Subfield symmetric spaces for finite special linear groups

Toshiaki Shoji and Karine Sorlin^{∗}

Graduate School of Mathematics Nagoya University Chikusa-ku, Nagoya 464-8602, Japan

Abstract. Let G be a connected algebraic group defined over a finite field F_{q}. For each irreducible character ρ of G(F_{q^{r}}), we denote by m_{r}(ρ) the multiplicity of 1_{G(F_{q})} in the restriction of ρ to G(F_{q}). In the case where G is reductive with connected center and is simple modulo center, Kawanaka determined m_{2}(ρ) for almost all cases, and then Lusztig gave a general formula for m_{2}(ρ). In the case where the center of G is not connected, such a result is not known. In this paper we determine m_{2}(ρ), up to some minor ambiguity, in the case where G is the special linear group.

We also discuss, for any r ≥ 2, the relationship between m_{r}(ρ) with the theory of Shintani descent in the case where G is a connected algebraic group.

0. Introduction

Let G be a connected reductive group defined over a finite field F_{q} with Frobenius map F. We consider the finite group G^{F^{2}} and its subgroup G^{F}. The quotient space G^{F^{2}}/G^{F} is regarded as an analogue of the symmetric space, and is called the subfield symmetric space over a finite field. The determination of spherical functions of G^{F^{2}}/G^{F} is almost equivalent to the determination of irreducible characters of the Hecke algebra H(G^{F^{2}}, G^{F}). For a class function f on G^{F^{2}}, we denote by m_{2}(f) the inner product of f with the induced character Ind_{G^{F}}^{G^{F^{2}}} 1. The classification of irreducible characters of H(G^{F^{2}}, G^{F}) and the determination of their degrees are equivalent to the determination of m_{2}(ρ) for all irreducible characters ρ of G^{F^{2}}.

In [K2], Kawanaka computed m_{2}(ρ) in the case where G is a classical group with connected center, or in the case where ρ is unipotent and the characteristic is good. Extending Kawanaka’s result, Lusztig gave in [L3] a closed formula for m_{2}(ρ) valid for any G which has the connected center and is simple modulo its center. He expects that his formula is still valid for G with disconnected center. In turn, Henderson studied in [H] the spherical functions of G^{F^{2}}/G^{F} by making use of the theory of perverse sheaves, and described them in the case where G = GL_{n}, in which case H(G^{F^{2}}, G^{F}) is abelian.

In this paper, we consider G = SL_{n} with the standard F_{q}-structure, which is the first example of the disconnected center case. Based on the parametrization of irreducible characters and the description of almost characters in [S3] (which is valid under some restriction on p, for example, p ≥ 2n), we determine m_{2}(ρ) (Theorem 5.3) for any irreducible characters, up to some minor ambiguity. In particular, we have m_{2}(ρ) ∈ {0, 1, 2}. We discuss the relationship between our result and Lusztig’s conjectural formula.

∗The second author would like to thank the JSPS for support which made this collaboration possible

Kawanaka’s main idea for the computation of m_{2}(ρ), beside the use of the results of Lusztig on m_{2}(R_{T} (θ)), is to connect it with the twisted Frobenius-Schur indicator through the twisting operator. In section 1, we generalize Kawanaka’s result, and discuss a connection of m_{2}(ρ) with Shintani descent. This leads to a formula for m_{2}(R_{x}) where R_{x} is an almost character of G^{F^{2}}, which is regarded as a counter part of Lusztig’s formula for m_{2}(χ_{A}) in [L3, 7], where χ_{A} is the characteristic function of character sheaves. In section 1, we also discuss a more general situation. We define m_{r}(ρ) as the multiplicity of an irreducible character ρ of G^{F^{r}} with the induced character Ind_{G^{F}}^{G^{F^{r}}} 1 for any integer r ≥ 2. We give some formula (Theorem 1.14) for m_{r}(R_{x}) though it is not so effective as the m_{2} case.

The subsequent sections are devoted to the computation of m_{2}(ρ) for the case where G = SL_{n}. We obtain the results by applying the results in section 1, together with the computation of m_{2}(ρ̃|_{G^{F^{2}}}) for irreducible characters ρ̃ of GL_{n}(F_{q^{2}}).

The authors are grateful to the referee for the careful reading of the paper, and the suggestion concerning the ambiguity in the first draft of this paper.

Some notations: All the representations considered in this paper are over Q̄_{l}. For a finite group Γ, we denote by Irr Γ the set of irreducible characters of Γ. If δ is an automorphism of Γ, we denote by (Irr Γ)^{δ} the set of F -stable irreducible characters of Γ.

Assume that Γ is an abelian group. In this case, we also use the notation Γ^{∧} to denote the set of irreducible characters of Γ, which coincides with the group Hom(Γ, Q̄_{l}^{∗}). If δ is an automorphism of Γ, we denote by Γ_{δ}^{∧} the subgroup of Γ^{∧} consisting of δ-fixed irreducible characters of Γ. Also in this case, we denote by Γ_{δ} the largest quotient of Γ on which δ acts trivially, i.e., Γ_{δ} is the quotient of Γ by the subgroup generated by g^{−1}δ(g) for g ∈ Γ. Don’t confuse Γ_{δ}^{∧} with (Γ^{∧})_{δ} for a finite group Γ^{∧}. The group (Γ^{∧})_{δ} does not occur in this paper.

