n−2} … s_{1}b.

□

For each w ∈ W, we define l(w) to be the length of an expression of w as a product of generators s ∈ S. Let w_{J} be the longest element of W_{J} = ⟨J⟩ for each J ⊆ S. In particular, w_{S} is the longest element of W.

For each w ∈ W, we define the linear character Ψ^{w} : U^{w} ⟶ ℂ by Ψ^{w}(u) = Ψ(wuw^{−1}) for all u ∈ U^{w} = w^{−1}Uw. Let Ψ_{w} be the restriction of Ψ^{w} to the subgroup U_{w_{S}w} = U ∩ U^{w} of U^{w}. So for all α ∈ U_{w_{S}w} = ∏_{α∈Pos(w)} X_{α} and t ∈ 𝔽_{q},

(5)  Ψ_{w}(x_{α}(t)) = \begin{cases}1 & if w(α) ∈ Φ^{+} \ Π \\ θ(t) & if w(α) ∈ Π, \end{cases}

by (1).

Note that If w = w_{S}s_{α} where α ∈ Π, then U ∩ U^{w} = X_{α} and Ψ_{w} is the linear character corresponding to the idempotent X̃_{α}.

Definition 3.3. For each w ∈ W, define

e_{w} := \frac{1}{|U_{w_{S}w}|} ∑_{u∈U_{w_{S} w}} Ψ_{w}(u^{−1})u.

Lemma 3.4. For all w ∈ W and u ∈ U_{w_{S}w}, we have

ue_{w} = e_{w}u = Ψ_{w}(u)e_{w}.

The proof is obvious.

For each w ∈ W, let

Neg(w) := {α ∈ Φ^{+} | w(α) ∈ Φ^{−} }
Pos(w) := {α ∈ Φ^{+} | w(α) ∈ Φ^{+} }.

Thus Pos(w) = Φ^{+} \ Neg(w) and Neg(w) = Pos(w_{S}w).

For each w ∈ W, we define

S(w) = { α ∈ Φ^{+} | w(α) ∈ Π },

the set of positive roots made simple by w, and

S′(w) = Pos(w) \ S(w).

Clearly, Pos(w) = S(w) ∪ S′(w). It is a well-known fact that U_{w_{S}w} = U ∩ U^{w} = ∏_{α∈Pos(w)} X_{α} (see Proposition 2.5.16, [3]). It follows from (5) that for each w ∈ W, the idempotent e_{w} can be written as

(6)  ∏_{\begin{matrix}α∈S′(w) \\ β∈S(w) \end{matrix}} \overline{X}_{α}X̃_{β}

All finite groups of Lie type satisfy the following commutator relation (see [3]). Let α, β be linearly independent roots. Then

(7)  [ X_{α}, X_{β} ] ⊆ ∏ X_{iα+jβ}

where the product is taken (in some order) over the set of all roots which are of the form iα + jβ, i, j > 0.

The Commutator Formula (7) shows that

[X_{γ}, X_{β}] ⊆ ∏_{α∈S′(w)} X_{α}

for each β, γ ∈ Pos(w). Consequently, the positions of X̃_{β} factors in the above expression can be varied arbitrarily. Hence by Lemma 3.1,

(8)  e_{w}\overline{X}_{α} = \overline{X}_{α}e_{w} = e_{w}
(9)  e_{w}X̃_{β} = X̃_{β}e_{w} = e_{w}

for all α ∈ S′(w) and β ∈ S(w).

Lemma 3.5. For each w ∈ W,

e_{w^{−1}}wb = wb.

The proof is straightforward and so is omitted.

The following fact is useful in proving Lemma 3.7.

Lemma 3.6. Let w ∈ W. For any sequence α_{1}, α_{2}, …, α_{k} of distinct positive roots,

U^{w} ∩ (X_{α_{1}}X_{α_{2}} … X_{α_{k}}) = X_{β_{1}}X_{β_{2}} … X_{β_{m}}

where β_{1}, β_{2}, …, β_{m} is the subsequence of α_{1}, α_{2}, …, α_{k} consisting of those α_{i} such that w(α_{i}) is positive.

The proof is omitted.

Lemma 3.7. Let w_{1}, w_{2} ∈ W. For any w ∈ W, e_{w_{1}}we_{w_{2}} ≠ 0 if and only if

α ∈ S(w_{1}) and w^{−1}(α) ∈ S(w_{2})

for all α ∈ Pos(w_{1}) ∩ Pos(w^{−1}) ∩ Pos(w_{2}w^{−1}). Equivalently, e_{w_{1}}we_{w_{2}} ≠ 0 if and only if

β ∈ S(w_{2}) and w(β) ∈ S(w_{1})

for all β ∈ Pos(w_{2}) ∩ Pos(w) ∩ Pos(w_{1}w)

Proof. Observe that

e_{w_{1}}we_{w_{2}} ≠ 0 if and only if Ψ_{w_{1}}(u) = Ψ_{w_{2}}(w^{−1}uw)

for all u ∈ U_{w_{S}w_{1}} ∩ wU_{w_{S}w_{2}}w^{−1}. By Lemma 3.6, we have

U_{w_{S}w_{1}} ∩ wU_{w_{S}w_{2}}w^{−1} = ∏_{α∈Pos(w_{1})∩Pos(w^{−1})∩Pos(w_{2}w^{−1})} X_{α}.

Hence e_{w_{1}}we_{w_{2}} ≠ 0 if and only if

(10)  Ψ_{w_{1}}(x_{α}(t)) = Ψ_{w_{2}}(w^{−1}x_{α}(t)w)  for all t ∈ 𝔽_{q}

for all α ∈ Pos(w_{1}) ∩ Pos(w^{−1}) ∩ Pos(w_{2}w^{−1}). By (5), if α ∈ Pos(w_{1}) ∩ Pos(w^{−1}) ∩ Pos(w_{2}w^{−1}) then (10) becomes

Ψ(x_{w_{1}(α)}(t)) = Ψ(x_{w_{2}w^{−1}(α)}(t)) for all t ∈ 𝔽_{q},

and this holds if and only if w_{1}(α) and w_{2}w^{−1}(α) are both in Π or both in Φ^{+} \ Π. Thus e_{w_{1}}we_{w_{2}} ≠ 0 if and only if the conditions α ∈ S(w_{1}) and w^{−1}(α) ∈ S(w_{2}) are equivalent for α ∈ Pos(w_{1}) ∩ Pos(w^{−1}) ∩ Pos(w_{2}w^{−1}).

The equivalent statement can be proved in a similar way. □

Corollary 3.8. Let w_{1}, w_{2} ∈ W. Then for any w ∈ W, e_{w_{1}}we_{w_{2}} = 0 if there exists α ∈ S(w_{1}) such that

w^{−1}(α) ∈ S′(w_{2})

or if there exists β ∈ S(w_{2}) such that

w(β) ∈ S′(w_{1}).