Contents

1. G(F_{q})-invariants in G(F_{q^{r}})-modules and Shintani descent.

2. Parametrization of irreducible characters of SL_{n}(F_{q^{2}}).

3. Almost characters of SL_{n}(F_{q^{2}}).

4. Determination of m_{2}(ρ_{ṡ,E}|_{G^{F^{2}}}).

5. Determination of m_{2}(ρ) for ρ ∈ Irr SL_{n}(F_{q^{2}}).

1. G(F_{q})-invariants in G(F_{q^{r}})-modules and Shintani descent

1.1. For any finite group Γ and an automorphism F : Γ → Γ, we denote by Γ/∼_{F} the set of F -twisted conjugacy classes in Γ, where x, y ∈ Γ are F -twisted conjugate if there exists z ∈ Γ such that y = z^{−1}xF (z). In the case where F acts trivially on Γ, the set Γ/∼_{F} coincides with the set of conjugacy classes in Γ, which we denote by Γ/∼.

For a connected algebraic group X defined over F_{q}, and two Frobenius maps F_{1}, F_{2} on X such that F_{1}F_{2} = F_{2}F_{1}, we define a norm map

N_{F_{1}/F_{2}} : X^{F_{1}}/∼_{F_{2}}→ X^{F_{2}}/∼_{F_{1}^{−1}}

as follows; for x ∈ X^{F_{1}}, we choose α ∈ X such that x = α^{−1}F_{2}(α), and put x′ = F_{1}(α)α^{−1}. Then x′ ∈ X^{F_{2}} and the correspondence x → x′ induces a bijective map N_{F_{1}/F_{2}}, which we call the norm map from X^{F_{1}}/∼_{F_{2}} to X^{F_{2}}/∼_{F_{1}^{−1}}.

For a finite set Y, we denote by C(Y) the Q̄_{l}-space of all Q̄_{l}-valued functions on Y. Then the norm map N_{F_{1}/F_{2}} induces a linear isomorphism

Sh_{F_{1}/F_{2}} = N_{F_{1}/F_{2}}^{∗−1} : C(X^{F_{1}}/∼_{F_{2}}) → C(X^{F_{2}}/∼_{F_{1}^{−1}}),

which is called the Shintani descent from X^{F_{1}} to X^{F_{2}}.

1.2. Let G be a connected algebraic group defined over a finite field F_{q} with Frobenius map F. We fix a positive integer r, and consider the group H = G×⋯×G (r-factors). H is endowed with the natural Frobenius map given by (g_{1}, …, g_{r}) ↦ (F (g_{1}), …, F (g_{r})), which we also denote by F. Let F′ = F ω : H → H be a twisted Frobenius map on H, where ω : H → H, (g_{1}, …, g_{r}) ↦ (g_{r}, g_{1}, …, g_{r−1}) is the cyclic permutation of factors. Since ω^{r} = 1 and F ω = ωF, we have (F′)^{rm} = F^{rm} for any m ≥ 1.

Lemma 1.3. The map G^{F^{rm}} → H^{F^{rm}}, x → (x, 1, …, 1) induces a bijection

f : G^{F^{rm}}/∼_{F^{r}} → H^{F^{rm}}/∼_{F′} . (1.3.1)

Proof. Take x = (x_{1}, …, x_{r}), y = (y_{1}, …, y_{r}) ∈ H^{F^{rm}}. If x and y are in the same class, there exists z = (z_{1}, …, z_{r}) such that y_{i} = z_{i}^{−1}x_{i}F (z_{i−1}) for i ∈ Z/rZ. Now assume that x = (x_{1}, 1, …, 1). Then z^{−1}xF′(z) = (y_{1}, 1, …, 1) for z ∈ G^{F^{rm}} if and only if z = (z_{1}, F (z_{1}), …, F^{r−1}(z_{1})). Moreover in this case, y_{1} = z_{1}^{−1}x_{1}F^{r}(z_{1}). This shows that the map f is well-defined, and is injective. It is easy to see that each F′-conjugacy class in H^{F^{rm}} contains a representative of the form (x_{1}, 1, …, 1). Hence f is surjective.

1.4. For each x ∈ G^{F^{rm}}, k ≥ 1, we put N_{k}(x) = xF (x) ⋯ F^{k−1}(x). Then the map G^{F^{rm}} → G^{F^{rm}}, x ↦ N_{k}(x) induces a map G^{F^{rm}}/∼_{F} → G^{F^{rm}}/∼_{F^{k}}, which we also denote by N_{k}. Let Δ(H) ≃ G be the diagonal subgroup of H. The inclusion Δ(H)^{F^{rm}} ↪ H^{F^{rm}} induces a map d : Δ(H)^{F^{rm}}/∼_{F} → H^{F^{rm}}/∼_{F′}. Then we have a commutative diagram

G^{F^{rm}}/∼_{F^{r}} ⟶^{f} H^{F^{rm}}/∼_{F′}
N_{r}↑  ↑^{d}
G^{F^{rm}}/∼_{F} ⟶^{f_{0}} Δ(H)^{F^{rm}}/∼_{F} ,

where f_{0} is the bijection induced from the isomorphism G \underrightarrow{∼} Δ(H). This follows from the following relation for x ∈ G^{F^{rm}},

(N_{r}(x), 1, …, 1) = y^{−1}(x, x, …, x)F′(y)

with y = (1, N_{1}(x), N_{2}(x), …, N_{r−1}(x)).