This is straightforward from Lemma 3.7.

Lemma 3.9. Let w_{1}, w_{2}, w ∈ W and h ∈ H. Then e_{w_{1}}hwe_{w_{2}} ≠ 0 if and only if e_{w_{1}}we_{w_{2}} ≠ 0 and h centralizes the root subgroup X_{β} for all β ∈ w(S(w_{2})) ∩ S(w_{1}).

This lemma can be proved by using the technique introduced in the proof of Lemma 3.7.

Observe that

(11)  Neg(s_{1}) = { α_{12} }
(12)  Neg(s_{n−1}) = { α_{n−1 n} }.
(13)  S(s_{1}) = {α_{13}, α_{34}, …, α_{n−1 n}},
(14)  S(s_{n−1}) = {α_{12}, α_{23}, …, α_{n−3 n−2}α_{n−2 n}},

Let Ω = {s_{i} | 2 ⩽ i ⩽ n − 2} and W_{Ω} = ⟨Ω⟩.

Lemma 3.10, Lemma 3.11, Lemma 3.12 and Lemma 3.13 are essentially proved for more general cases in [12].

Lemma 3.10. The set

{ w ∈ W_{Ω} | e_{s_{1}}we_{s_{n−1}} ≠ 0 }

consists of only s_{2}s_{3} … s_{n−2}.

Proof. Suppose that w ∈ W_{Ω} with e_{σ_{k}^{−1}}we_{(σ′_{l})^{−1}} ≠ 0. Note that w(1) = 1 and w(n) = n, since w ∈ W_{Ω}. Clearly,

P ((\begin{matrix}1 & 2 & 3 & … & n − 2 & n − 1 & n \\ 1 & 3 & 4 & … & n − 1 & 2 & n \end{matrix})) = s_{2}s_{3} … s_{n−2}.

We use induction on j to show that

w(j) = 1 + j for all j ∈ {2, 3, …, n − 2},.

By (14), α_{1 3} ∈ S(s_{1}). Then

w^{−1}(α_{1 3}) = α_{1 w^{−1}(3)} ∈ Φ^{+},

and by (12), we see that α_{1 w^{−1}(3)} ∉ Neg(s_{n−1}). So we must have α_{1 w^{−1}(3)} ∈ Pos(s_{n−1}). Lemma 3.7 gives that α_{1 w^{−1}(3)} ∈ S(s_{n−1}). By (14), w^{−1}(3) = 2 and so w(2) = 3.

Suppose now that j ∈ {3, 4, …, n − 2} and that w(m) = 1 + m for all m ∈ {2, 3, …, j − 1} (induction hypothesis). In particular, j = w(j − 1) ⩽ n − 1 since w(n) = n. By (13), α_{j j+1} ∈ S(s_{1}). Now

w^{−1}(α_{j j+1}) = α_{w^{−1}(j) w^{−1}(j+1)} = α_{j−1 w^{−1}(j+1)},

and this must be a positive root since w^{−1}(j + 1) ≠ w^{−1}(1) = 1 and

w^{−1}(j + 1) ∉ {w^{−1}(3), w^{−1}(4), …, w^{−1}(j − 1)} = {2, 3, …, j − 2}.

By (12), α_{j−1 w^{−1}(j+1)} ∉ Neg(s_{n−1}) as j−1 < n−3. So we must have α_{j−1 w^{−1}(j+1)} ∈ Pos(s_{n−1}). Lemma 3.7 gives that α_{j−1 w^{−1}(j+1)} ∈ S(s_{n−1}). Since j − 1 ⩽ n − 3, it follows from (14) that w^{−1}(j + 1) = j, completing the induction. □

Write w_{1} = s_{2}s_{3} … s_{n−2}.

For each k ∈ ℤ^{+} with 0 ⩽ k ⩽ n, and each x ∈ 𝔽_{q}^{∗}, let d_{k}(x) = (h_{ij}) ∈ H, where

h_{ij} = \begin{cases}1 & if i = j ≠ k \\ x & if i = j = k \end{cases}

Let D_{k} = { d_{k}(x) ∈ H | x ∈ 𝔽_{q}^{∗} }.

Lemma 3.11. With above notation, we have

{ h ∈ D_{2}D_{3} … D_{n} | e_{s_{1}}hw_{1}e_{s_{n−1}} ≠ 0 } = D_{2}.

Proof. Lemma 3.9 gives that e_{s_{1}}hw_{1}e_{s_{n−1}} ≠ 0 if and only if h centralizes X_{α} for all α ∈ S(s_{1}) ∩ w_{1}(S(s_{n−1})). By (14) and Lemma 3.10, we have

S(s_{1}) ∩ w_{1}(S(s_{n−1})) = {α_{13}, α_{34}, …, α_{n−1 n} }

Let h = d_{2}(y_{2})d_{3}(y_{3}) … d_{n}(y_{n}) ∈ D_{2}D_{3} … D_{n} where y_{i} ∈ 𝔽_{q}^{∗} for 2 ⩽ i ⩽ n. For any t ∈ 𝔽_{q}, h^{−1}x_{α_{1 3}}(t)h = x_{α_{1 3}}(t) if and only if y_{3} = 1. For 3 ⩽ i ⩽ n − 1,

h^{−1}x_{α_{i i+1}}(t)h = x_{α_{i i+1}}(t)

if and only if y_{i} = y_{i+1}, as required. □

Lemma 3.12. For any h ∈ D_{2},

e_{s_{1}}hw_{1}s_{n−1}b = ∏_{\begin{matrix}α∈S′(s_{1})∩Neg(s_{n−1}w_{1}^{−1}) \\ β∈S(s_{1})∩Neg(s_{n−1}w_{1}^{−1}) \end{matrix}} \overline{X}_{α}X̃_{β}hw_{1}s_{n−1}b.

Proof. By (6), we have

e_{s_{1}}hw_{1}s_{n−1}b = ∏_{\begin{matrix}α∈S′(s_{1}) \\ β∈S(s_{1}) \end{matrix}} \overline{X}_{α}X̃_{β}hw_{1}s_{n−1}b.

As w^{−1}X_{α}w = X_{w^{−1}(α)} for all w ∈ W and all α ∈ Φ,

e_{s_{1}}hw_{1}s_{n−1}b = ∏_{α, β, γ, μ} \overline{X}_{α}X̃_{β} \overline{X}_{γ} X̃_{μ} hw_{1}s_{n−1}b

= ∏_{α, β, μ} \overline{X}_{α}X̃_{β}X̃_{μ} hw_{1} ∏_{γ} \overline{X}_{s_{n−1}w_{1}^{−1}(γ)} b

where

α ∈ S′(s_{1}) ∩ Neg(s_{n−1}w_{1}^{−1})
β ∈ S(s_{1}) ∩ Neg(s_{n−1}w_{1}^{−1})
γ ∈ S′(s_{1}) ∩ Pos(s_{n−1}w_{1}^{−1})
μ ∈ S(s_{1}) ∩ Pos(s_{n−1}w_{1}^{−1}).