1.5. Concerning the norm maps, we have the following commutative diagram.

G^{F^{rm}}/∼_{F^{r}} ⟶^{N_{F^{rm}/F^{r}}} G^{F^{r}}/∼
N_{r}↑  ↑^{j}
G^{F^{rm}}/∼_{F} ⟶^{N_{F^{rm}/F}} G^{F}/∼,

where j is the map induced from the inclusion G^{F} ↪ G^{F^{r}}. We show (1.5.1). Let x̂ = G^{F^{rm}} and take α ∈ G such that x̂ = α^{−1}F (α). Then N_{F^{rm}/F} (x̂) is represented by x = F^{rm}(α)α^{−1}. On the other hand, since x̂′ = N_{r}(x̂) = α^{−1}F^{r}(α), we see that N_{F^{rm}/F^{r}}(x̂′) is represented by F^{rm}(α)α^{−1} which coincides with j(x). This shows the commutativity.

1.6. Let σ′ = F′|_{H^{F^{rm}}}, and H̃^{F^{rm}} be the semidirect product of H^{F^{rm}} with the cyclic group⟨σ′⟩ of order rm generated by σ′. For a character χ of G^{F^{rm}}, we define the character F (χ) by F (χ)(F (g)) = χ(g), and similarly for H. An irreducible character ψ of H^{F^{rm}} is F′-stable if and only if ψ is of the form that

ψ = χ ⊗ F (χ) ⊗ ⋯ ⊗ F^{r−1}(χ) (1.6.1)

for some F^{r}-stable irreducible character χ on G^{F^{rm}}. Let V_{i} (1 ≤ i ≤ r) be an irreducible G^{F^{rm}}-module for the irreducible character F^{i−1}(χ). Then there exists a linear isomorphism T_{i} : V_{i} → V_{i+1} such that T_{i} ∘ g = F (g) ∘ T_{i} for any g ∈ G^{F^{rm}} with V_{r+1} = V_{1} and that (T_{r}T_{r−1} ⋯ T_{1})^{m} = 1. Let ψ be as in (1.6.1). Then ψ is afforded by the H^{F^{rm}}-module V_{1} ⊗ V_{2} ⊗ ⋯ ⊗ V_{r}. Let us define an action of σ′ on V_{1} ⊗ ⋯ ⊗ V_{r} by

σ′ = ω ∘ (T_{1} ⊗ T_{2} ⊗ ⋯ ⊗ T_{r}),

where ω is the cyclic permutation of factors given by

ω(x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{r}) = x_{r} ⊗ x_{1} ⊗ ⋯ ⊗ x_{r−1}.

Then we have σ′ ∘ h = F′(h) ∘ σ′ for h ∈ H^{F^{rm}}, and so V_{1} ⊗ ⋯ ⊗ V_{r} can be extended to an H̃^{F^{rm}}-module. We denote by ψ̃ the corresponding extension of ψ to H̃^{F^{rm}}.

Let σ = F |_{G^{F^{rm}}}, and we consider G^{F^{rm}}⟨σ⟩ the semidirect product of G^{F^{rm}} with the cyclic group ⟨σ⟩ of order rm generated by σ. We define an action of σ^{r} on V_{1} by σ^{r} = T_{r}T_{r−1} ⋯ T_{1}. Then σ^{r} ∘ g = F^{r}(g) ∘ σ^{r} for any g ∈ G^{F^{rm}}, and the G^{F^{rm}}-module V_{1} can be extended to a G^{F^{rm}}⟨σ^{r}⟩-module Ṽ_{1}. We denote by χ̃ the corresponding extension of χ to G^{F^{rm}}⟨σ^{r}⟩. We show the following lemma.

Lemma 1.7. Let h = (g, 1, …, 1) ∈ H^{F^{rm}} with g ∈ G^{F^{rm}}. Let χ be an F^{r}-stable irreducible character of G^{F^{rm}}. Then for ψ = χ ⊗ F (χ) ⊗ ⋯ ⊗ F^{r−1}(χ) ∈ Irr H^{F^{rm}}, we have

ψ̃(hσ′) = χ̃(gσ^{r}).

Proof. Let v_{1}^{(1)}, …, v_{n}^{(1)} be a basis of V_{1}. We define a basis v_{1}^{(i+1)}, …, v_{n}^{(i+1)} of V_{i+1} inductively by v_{j}^{(i+1)} = T_{i}(v_{j}^{(i)}) for i = 1, 2, … r − 1. Then we have

T_{r}(v_{j}^{(r)}) = T_{r} ⋯ T_{1}(v_{j}^{(1)}) = σ^{r}v_{j}^{(1)}.

It follows that

hσ′ · v_{i_{1}}^{(1)} ⊗ v_{i_{2}}^{(2)} ⊗ ⋯ ⊗ v_{i_{r}}^{(r)} = (gσ^{r}v_{i_{r}}^{(1)}) ⊗ v_{i_{1}}^{(2)} ⊗ ⋯ ⊗ v_{i_{r−1}}^{(r)},

and we have

ψ̃(hσ′) = Tr (hσ′, V_{1} ⊗ ⋯ ⊗ V_{r}) = Tr (gσ^{r}, V_{1}) = χ̃(gσ^{r}).

This proves the lemma.

1.8. Let χ be an F^{r}-stable irreducible character of G^{F^{rm}}, and χ̃ be its extension to G^{F^{rm}}⟨σ^{r}⟩ as in the previous lemma. Let G^{F^{rm}}σ^{r}/ ∼ be set of conjugacy classes in G^{F^{rm}}⟨σ^{r}⟩ contained in the coset G^{F^{rm}}σ^{r}. Under the natural bijection G^{F^{rm}}/∼_{F^{r}}≃ G^{F^{rm}}σ^{r}/∼ via x ↔ xσ^{r}, we have an isomorphism C(G^{F^{rm}}/∼_{F^{r}}) ≃ C(G^{F^{rm}}σ^{r}/∼). Thus χ̃|_{G^{F^{rm}}σ^{r}} defines an element in the space C(G^{F^{rm}}/∼_{F^{r}}). Put

R(m) ; χ̃ = Sh_{F^{rm}/F^{r}}(χ̃|_{G^{F^{rm}}σ^{r}}).

Hence R(m) ; χ̃ is a class function on G^{F^{r}}. We have the following formula.

Proposition 1.9. Under the notation as above,

|G^{F^{rm}}|^{−1} ∑_{ĝ∈G^{F^{rm}}} χ̃(N_{r}(ĝ)σ^{r}) = |G^{F} |^{−1} ∑_{g∈G^{F}} R(m) ; χ̃(g). (1.9.1)

Proof. Take ĝ ∈ G^{F^{rm}}. Write ĝ as ĝ = α^{−1}F (α) and put g = F^{rm}(α)α^{−1}. Then g ∈ G^{F}, and we see that χ̃(N_{r}(ĝ)σ^{r}) = R(m) ; χ̃(g) by (1.5.1). Moreover, it is known that

♯{x ∈ G^{F^{rm}} | x^{−1}ĝF (x) = ĝ} = ♯{y ∈ G^{F} | y^{−1}gy = g}.

The formula (1.9.1) is immediate from these two facts.

1.10. Let c_{r}^{(m)}(χ̃) be the left hand side of (1.9.1), i.e.,

c_{r}^{(m)}(χ̃) = |G^{F^{rm}}|^{−1} ∑_{ĝ∈G^{F^{rm}}} χ̃(N_{r}(ĝ)σ^{r}). (1.10.1)

Then c_{r}^{(m)}(χ̃) is a generalization of the twisted Frobenius-Schur indicator discussed in Kawanaka and Matsuyama [KM]. In the case where m = 1, we simply write c_{r}^{(1)}(χ̃) as c_{r}(χ). Note that in this case, the extension does not enter the formula, and we have

c_{r}(χ) = |G^{F^{r}}|^{−1} ∑_{g∈G^{F^{r}}} χ(N_{r}(g)).

If r = 2, c_{2}(χ) coincides with the Frobenius-Schur indicator defined in [KM]. Let us define, for a class function f of G^{F^{r}},

m_{r}(f) =⟨f, Ind_{G^{F}}^{G^{F^{r}}} 1⟩ = |G^{F} |^{−1} ∑_{x∈G^{F}} f(x). (1.10.2)

Then the identity (1.9.1) can be rewritten as

c_{r}^{(m)}(χ̃) = m_{r}(R(m) ; χ̃). (1.10.3)

We note that (1.10.3) is a generalization of the formula due to Kawanaka [K2, (1.1)]. In fact, in the case where m = 1, the Shintani descent Sh_{F^{r}/F^{r}} coincides with the inverse of the twisting operator t_{1}^{∗} on C(G^{F^{r}}/∼) given in [K2], and so we have R_{χ}^{(1)} = t_{1}^{∗−1}χ. Then (1.10.3) implies the following.

Corollary 1.11. Let the notations be as above. Then we have c_{r}(χ) = m_{r}(t_{1}^{∗−1}χ).

In the case where r = 2, this formula is nothing but the formula (1.1) in [K2].

1.12. By Lemma 1.7, χ̃(N_{r}(ĝ)σ^{r}) = ψ̃(hσ′) with h = (N_{r}(ĝ), 1, …, 1) ∈ H^{F^{rm}}. As in 1.4, h is F′-conjugate to (ĝ, …, ĝ) ∈ Δ(H)^{F^{rm}}, and so

ψ̃(hσ′) = ψ̃((ĝ, …, ĝ)σ′).