By Lemma 3.10, e_{s_{1}}w_{1}e_{s_{n−1}} ≠ 0, and so, by Lemma 3.7 if γ ∈ S′(s_{1})∩Pos(s_{n−1}w_{1}^{−1}) then w_{1}^{−1}(γ) ∈ S′(s_{n−1}). So

\overline{X}_{s_{n−1}w_{1}^{−1}(γ)}b = b (by Lemma 3.1 (iii)).

Similarly, by Lemma 3.7, if μ ∈ S(s_{1})∩Pos(s_{n−1}w_{1}^{−1}) then w_{1}^{−1}(μ) ∈ S(s_{n−1}), and, by Lemma 3.11, X̃_{μ}h = hX̃_{μ}. Thus we have

e_{s_{1}}hw_{1}s_{n−1}b = ∏_{α, β} \overline{X}_{α}X̃_{β}hw_{1}s_{n−1} ∏_{μ} X̃_{s_{n−1}w_{1}^{−1}(μ)} b = ∏_{α, β} \overline{X}_{α}X̃_{β}hw_{1}s_{n−1} b

by Lemma 3.1(iii). This completes the proof. □

Lemma 3.13. For any w ∈ W_{Ω}, h ∈ H and u_{1}, u_{2} ∈ U, we have

e_{s_{1}}u_{1}hwu_{2}e_{s_{n−1}} = κ e_{s_{1}}hwe_{s_{n−1}}

for some κ ∈ ℂ^{∗}.

Proof. Lemma 3.4 implies that

e_{s_{1}}u_{1}h = κ_{1} e_{s_{1}}hu′_{1},

for some κ_{1} ∈ ℂ^{∗} and some u′_{1} ∈ U_{s_{1}} = U ∩ U^{w_{S}s_{1}} = ∏_{α∈Neg(s_{1})} X_{α}. If w ∈ W_{Ω} and α ∈ Neg(s_{1}) then by (11) we see that w^{−1}(α) ∈ Φ^{+}. Hence w^{−1}u′_{1}w ∈ U and so

(15)  e_{s_{1}}u_{1}hwu_{2}e_{s_{n−1}} = κ_{1}e_{s_{1}}hw(w^{−1}u′_{1}w)u_{2}e_{s_{n−1}} = κ_{1}e_{s_{1}}hwu_{3}e_{s_{n−1}}

for some u_{3} ∈ U. By Lemma 3.4, we have

u_{3}e_{s_{n−1}} = κ_{2}u′_{3}e_{s_{n−1}}

for some κ_{2} ∈ ℂ^{∗} and u′_{3} ∈ U_{s_{n−1}} = ∏_{α∈Neg(s_{n−1})} X_{α}. Now since w ∈ W_{Ω} and α ∈ Neg(s_{n−1}), by (11), w(α) = α_{i n} for some i ⩾ 2 and so by (11) and (12) it follows that w(α) ∉ Neg(s_{1}), whence w(α) ∈ Pos(s_{1}). Thus

wu′_{3}w^{−1} ∈ ∏_{γ∈Pos(s_{1})} X_{γ} = U_{w_{S}s_{1}}

10 I. TULUNAY

So

(16)  e_{s_{1}}hwu_{3}e_{s_{n−1}} = κ_{2}e_{s_{1}}h(wu′_{3}w^{−1})we_{s_{n−1}} = κ′e_{s_{1}}hwe_{s_{n−1}}

for some κ′ ∈ ℂ^{∗}, by Lemma 3.4. Combining (15) and (16) gives the result. □ Theorem 3.14.

s_{1}b = ∑_{h∈D_{2}} κ(h) ∏_{3⩽i⩽n} \overline{X}_{α_{2 i}}hs_{2}s_{3} … s_{n−2}s_{n−1}b

where κ(h) is a constant depending on h. Proof. By [Theorem 2.3, [11]], we have

s_{1}b ∈ ℂCs_{n−1}b,

where

C = \{( \begin{matrix}1 & ∗ … ∗ & ∗ \\ 0 ⋮ 0 & GL_{n−2}(q) & ∗ ; ⋮ ; ∗ \\ 0 & 0 … 0 & 1 \end{matrix} )\} .

Then for some scalar κ(c), by Bruhat decomposition,

s_{1}b = ∑_{c∈C} κ(c) cs_{n−1}b = ∑_{\begin{matrix}u_{1}, u_{2}∈U \\ h∈D_{2}D_{3}...D_{n−1} \\ w∈W_{Ω} \end{matrix}} κ(u_{1}hwu_{2}) u_{1}hwu_{2}s_{n−1}b

Lemma 3.5 implies that (for some scalars κ′(hw), κ(h) )

s_{1}b = ∑_{\begin{matrix}u_{1}, u_{2}∈U \\ h∈D_{2}D_{3}...D_{n−1} \\ w∈W_{Ω} \end{matrix}} κ(u_{1}hwu_{2}) e_{s_{1}}u_{1}hwu_{2}e_{s_{n−1}}s_{n−1}b
= ∑_{\begin{matrix}h∈D_{2}D_{3}...D_{n−1} \\ w∈W_{Ω} \end{matrix}} κ′(hw) e_{s_{1}}hwe_{s_{n−1}}s_{n−1}b  by Lemma 3.13.

By Lemma 3.10, Lemma 3.11 and Lemma 3.5,

s_{1}b = ∑_{h∈D_{2}} κ(h)e_{s_{1}}hw_{1}e_{s_{n−1}}s_{n−1}b

Lemma 3.12 gives

s_{1}b = ∑_{h∈D_{2}} κ_{1}(h) ∏_{\begin{matrix}α∈S′(s_{1})∩Neg(s_{n−1}w_{1}^{−1}) \\ β∈S(s_{1})∩Neg(s_{n−1}w_{1}^{−1}) \end{matrix}} \overline{X}_{α}X̃_{β}hw_{1}s_{n−1}b.

Observe that

Neg(s_{n−1}w_{1}^{−1}) = { α_{23}, α_{24}, …, α_{2 n} }.

CUSPIDAL MODULES AS SUMMANDS OF A GELFAND-GRAEV MODULE 11

So S(s_{1}) ∩ Neg(s_{n−1}w_{1}^{−1}) is empty and

S′(s_{1}) ∩ Neg(s_{n−1}w_{1}^{−1}) = Neg(s_{n−1}w_{1}^{−1}) = { α_{2 i} | 3 ⩽ i ⩽ n }.