On the other hand, under the isomorphism Δ(H)^{F^{rm}} ≃ G^{F^{rm}}, V_{1} ⊗ ⋯ ⊗ V_{r} is an G^{F^{rm}}-module, and its character χF (χ) ⋯ F^{r−1}(χ) is F -stable. Moreover, we have σ′ ∘ g = F (g) ∘ σ′ on V_{1} ⊗ ⋯ ⊗ V_{r} for any g ∈ G^{F^{rm}}. This implies that the action of σ′ defines a structure of G^{F^{rm}}⟨σ⟩-module on V_{1} ⊗ ⋯ ⊗ V_{r}, where σ acts by σ′ on it. We denote the character of this module by ψ̃_{0}, which is an extension of χF (χ) ⋯ F^{r−1}(χ). Thus, we have

ψ̃((ĝ, …, ĝ)σ′) = ψ̃_{0}(ĝσ).

Now (1.9.1) can be rewritten as

m_{r}(R(m) ; χ̃) = |G^{F^{rm}}|^{−1} ∑_{ĝ∈G^{F^{rm}}} ψ̃_{0}(ĝσ). (1.12.1)

1.13. Let k be a positive integer. We define an inner product on C(G^{F^{k}}σ/∼) by

⟨f, h⟩_{G^{F^{k}} σ} = |G^{F^{k}}|^{−1} ∑_{x∈G^{F^{k}}} f(xσ)\overline{h(xσ)}

for f, h ∈ C(G^{F^{k}}σ/∼). Then the following orthogonality relations are known. For any F -stable irreducible characters χ, χ′ of G^{F^{k}} and their extensions χ̃, χ̃′ to G^{F^{k}}⟨σ⟩,

(1.13.1) ⟨χ̃, χ̃′⟩_{G^{F^{k}} σ} = \begin{cases}θ(σ) & if χ̃ = θ ⊗ χ̃′ with θ ∈ Irr ⟨σ⟩, \\ 0 & if χ′ ≠ χ. \end{cases}

Here in the left hand side, χ̃, χ̃′ are regarded as functions on G^{F^{rm}}σ by restriction. For any f ∈ C(G^{F^{k}}σ/∼), we put

M̃_{k}(f) =⟨f, ˜1⟩ _{G^{F^{k}} σ} = |G^{F^{k}}|^{−1} ∑_{x∈G^{F^{k}}} f(xσ),

where ˜1 means the restriction of the unit character of G^{F^{k}}⟨σ⟩ to G^{F^{k}}σ. We also put, for a class function h of G^{F^{k}},

M_{k}(h) =⟨h, 1⟩ _{G^{F^{k}}} = |G^{F^{k}}|^{−1} ∑_{x∈G^{F^{k}}} h(x).

The following statement is immediate from (1.13.1).

(1.13.2) Let ρ be an F -stable character of G^{F^{k}}, and ρ̃ its extension to G̃^{F^{k}}. Then we have |M̃_{k}(ρ̃)| ≤ M_{k}(ρ). Moreover, if M_{k}(ρ) = 1, then M̃_{k}(ρ̃) is a k-th root of unity.

We have the following theorem.

Theorem 1.14. Let χ be an F^{r}-stable irreducible character of G^{F^{rm}}, and χ̃ an extension of χ to G^{F^{rm}}⟨σ^{r}⟩. Let ψ̃_{0} be the extension of χF (χ) ⋯ F^{r−1}(χ) to G^{F^{rm}}⟨σ⟩ as in

1.12. Put Sh_{F^{rm}/F^{r}}(χ̃|_{G^{F^{rm}}σ^{r}}) = R(m) ; χ̃.

(i) We have c_{r}^{(m)}(χ̃) = m_{r}(R(m) ; χ̃) = M̃_{rm}(ψ̃_{0}). In particular,

|m_{r}(R(m) ; χ̃)| ≤ M_{rm}(χF (χ) ⋯ F^{r−1}(χ)).

Furthermore, if M_{rm}(χF (χ) ⋯ F^{r−1}(χ)) = 1, we have |m_{r}(R(m) ; χ̃)| = 1. (ii) Assume that r = 2. Then there exists a 2m-th root of unity ζ such that

m_{2}(R(m) ; χ̃) = \begin{cases}ζ & if \overline{χ} = F (χ), \\ 0 & otherwise, \end{cases}

where \overline{χ} is the complex conjugate of the character χ.

Proof. The equality m_{r}(R(m) ; χ̃) = M̃_{rm}(ψ̃_{0}) in (i) follows from (1.12.1). The inequality in (i) follows from (1.13.2). Assume that r = 2. Then we have

M_{2m}(χF (χ)) =⟨χF (χ), 1⟩_{G^{F^{2m}}} =⟨F (χ), \overline{χ}⟩_{G^{F^{2m}}} = \begin{cases}1 & if F (χ) = \overline{χ}, \\ 0 & otherwise. \end{cases} .

So the assertion (ii) follows from (1.13.2). This proves the theorem.