This completes the proof. □

For 2 ⩽ i < j ⩽ n − 1, let

Υ_{i j} = ∑_{t∈𝔽_{q}^{∗}} x_{i j}(t^{−1})d_{i}(−t^{−1})d_{j}(t)
(17)  Υ̃_{i j} = ∑_{t∈𝔽_{q}^{∗}} θ(t^{−1})x_{i j}(t^{−1})d_{i}(−t^{−1})d_{j}(t).

We write Υ_{i} for Υ_{i i+1} and Υ̃_{i} for Υ̃_{i i+1} for all 2 ⩽ i ⩽ n − 1. Lemma 3.15. For all 2 ⩽ i ⩽ n − 2,

\overline{X}_{i+1 j}Υ_{i}s_{i}s_{i+1} … s_{n−1}b = \begin{cases}X̃_{i i+2}Υ_{i}s_{i}s_{i+1} … s_{n−1}b & if j = i + 2 \\ \overline{X}_{i j}Υ_{i}s_{i}s_{i+1} … s_{n−1}b & if i + 3 ⩽ j ⩽ n. \end{cases}

Proof. By the definition of \overline{X}_{i+1 i+2} and Υ_{i}, \overline{X}_{i+1 i+2}Υ_{i}s_{i}s_{i+1} … s_{n−1}b

= \frac{1}{q} ∑_{\begin{matrix}t∈𝔽_{q}^{∗} \\ l∈𝔽_{q} \end{matrix}} x_{i+1 i+2}(l)x_{i i+1}(t^{−1})d_{i}(−t^{−1})d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}b

= \frac{1}{q} ∑_{\begin{matrix}t∈𝔽_{q}^{∗} \\ l∈𝔽_{q} \end{matrix}} x_{i i+1}(t^{−1})x_{i i+2}(−lt^{−1})x_{i+1 i+2}(l)d_{i}(−t^{−1})d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}b.

Note that x_{i+1 i+2}(l) commutes with d_{i}(−t^{−1}). Since

x_{i+1 i+2}(l)d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}b = d_{i+1}(t)x_{i+1 i+2}(lt^{−1})s_{i}s_{i+1} … s_{n−1}b
= d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}x_{i i+1}(lt^{−1})b
= θ(lt^{−1})d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}b,

we have \overline{X}_{i+1 i+2}Υ_{i}s_{i}s_{i+1} … s_{n−1}b

= ∑_{t∈𝔽_{q}^{∗}} (\frac{1}{q} ∑_{l∈𝔽_{q}} θ(lt^{−1})x_{i i+2}(−lt^{−1})) x_{i i+1}(t^{−1})d_{i}(−t^{−1})d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}b

= X̃_{i i+2}Υ_{i}s_{i}s_{i+1} … s_{n−1}b, since lt^{−1} runs through all elements of 𝔽_{q} as l does.

12 I. TULUNAY

Now suppose that i + 3 ⩽ j ⩽ n. Then \overline{X}_{i+1 j}Υ_{i}s_{i}s_{i+1} … s_{n−1}b

= \frac{1}{q} ∑_{\begin{matrix}t∈𝔽_{q}^{∗} \\ l∈𝔽_{q} \end{matrix}} x_{i+1 j}(l)x_{i i+1}(t^{−1})d_{i}(−t^{−1})d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}b

= \frac{1}{q} ∑_{\begin{matrix}t∈𝔽_{q}^{∗} \\ l∈𝔽_{q} \end{matrix}} x_{i i+1}(t^{−1})x_{i j}(−lt^{−1})x_{i+1 j}(l)d_{i}(−t^{−1})d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}b.

Note that x_{i+1 j}(l) commutes with d_{i}(−t^{−1}), and

x_{i+1 j}(l)d_{i+1}(t) = d_{i+1}(t)x_{i+1 j}(lt^{−1}).

Since x_{i+1 j}(lt^{−1})s_{i}s_{i+1} … s_{n−1}b = s_{i}s_{i+1} … s_{n−1}b, \overline{X}_{i+1 j}Υ_{i}s_{i}s_{i+1} … s_{n−1}b

= ∑_{t∈𝔽_{q}^{∗}} (\frac{1}{q} ∑_{l∈𝔽_{q}} x_{i j}(−lt^{−1})) x_{i i+1}(t^{−1})d_{i}(−t^{−1})d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}b

= \overline{X}_{i j}Υ_{i}s_{i}s_{i+1} … s_{n−1}b, as required. □

The following corollary is an immediate consequence of the above lemma. Corollary 3.16. For all 2 ⩽ i ⩽ n − 2,

( ∏_{j=i+2}^{n} \overline{X}_{i+1 j}) Υ_{i}s_{i}s_{i+1} … s_{n−1}b = X̃_{i i+2} ( ∏_{j=i+3}^{n} \overline{X}_{i j}) Υ_{i}s_{i}s_{i+1} … s_{n−1}b.

Lemma 3.17. For any t ∈ 𝔽_{q}^{∗} and any i with 1 ⩽ i ⩽ n − 1,

s_{i}x_{i i+1}(t)s_{i} = x_{i i+1}(t^{−1})d_{i}(−t^{−1})d_{i+1}(t)s_{i}x_{i i+1}(t^{−1}).

Proof. Observe that for any t ∈ 𝔽_{q}^{∗},

(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})(\begin{matrix}1 & t \\ 0 & 1 \end{matrix})(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) = (\begin{matrix}1 & t^{−1} \\ 0 & 1 \end{matrix})(\begin{matrix}−t^{−1} & 0 \\ 0 & t \end{matrix})(\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})(\begin{matrix}1 & t^{−1} \\ 0 & 1 \end{matrix})

from which the result follows. □ Lemma 3.18. For 1 ⩽ i ⩽ n − 2,

s_{i}\overline{X}_{i i+1}s_{i}s_{i+1} … s_{n−1}b = \frac{1}{q}s_{i+1}s_{i+2} … s_{n−1}b + \frac{1}{q}Υ_{i}s_{i}s_{i+1} … s_{n−1}b,

and s_{n−1}\overline{X}_{n−1 n}s_{n−1}b = \frac{1}{q}b + \frac{1}{q} Υ̃_{n−1} s_{n−1}b.

CUSPIDAL MODULES AS SUMMANDS OF A GELFAND-GRAEV MODULE 13

Proof. Lemma 3.17 implies that

s_{i}\overline{X}_{i i+1}s_{i} = \frac{1}{q} ∑_{t∈𝔽_{q}} s_{i}x_{i i+1}(t)s_{i}

= \frac{1}{q} + \frac{1}{q} ∑_{t∈𝔽_{q}^{∗}} x_{i i+1}(t^{−1})d_{i}(−t^{−1})d_{i+1}(t)s_{i}x_{i i+1}(t^{−1}) (18)

for all 1 ⩽ i ⩽ n − 1. Assume that 1 ⩽ i ⩽ n − 2. Since for all t ∈ 𝔽_{q}^{∗}

x_{i i+1}(t^{−1})s_{i+1}s_{i+2} … s_{n−1}b = s_{i+1}s_{i+2} … s_{n−1}x_{i n}(t^{−1})b = s_{i+1}s_{i+2} … s_{n−1}b, we have

s_{i}\overline{X}_{i i+1}s_{i}s_{i+1} … s_{n−1}b = \frac{1}{q}s_{i+1}s_{i+2} … s_{n−1}b

+ \frac{1}{q} ∑_{t∈𝔽_{q}^{∗}} x_{i i+1}(t^{−1})d_{i}(−t^{−1})d_{i+1}(t)s_{i}x_{i i+1}(t^{−1})s_{i+1}s_{i+2} … s_{n−1}b

= \frac{1}{q}s_{i+1}s_{i+2} … s_{n−1}b + \frac{1}{q} ∑_{t∈𝔽_{q}^{∗}} x_{i i+1}d_{i}(−t^{−1})d_{i+1}(t)s_{i}s_{i+1} … s_{n−1}b

= \frac{1}{q}s_{i+1}s_{i+2} … s_{n−1}b + \frac{1}{q}Υ_{i}s_{i}s_{i+1} … s_{n−1}b.

As x_{n−1 n}(t^{−1})b = θ(t^{−1})b,

s_{n−1}\overline{X}_{n−1 n}s_{n−1}b = \frac{1}{q}b + \frac{1}{q}Υ̃_{n−1} s_{n−1}b,

by (18). □

It is known that s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1} (19) s_{i}s_{j} = s_{j}s_{i} if |i − j| > 1 (20) for all i, j such that 1 ⩽ i ⩽ n − 2 and 1 ⩽ j ⩽ n − 1.

For each k ∈ ℤ with 1 ⩽ k ⩽ n − 2, we define an element τ_{k} of W by

τ_{k} := s_{n−k}s_{n−k+1} … s_{n−2}s_{n−1}s_{n−2} … s_{n−k+1}s_{n−k}.

Note that τ_{k}^{−1} = τ_{k}.

Observe that by (19) and (20)

τ_{k} = s_{n−k}s_{n−k+1} … s_{n−2}s_{n−1}s_{n−2} … s_{n−k+1}s_{n−k}
= s_{n−k}s_{n−k+1} … s_{n−3}(s_{n−1}s_{n−2}s_{n−1})s_{n−3} … s_{n−k+1}s_{n−k}
= s_{n−1}s_{n−k}s_{n−k+1} … s_{n−3}s_{n−2}s_{n−3} … s_{n−k+1}s_{n−k}s_{n−1}
…

= s_{n−1}s_{n−2} … s_{n−k+1}s_{n−k}s_{n−k+1} … s_{n−2}s_{n−1} (21) for all k with 1 ⩽ k ⩽ n − 2.

For all y ∈ 𝔽_{q}^{∗}, write κ(y) = κ(d_{2}(y)) (the constant appearing in Theorem 3.14). Let

(22)  γ_{i} = ∑_{y∈𝔽_{q}^{∗}} κ(y)d_{i}(y).

Theorem 3.19. We have

s_{n−1}s_{n−2} … s_{1}b = \frac{1}{q^{n−2}} γ_{n}b + γ_{n} ∑_{k=2}^{n−1} E_{k}τ_{n−k}b

where

E_{k} = \begin{cases}\frac{1}{q^{k−1}} X̃_{k k+1} (∏_{j=k+2}^{n−1} \overline{X}_{k j}) Υ_{k n} & if 2 ⩽ k ⩽ n − 2 \\ \frac{1}{n−2} Υ̃_{n−1 n} & if k = n − 1 \end{cases}

and Υ̃_{i j} as in (17). Proof. Theorem 3.14 implies that

(23)  s_{n−1}s_{n−2} … s_{1}b = (s_{n−1}s_{n−2} … s_{2}) (γ_{2} (∏_{j=3}^{n} \overline{X}_{2 j}) s_{2}s_{3} … s_{n−1}b)

by (22). Now

s_{2}γ_{2} (∏_{j=3}^{n} \overline{X}_{2 j}) s_{2}s_{3} … s_{n−1}b = γ_{3} (∏_{j=4}^{n} \overline{X}_{3 j}) (s_{2}\overline{X}_{23}s_{2})s_{3}s_{4} … s_{n−1}b

= γ_{3} \frac{1}{q} (∏_{j=4}^{n} \overline{X}_{3 j}) s_{3}s_{4} … s_{n−1}b + γ_{3} \frac{1}{q} (∏_{j=4}^{n} \overline{X}_{3 j}) Υ_{2 3}s_{2}s_{3} … s_{n−1}b

by Lemma 3.18, provided n > 3. If n = 3 then (by Lemma 3.18) the same expression is obtained with Υ_{2 3} replaced by Υ̃_{2 3}, and so

s_{n−1}s_{n−2} … s_{1}b = s_{2}s_{1}b = \frac{γ_{3}}{q} + \frac{γ_{3}}{q} Υ̃_{2 3}s_{2}b,

as desired. For n > 3 the second term above can be written as

\frac{γ_{3}}{q} X̃_{24} (∏_{j=5}^{n} \overline{X}_{2 j}) Υ_{2 3}s_{2}s_{3} … s_{n−1}b

by Corollary 3.16, and so multiplying s_{n−1}s_{n−2} … s_{3} we obtain

s_{n−1}s_{n−2} … s_{1}b = \frac{1}{q}(s_{n−1}s_{n−2} … s_{3})γ_{3} (∏_{j=4}^{n} \overline{X}_{3 j}) s_{3}s_{4} … s_{n−1}b

+ \frac{γ_{n}}{q} X̃_{23} (∏_{j=4}^{n−1} \overline{X}_{2 j}) Υ_{2 n}s_{n−1} … s_{3}s_{2}s_{3} … s_{n−1}b

= \frac{1}{q}(s_{n−1}s_{n−2} … s_{3})γ_{3} (∏_{j=4}^{n} \overline{X}_{3 j}) s_{3}s_{4} … s_{n−1}b + γ_{n}E_{2}τ_{n−2}b.

The first term here is similar to the right hand side of (23), and so by calculations parallel to those above it can be expressed as

\frac{1}{q^{2}} (s_{n−1}s_{n−2} … s_{4})γ_{4} (∏_{j=5}^{n} \overline{X}_{4 j}) + γ_{n}E_{3}τ_{n−3}b

(where the first term is just \frac{1}{q^{2}} γ_{4}b in the case n = 4). Repeating the process a further n − 4 times yields the desired formula. □

Lemma 3.20. For all k with 1 ⩽ k ⩽ n − 2

( ∏_{1⩽i⩽n−2} \overline{X}_{i n}) τ_{k}b = 0.