1.15. In the case where r = 2, we determine the quantity ζ = c_{2}^{(m)}(χ̃) more explicitly. Let χ be an F^{2}-stable irreducible character of G^{F^{2m}} and χ̃ its extension to G^{F^{2m}}⟨σ^{2}⟩ as in the theorem. Let us assume that F (χ) = \overline{χ}. We follow the setting in 1.6. In particular V_{1} (resp. V_{2}) is a G^{F^{2m}}-module affording χ (resp. F (χ)). Since F (χ) = \overline{χ}, the subspace W = (V_{1} ⊗ V_{2})^{G^{F^{2m}}} of G^{F^{2m}}-invariant vectors in V_{1} ⊗ V_{2} is of dimension 1. The map σ′ : V_{1} ⊗ V_{2} → V_{1} ⊗ V_{2}, v_{1} ⊗ v_{2} ↦ T_{2}(v_{2}) ⊗ T_{1}(v_{1}) preserves the space W, and the eigenvalue of σ′ on W coincides with ζ = c_{2}^{(m)}(χ̃). The map σ^{2} = T_{2}T_{1} : V_{1} → V_{1} extends the G^{F^{2m}}-module V_{1} to the G^{F^{2m}}⟨σ^{2}⟩-module Ṽ_{1} affording the character χ̃.

The G^{F^{2m}}-module V_{2} can be identified with V_{1} by replacing the action of g ∈ G^{F^{2m}} by F (g). Under this identification, we may take T_{1} = Id_{V_{1}} and T_{2} = σ^{2} on V_{1}. Hence we have σ′(v_{1} ⊗ v_{2}) = σ^{2}(v_{2}) ⊗ v_{1}. Now the averaging operator V_{1} ⊗ V_{2} → W, v ↦ |G^{F^{2m}}|^{−1} ∑_{g∈G^{F^{2m}}} g · v determines a bilinear form B : V_{1} × V_{1} → Q̄_{l} (up to scalar) having the following properties.

(1.15.1)  B(g · v_{1}, F (g) · v_{2}) = B(v_{1}, v_{2}) for g ∈ G^{F^{2m}}, v_{1}, v_{2} ∈ V_{1}
B(σ^{2}(v_{2}), v_{1}) = ζB(v_{1}, v_{2}) for v_{1}, v_{2} ∈ V_{1}.

Conversely, if there exists such a bilinear form on V_{1}, this form coincides with B up to scalar. Hence ζ determines the value c_{2}^{(m)}(χ̃).

The extension χ̃ of χ is determined by the choice of T_{1}, T_{2} such that (T_{2}T_{1})^{m} = Id_{V_{1}}. If we replace T_{1} by a scalar multiple ξT_{1} for an m-th root of unity ξ, it gives a different extension of χ̃′ of χ. By changing χ̃ by χ̃′, the eigenvalue ζ of σ′ on W is replaced by ξζ. Summing up the above arguments, we have the following refinement of Theorem 1.14, which is a generalization of Theorem 2.1.3 in [K2].

Corollary 1.16. Let χ be an F^{2}-stable irreducible character of G^{F^{2m}} and χ̃ an extension of χ to G^{F^{2m}}⟨σ^{2}⟩.

(i) We have

c_{2}^{(m)}(χ̃) = \begin{cases}ζ & if F (χ) = \overline{χ}, \\ 0 & otherwise, \end{cases}

where ζ is an 2m-th root of unity. (ii) Assume that F (χ) = \overline{χ}. Let ζ_{0} be a primitive 2m-th root of unity in Q̄_{l}. Then

there exists a unique extension χ̃ of χ such that c_{2}^{(m)}(χ̃) = 1 or ζ_{0}. Let V_{1} be the G^{F^{2m}}⟨σ^{2}⟩-module affording χ̃. Then c_{2}^{(m)}(χ̃) = 1 (resp ζ_{0}) if and only if there exists a non-zero bilinear form B(·, ·) on V_{1} satisfying (1.15.1) with ζ = 1 (resp. ζ = ζ_{0}).

1.17. In the case where G is a connected reductive group with connected center, Lusztig defined in [L1] almost characters of G^{F}. In the case where G is a special linear group SL_{n} with F of split type, almost characters are also formulated in [S3]. In either case, the set of almost characters coincides with the set of Sh_{F^{m}/F} (χ̃|_{G^{F^{m}}σ}), up to an m-th root of unity multiple, for sufficiently divisible m, where χ runs over all the F -stable irreducible characters of G^{F^{m}}. We denote by R_{χ} the almost character of G^{F} corresponding to χ. As a corollary to Theorem 1.14, we have the following result.

Corollary 1.18. Assume that G is either a connected reductive group with connected center, or SL_{n} with F of split type. Let R_{χ} be the almost character of G^{F^{2}} associated to an F^{2}-stable irreducible character χ of G^{F^{2m}}. Then we have

(1.18.1) m_{2}(R_{χ}) = \begin{cases}ζ & if F (χ) = \overline{χ}, \\ 0 & otherwise, \end{cases}

where ζ is a certain 2m-th root of unity.