Proof. Since τ_{k} interchanges n − k and n,

\overline{X}_{n−k−1 n}τ_{k}b = τ_{k}\overline{X}_{n−k−1 n−k}b = 0

by Lemma 3.1(iii). □

Corollary 3.21. We have

∏_{1⩽i⩽n−2} \overline{X}_{i n}s_{n−1}s_{n−2} … s_{1}b = \frac{1}{q^{n−2}} γ_{n}b

Proof. Multiply the equation given in Theorem 3.19 by ∏_{1⩽i⩽n−2} \overline{X}_{i n}. Since ∏_{1⩽i⩽n−2} \overline{X}_{i n} is a subgroup normalized by U and H,

( ∏_{1⩽i⩽n−2} \overline{X}_{i n}) γ_{n}E_{k}τ_{n−k}b = γ_{n}E_{k} (∏_{i=1}^{n−2} \overline{X}_{i n}) τ_{n−k}b = 0

for all k ∈ {2, 3, …, n − 1}, by Lemma 3.20. Hence

∏_{1⩽i⩽n−2} \overline{X}_{i n}s_{n−1}s_{n−2} … s_{1}b = \frac{1}{q^{n−2}} γ_{n}b,

as required. □

Proof of Lemma 2.2. By Lemma 3.2,

es_{n−1}s_{n−2} … s_{1}b = X̃_{n−1, n} ∏_{1⩽i⩽n−2} \overline{X}_{i n}s_{n−1}s_{n−2} … s_{1}b.

Multiplying the equation in Corollary 3.21 by X̃_{n−1 n} gives

es_{n−1}s_{n−2} … s_{1}b = \frac{1}{q^{n−2}} X̃_{n−1, n}γ_{n}b.

If t ∈ 𝔽_{q}^{∗} then

X̃_{n−1 n}d_{n}(t)b = \frac{1}{q} ∑_{l∈𝔽_{q}} θ(−l)x_{n−1 n}(l)d_{n}(t)b

= \frac{1}{q} ∑_{l∈𝔽_{q}} θ(−l)d_{n}(t)x_{n−1 n}(tl)b

= \frac{1}{q} ∑_{l∈𝔽_{q}} θ(−l)θ(tl)d_{n}(t)b

= \begin{cases}b & if t = 1 \\ 0 & if t ≠ 1, \end{cases}

since the character θ^{t} of 𝔽_{q} defined by θ^{t}(l) = θ(lt) is not equivalent to θ when t ≠ 1. Hence

X̃_{n−1 n}γ_{n}b = κ(1)b,

from which the result follows.

Now the aim is to determine the constant k(1), (where 1 ∈ 𝔽_{q}^{∗}) by using the character theory of finite general linear groups.

4. CONJUGACY CLASSES AND CUSPIDAL CHARACTERS

This section is devoted to give some well-known facts about conjugacy classes and cuspidal characters of G.

Let f be an irreducible monic polynomial over 𝔽_{q} of degree d ⩾ 1 such that f ≠ x. For each such f fix a matrix P_{f} ∈ G_{k} whose characteristic polynomial is equal to f (such a matrix always exists). Let μ = (μ_{1}, μ_{2}, …, μ_{r}) be a partition with r parts. Define |μ| = μ_{1} + μ_{2} + … + μ_{r}. For each such f and μ define

P_{μ}(f) = (\begin{matrix}P_{f, μ_{1}} & 0 & … & 0 \\ 0 & P_{f, μ_{2}} & … & 0 \\ … & … & … & … \\ 0 & 0 & … & P_{f, μ_{r}} \end{matrix} ) ,

where for each μ_{i}

P_{f, μ_{i}} = (\begin{matrix}P_{f} & I_{d} & … & 0 & 0 \\ 0 & P_{f} & … & 0 & 0 \\ … & … & … & … & … \\ 0 & 0 & … & P_{f} & I_{d} \\ 0 & 0 & … & 0 & P_{f} \end{matrix})  (an analog of a Jordan Block)

and I_{d} is the identity matrix of order d. Clearly P_{f, μ_{i}} is of degree μ_{i}d and P_{μ}(f) is of degree |μ|d. Let {f_{1}, f_{2}, …, f_{N}} be a set of distinct irreducible monic polynomials and {ρ_{1}, ρ_{2}, …, ρ_{N}} be a set of partitions satisfying the following conditions:

(i) ∑_{1⩽i⩽N} |ρ_{i}| deg(f_{i}) = n, (ii) f_{i} ≠ x for all i, 1 ⩽ i ⩽ N. Consider the following matrix

(24)  P = (\begin{matrix}P_{ρ_{1}}(f_{1}) & 0 & … & 0 \\ 0 & P_{ρ_{2}}(f_{2}) & … & 0 \\ … & … & … & … \\ 0 & 0 & … & P_{ρ_{N}} (f_{N}) \end{matrix}) ∈ G.

Note that the characteristic polynomial of P is equal to f_{1}^{|ρ_{1}|}f_{2}^{|ρ_{2}|} … f_{N}^{|ρ_{N} |}.

Theorem 4.1. The set of matrices of the form (24) for all possible choices of the f_{i} and the ρ_{i} satisfying conditions (i) and (ii) above is a complete system of representatives of the conjugacy classes of G.

Denote the class of G containing the element of P in (24) by

c = [f_{1}^{ρ_{1}}f_{2}^{ρ_{2}} … f_{N}^{ρ_{N}}].

Let C be the set of all conjugacy classes of G. Define the following subsets of C:

C_{>1} := { [f_{1}^{ρ_{1}}f_{2}^{ρ_{2}} … f_{N}^{ρ_{N}}] | N > 1 }
C_{1} := { [f^{ρ}] | |ρ| deg f = n }
C^{s} := { [f_{1}^{(e_{1})}f_{2}^{(e_{2})} … f_{N}^{(e_{N})}] | N ⩾ 1 }
C_{1}^{(e)} := \{ [f^{(e)}] ∈ C_{1} | e|n, deg(f) = \frac{n}{e} \} = C^{s} ∩ C_{1}

for each 1 ⩽ e ⩽ n such that e divides n. Thus g ∈ C_{1} if and only if the characteristic polynomial of g is a power of an irreducible, and g ∈ C_{>1} if and only if the characteristic polynomial of g is divisible by more than one irreducible. Furthermore, C^{s} is the subset of C for which the partitions (ρ_{1}, ρ_{2}, …, ρ_{N}) have only one part and C_{1}^{(e)} is the subset of C_{1} for which the partition ρ has just one part. Note that g ∈ G lies in a class in the set C^{s} (the set of regular classes) if and only if the minimal polynomial of g equals its characteristic polynomial.

Let

C_{f} = (\begin{matrix}0 & 0 & 0 & … & 0 & −a_{0} \\ 1 & 0 & 0 & … & 0 & −a_{1} \\ 0 & 1 & 0 & … & 0 & −a_{2} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & … & 0 & −a_{n−2} \\ 0 & 0 & 0 & … & 1 & −a_{n−1} \end{matrix})

be the companion matrix of a monic polynomial f = x^{n} + a_{n−1}x^{n−1} + … + a_{0}. Note that if λ is one of the roots of f over 𝔽_{q} then

Tr(C_{f}) = −a_{n−1} = Tr(λ)

Tr(C_{f^{e}}) = e Tr(C_{f}). (25)

The following proposition can be found in [10].

Proposition 4.2. For a monic polynomial f of degree n, the minimal polynomial and the characteristic polynomial of the companion matrix C_{f} are both equal to f. Clearly det C_{f} = (−1)^{n}a_{0}.

Proposition 4.2 implies that C_{f} is in C^{s}.

Let ϕ_{r}(x) = (x − 1)(x^{2} − 1) … (x^{r−1} − 1). A formula for an irreducible cuspidal character ξ_{χ} of G can be given by

ξ_{χ}(c) = \begin{cases}(−1)^{r+n}ϕ_{r}(q^{d}) ∑_{0⩽i⩽d−1} χ(λ^{q^{i}}) & if c = [f^{ρ}] ∈ C_{1} \\ 0 & if c ∈ C_{>1}, \end{cases}

where r is the number of the parts of the partition ρ, deg(f) = d and λ is one of the roots of f over 𝔽_{q^{n}}. (See [7] or [9]). Note also that λ, λ^{q}, …, λ^{q^{d−1}} give all roots of f.

In particular,

(26) ξ_{χ}(C_{f}) = \begin{cases}(−1)^{n+1} (χ(λ_{1}) + χ(λ_{2}) + … + χ(λ_{d})) & if C_{f} ∈ C_{1}^{(n/d)}, \\ 0 & otherwise \end{cases}

where λ_{1}, λ_{2}, …, λ_{d} are all roots of f.

5. THE BESSEL FUNCTION

The following definition was given by Gel’fand in his paper [7]. Also see G.James’s paper [9].

Definition 5.1. Define J : G ⟶ ℂ by

J (g) = \frac{1}{|U|} ∑_{u∈U} Ψ(u^{−1})ξ_{χ}(gu).

Then the map J is called the Bessel function of the cuspidal representation with respect to Ψ.

Consider the subgroup

A := { (a_{ij}) ∈ | a_{nn} = 1, a_{ni} = 0, if 1 ⩽ i ⩽ k − 1 }

of G. Let l = [A : U] and A/U = {1 = a_{1}, a_{2}, …, a_{l} } be a set of left coset representatives of U in A. It is known that

{ a_{i}b | 1 ⩽ i ⩽ l }

forms a basis for an irreducible cuspidal module M (see for example [7], [9], [2]). The following lemma is essentially proved in [7], [9] and [1].

Lemma 5.2. For arbitrary g ∈ G, let

ga_{j}b = ∑_{1⩽i⩽l} J (a_{i}^{−1}ga_{j})a_{i}b

for all j, 1 ⩽ j ⩽ l. Then this action defines a representation of G in M and this representation has character ξ_{χ}.

Let J_{n−1} = {s_{1}, s_{2}, …, s_{n−2}} and W_{J_{n−1}} = ⟨J_{n−1}⟩. So W ∩ A = W_{J_{n−1}}. By using Bruhat decomposition, it can be seen that

A = ˙⋃_{w∈W_{J_{n−1}}}(U_{n−1})_{w}H_{n−1}wU,

where (U_{n−1})_{w} = U_{n−1} ∩ (U_{n−1})^{w_{J_{n−1}} w}. So the set

{ uhw | u ∈ (U_{n−1})_{w}, h ∈ H_{n−1}, w ∈ W_{J_{n−1}} }

forms a set of left coset representatives of U in A. In particular, s_{n−2}s_{n−3} … s_{1} is in the set of left coset representatives of U in A. Applying Lemma 5.2 with a_{j} = s_{n−2}s_{n−3} … s_{1}, and s_{n−1} in place of g, we find that the coefficient of b in s_{n−1}s_{n−2} … s_{1}b is

J (s_{n−1}s_{n−2} … s_{1}).

However, by Theorem 3.19, the coefficient of b in s_{n−1}s_{n−2} … s_{1}b is

\frac{1}{q^{n−2}} κ(1).

Hence

J (s_{n−1}s_{n−2} … s_{1}) = \frac{1}{q^{n−2}} κ(1).

So the task is now to calculate Λ(χ) = J (s_{n−1}s_{n−2} … s_{1}).

The following property of J is known (see [7]):

(27)  \overline{J (g)} = J (g^{−1})

for all g ∈ G.

Let σ = s_{1}s_{2} … s_{n−1}. Then σ is the companion matrix of f = x^{n} − 1.

Lemma 5.3. For any monic polynomial f of degree n, the centralizer C_{U}(C_{f}) of C_{f} in U is trivial.

Proof. Suppose that u = (u_{ij}) is upper unitriangular and commutes with C_{f}. Comparing the (i, j) entries of C_{f}u and uC_{f} as i ranges from 1 to n − 1, and j from i to n − 1 we find successively

u_{12} = 0, u_{23} = 0, …, u_{1 n} = 0, u_{23} = u_{12} = 0, u_{24} = u_{13} = 0,

and so on. Finally we obtain that u is the identity. □

The following results are essentially proved for the general case f = x^{n} − z for all z ∈ 𝔽_{q}^{∗} in [12].

Lemma 5.4. |U ∩ σ^{−1}Uσ| = q^{\frac{1}{2} (n−1)(n−2)}.

The proof is straightforward and so is omitted.

Lemma 5.5. If f is any monic polynomial of degree n with f(0) = −1 then there are exactly q^{\frac{1}{2} (n−1)(n−2)} elements conjugate to C_{f} in the coset Uσ. Furthermore, every element of Uσ is conjugate to C_{f} for some polynomial f. Similarly the double coset UσU contains q^{\frac{1}{2} n(n−1)} elements of each of these classes and no other elements.

Proof. It is clear (by elementary column operations) that for any such f there is a u ∈ U with C_{f} = σu ∈ UσU. By Lemma 5.3, in the conjugation action of U on UσU the orbit containing C_{f} has length |U|, and since the number of polynomials f satisfying the given conditions is q^{n−1}, this accounts for q^{n−1}|U| elements of UσU. But by Lemma 5.4 we see that

|UσU| = |U||U : U ∩ σ^{−1}Uσ| = |U|q^{n−1}

and so UσU contains no further elements. Thus UσU contains exactly |U| elements conjugate to C_{f}. If C is any conjugacy class then, for each u ∈ U,

C ∩ Uσu = u^{−1}(C ∩ Uσ)u

and so all cosets Ug contained in UσU contain the same number of elements of C. So the number of elements of Uσ conjugate to C_{f} is

|U|\frac{|Uσ|}{|UσU|} = q^{\frac{1}{2} (n−1)(n−2)},

as claimed. □

Proposition 5.6.