Remark 1.19. In [L3, Prop. 7.2], Lusztig proved a formula concerning the characteristic functions of character sheaves as follows. Let A be an F^{2}-stable character sheaf of a connected reductive group G. We denote by χ_{A,ϕ_{A}} ∈ C(G^{F^{2}}/∼) the characteristic function of A with respect to an isomorphism ϕ_{A} : (F^{2})^{∗}A \underrightarrow{∼} A. Then under the assumption that q is sufficiently large (and that χ_{A,ϕ_{A}} can be written as a linear combination of irreducible characters with cyclotomic integers coefficients), there exists a choice of ϕ_{A} such that

(1.19.1) m_{2}(χ_{A,ϕ_{A}}) = \begin{cases}(−1)^{dim supp A} & if F^{∗}(A) ≃ DA, \\ 0 & otherwise, \end{cases}

where DA is the Verdier dual of A. Since the proof depends on the asymptotic behavior of q → ∞, the condition on q is considerably large. In the case where G has a connected center, using the description of m_{2}(χ) for any irreducible character χ of G^{F^{2}} in [L3], (1.18.1) can be verified directly. In [S2], it was shown that almost characters coincide with the characteristic functions of character sheaves whenever G has a connected center. A similar result was also shown in [S4] for SL_{n} with F of split type. Hence the formula (1.18.1) is a counter part of (1.19.1) to almost characters, which works without any assumption on q. Also, Theorem 1.14 (ii) is regarded as an extension of (1.19.1) to arbitrary connected algebraic groups.

1.20. As a special case of the situation discussed in Theorem 1.14 (i), we consider the case where G = GL_{n} with the standard or non-standard Frobenius map F over F_{q}. Irreducible characters of G^{F^{r}} are described as follows. Let G^{∗} ≃ GL_{n} be the dual group of G. For each F^{r}-stable semisimple class {s}, choose a representative s ∈ G^{∗F^{r}}. Let T^{∗} be a maximally split maximal torus in Z_{G^{∗}}(s). Let W = N_{G^{∗}}(T^{∗})/T^{∗} be the Weyl group of G^{∗}, and put W_{s} = {w ∈ W | w(s) = s}. Then W_{s} is the Weyl group of Z_{G^{∗}}(s), and F^{r} acts naturally on W_{s}, which we denote by δ. Let (Irr W_{s})^{δ} be the set of F^{r}-stable irreducible representations of W_{s}. For each E ∈ (Irr W_{s})^{δ}, we fix an extension Ẽ of E to the semidirect group W_{s}⟨δ⟩, where ⟨δ⟩ is the infinite cyclic group with generator δ. Put

R_{s,Ẽ} = |W_{s}|^{−1} ∑_{w∈W_{s}} Tr (wδ, Ẽ)R_{T_{w}^{∗}}(s),

where R_{T_{w}^{∗}}(s) denotes the Deligne-Lusztig character R_{T_{w}}(θ) under the natural correspondence (s, T_{w}^{∗}) ↔ (θ, T_{w}).

It is known, under a suitable choice of the extension, ±R_{s,Ẽ} gives rise to an irreducible character of G^{F^{r}}, which we denote by ρ_{s,E}. Then the set Irr G^{F^{r}} of irreducible characters of G^{F^{r}} is given as

Irr G^{F^{r}} = ∐_{{s}}{ρ_{s,E} | E ∈ (Irr W_{s})^{δ}},

where {s} runs over F^{r}-stable semisimple conjugacy classes in G^{∗}.

Let (s, T^{∗}) be as above. We choose an F^{r}-stable maximal torus T of G which is dual to T^{∗}, and let B be a Borel subgroup of G containing T. We choose an integer m > 0 such that F^{mr} leaves B invariant. One can find a linear character θ of T^{F^{rm}} corresponding to s ∈ T^{∗F^{rm}}. Then we have

End_{G^{F^{rm}}}(Ind_{B^{F^{rm}}}^{G^{F^{rm}}} θ̃) ≃ Q̄_{l}[W_{s}],

where θ̃ is the lift of θ to the linear character of B^{F^{rm}}. Let us denote by χ_{θ,E} the irreducible constituent of Ind_{B^{F^{rm}}}^{G^{F^{rm}}} θ̃ corresponding to E ∈ Irr W_{s}. Then χ_{θ,E} is F^{r}- stable if and only if E ∈ (Irr W_{s})^{δ}, and in which case, Sh_{F^{rm}/F^{r}}(χ̃_{θ,E}|_{G^{F^{rm}}σ^{r}}) coincides with ρ_{s,E} up to a scalar multiple. Thus under this setting, Theorem 1.14 (i) can be rewritten as follows.

Corollary 1.21. Let G = GL_{n} with the standard or non-standard Frobenius map F. Then for each ρ_{s,E} ∈ Irr G^{F^{r}}, we have

m_{r}(ρ_{s,E}) ≤ M_{rm}(χ_{θ,E}F (χ_{θ,E}) ⋯ F^{r−1}(χ_{θ,E})).

Moreover, if M_{rm}(χ_{θ,E}F (χ_{θ,E}) ⋯ F^{r−1}(χ_{θ,E})) = 1, we have m_{r}(ρ_{s,E}) = 1.