J (σ) = \frac{(−1)^{n+1}}{q^{n−1}} ∑_{\begin{matrix}λ∈𝔽_{q^{n}} \\ N(λ)=(−1)^{n+1} \end{matrix}} θ(− Tr(λ))χ(λ).

Proof. By Definition 5.1, we have

J (σ) = \frac{1}{|U|} ∑_{u∈U} Ψ(u^{−1})ξ_{χ}(σu)

= \frac{1}{|U|} ∑_{u∈U} Ψ(u^{−1})ξ_{χ}(uσ)

= \frac{1}{|U|} ∑_{u∈U} Ψ(u)ξ_{χ}(u^{−1}σ).

For any u ∈ U,

u^{−1}σ = (\begin{matrix}−u_{12} & ∗ & ∗ & … & ∗ & z \\ 1 & −u_{23} & ∗ & … & ∗ & 0 \\ 0 & 1 & −u_{34} & … & ∗ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 0 & … & −u_{n−1 n} & 0 \\ 0 & 0 & 0 & … & 1 & 0 \end{matrix}) .

Then clearly Tr(u^{−1}σ) = −u_{12} − u_{23} − … − u_{n−1 n}. So we have

J (σ) = \frac{1}{|U|} ∑_{u^{−1}∈U} θ(− Tr(u^{−1}σ))ξ_{χ}(u^{−1}σ)

= \frac{q^{\frac{1}{2} (n−1)(n−2)}}{|U|} ∑ θ(− Tr(C_{f}))ξ_{χ}(C_{f})

by Lemma 5.5, where f ranges over all monic polynomials of degree n with f(0) = −1. By (26), ξ_{χ}(C_{f}) = 0 unless C_{f} ∈ C_{1}^{(e)}; that is, unless f is a power of an irreducible. For each divisor d of n, let P(d) be the set of monic irreducible polynomials f_{1} of degree d such that (f_{1}(0))^{e} = −1 where de = n. Then we have

J (σ) = \frac{1}{q^{n−1}} ∑_{de=n} ∑_{f_{1}∈P(d)} θ(− Tr(C_{f_{1}^{e}}))ξ_{χ}(C_{f_{1}^{e}})

= \frac{1}{q^{n−1}} ∑_{de=n} ∑_{f_{1}∈P(d)} θ(−e Tr(C_{f_{1}})ξ_{χ}(C_{f_{1}^{e}}) by (25),

= \frac{(−1)^{n+1}}{q^{n−1}} ∑_{de=n} ∑_{f_{1}∈P(d)} θ(−e Tr(C_{f_{1}})) (χ(λ_{1}) + χ(λ_{2}) + … χ(λ_{d−1}))

by (26), the λ_{i} being the roots of f_{1}. Note that each of these roots is an element λ ∈ 𝔽_{q^{d}} such that 𝔽_{q}(λ) = 𝔽_{q^{d}} and N(λ) = N_{𝔽_{q^{d}}:𝔽_{q}}(λ) = (−1)^{n+1}; moreover each such λ is a root of some f_{1} ∈ P(d); and Tr_{𝔽_{q^{d}}:𝔽_{q}}(λ) = Tr(C_{f_{1}}). Thus

J (σ) = \frac{(−1)^{n+1}}{q^{n−1}} ∑_{de=n} ∑_{\begin{matrix}λ∈𝔽_{q^{d}} \\ 𝔽_{q}(λ)=𝔽_{q^{d}} \\ N(λ)=(−1)^{n+1} \end{matrix}} θ(−e Tr_{𝔽_{q^{d}}:𝔽_{q}}(λ))χ(λ)

= \frac{(−1)^{n+1}}{q^{n−1}} ∑_{de=n} ∑_{\begin{matrix}λ∈𝔽_{q^{d}} \\ 𝔽_{q}(λ)=𝔽_{q^{d}} \\ N(λ)=(−1)^{n+1} \end{matrix}} θ(− Tr(λ))χ(λ)

= \frac{(−1)^{n+1}}{q^{n−1}} ∑_{\begin{matrix}λ∈𝔽_{q^{n}} \\ N(λ)=(−1)^{n+1} \end{matrix}} θ(− Tr(λ))χ(λ),

by using the fact that if 𝔽_{q^{d}} = 𝔽_{q}(λ) is an extension of degree d over 𝔽_{q} (where 1 < d < n and d divides n) then 𝔽_{q} ⊆ 𝔽_{q^{d}} ⊆ 𝔽_{q^{n}} and

Tr_{𝔽_{q^{n}}:𝔽_{q}}(λ) = \frac{n}{d} Tr_{𝔽_{q^{d}}:𝔽_{q}}(λ).

□

Corollary 5.7.

Λ(χ) = \frac{(−1)^{n+1}}{q^{n−1}} ∑_{\begin{matrix}λ∈𝔽_{q^{n}} \\ N(λ)=(−1)^{n+1}z \end{matrix}} θ(Tr(λ))χ(λ^{−1})

Proof. It follows from

Λ(χ) = J (σ^{−1}) = \overline{J (σ)},

by (27). □

References

[1] Alperin, J.; James, G.D. Bessel functions on finite groups. J. Algebra 1995 171, 524–530.

[2] Carter, R.W. Cuspidal matrix representations for GL_{2}(q) and GL_{3}(q). Proc.London Math.Soc. 1992 64(3), 487–523.

[3] Carter, R.W. Finite groups of Lie type; conjugacy classes and irreducible characters; Wiley-Interscience, New York, 1985.

[4] Curtis, C.W.; Reiner, I. Methods of representation theory with application to finite groups and orders; John Wiley, New York, 1981; Vol.1.

[5] Curtis, C.W.; Reiner, I. Methods of representation theory with application to finite groups and orders; John Wiley, New York, 1981; Vol.2.

[6] Curtis, C.W.; Reiner, I. Representation theory of finite groups and associative algebras; Interscience, New York, 1962.

[7] Gel’fand, S.I. Representations of the full linear group over a finite field. Math. USSR. Sbornik 1970 12 (1), 13–39.

[8] Green, J.A. The characters of the finite general linear groups. Trans. Amer. Math. Soc. 1955 80, 402–447.

[9] James, G.D. The irreducible representations of the finite general linear groups. Proc. London Math. Soc. 1986 52(3), 236–268.

[10] MacLane, S.; Birkhoff, G. Algebra, (Second edition); Macmillan Publishing Co., Inc. 1979.

[11] Tulunay, I. Restriction of a cuspidal module for finite general linear groups. J. Algebra to appear.

[12] Tulunay, I. Cuspidal modules of finite general linear groups, PhD thesis, University of Sydney, 2001.

School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

E-mail address: ilknur@maths.usyd.edu.au