2. Parametrization of irreducible characters of SL_{n}(F_{q^{2}})

2.1. In the remainder of this paper, we assume that G̃ = GL_{n} and G = SL_{n} with Frobenius maps F with respect to the standard F_{q}-structures. We assume that p is large enough so that the results in [S3] can be applied. (Although there was no assumption for p in [S3], it should be changed. Actually, Kawanaka’s construction of generalized Gelfand-Graev characters of GL_{n} or SL_{n} requires no assumptions for p. However, our construction (cf. [S3, 2.3]) depends on the Dynkin-Kostant theory, which requires p is not too small.) For example p ≥ 2n is enough in our case. Let G̃^{∗} (resp. G^{∗}) be the dual group of G̃ (resp. G). Then G̃^{∗} ≃ GL_{n}, and G^{∗} ≃ G̃^{∗}/Z̃^{∗}, where Z̃^{∗} is the center of G̃^{∗}. The inclusion map G ↪ G̃ induces a natural surjection π : G̃^{∗} → G^{∗}. As in the case of G̃, the set Irr G^{F^{2}} is partitioned as

Irr G^{F^{2}} = ∐_{{s}} E(G^{F^{2}}, {s}),

where {s} runs over F^{2}-stable semisimple classes in G^{∗}. Take s such that F^{2}(s) = s. Let T^{∗} be an F^{2}-stable maximal torus of Z_{G^{∗}}(s) such that T^{∗} is contained in an F^{2}- stable Borel subgroup of Z_{G^{∗}}(s). Let T̃^{∗} be an F^{2}-stable maximal torus of G̃^{∗} such that π(T̃^{∗}) = T^{∗}. Then W = NG̃^{∗}(T̃^{∗})/T̃^{∗} is naturally identified with N_{G^{∗}}(T^{∗})/T^{∗}. Put

W_{s} = N_{Z_{G^{∗}}(s)}(T^{∗})/T^{∗}, W_{s}^{0} = N_{Z_{G^{∗}}^{0}(s)}(T^{∗})/T^{∗}.

Then W_{s}^{0} is the Weyl group of Z_{G^{∗}}^{0}(s). Now W_{s} can be decomposed as W_{s} ≃ W_{s}^{0} ⋊ Ω_{s}, where Ω_{s} = Z_{G^{∗}}(s)/Z_{G^{∗}}^{0}(s) is a cyclic group. If we choose ṡ ∈ T̃^{∗} such that π(ṡ) = s, then W_{s}^{0} is naturally identified with Wṡ = {w ∈ W | w(ṡ) = ṡ}.

F^{2} acts naturally on W_{s}. We denote by δ this action and consider the semidirect product W_{s}⟨δ⟩, where δwδ^{−1} = F^{2}(w). δ stabilizes W_{s}^{0} and Ω_{s}.

2.2. For each E ∈ Irr W_{s}^{0}, let Ω_{s}(E) be the stabilizer of E in Ω_{s}. Assume that the Ω_{s}-orbit of E is δ-stable. Put

Ω̃_{s}(E) = {u ∈ Ω_{s} | ^{uδ}E = E}.

Then one can write Ω̃_{s}(E) = Ω_{s}(E)a for some a ∈ Ω_{s}. Since Ω_{s} is abelian, Ω_{s}(E) is δ-stable, and Ω_{s}(E) acts on Ω̃_{s}(E) by (z, u) ↦ z^{−1}uδ(z) for z ∈ Ω_{s}(E) and u ∈ Ω̃_{s}(E). We denote by Ω̃_{s}(E)_{δ} the set of equivalent classes under this action. It is easy to see that Ω̃_{s}(E)_{δ} can be identified with the set Ω_{s}(E)_{δ}a, where Ω_{s}(E)_{δ} is the largest quotient of Ω_{s}(E) on which δ acts trivially. Let \overline{Irr} W_{s}^{0} be the set of Ω_{s}-orbits in the set Irr W_{s}^{0}. We denote by (\overline{Irr} W_{s}^{0})^{δ} the set of δ-stable orbits in Irr W_{s}^{0}. By abuse of notation, (\overline{Irr} W_{s}^{0})^{δ} means also a set of representatives for the δ-stable Ω_{s}-orbits in Irr W_{s}^{0}.

For each pair (s, E) with E ∈ (\overline{Irr} W_{s}^{0})^{δ}, put

\overline{M}_{s,E} = Ω_{s}^{δ}(E)^{∧} × Ω̃_{s}(E)_{δ},

where Ω_{s}^{δ} = {u ∈ Ω_{s}, δ(u) = u} and Ω_{s}^{δ}(E) is the stabilizer of E in Ω_{s}^{δ}, and Ω_{s}^{δ}(E)^{∧} is the set of irreducible characters of Ω_{s}^{δ}(E). It is known by [S3] that there exists a natural bijection

E(G^{F^{2}}, \{s\}) = ∐_{E∈(\overline{Irr} W_{s}^{0})^{δ}} \overline{M}_{s,E} (2.2.1)

We denote by ρ_{η,z} the irreducible character of G^{F^{2}} corresponding to (η, z) ∈ \overline{M}_{s,E}.

The above parametrization satisfies the following properties. The set of G^{∗F^{2}}- conjugacy classes in the set {s}^{F^{2}} is in bijection with (Ω_{s})_{δ}. For each x ∈ (Ω_{s})_{δ}, take a representative ẋ ∈ Ω_{s}, and let ẍ ∈ Z_{G^{∗}}(s) be a representative of ẋ ∈ Ω_{s} = Z_{G^{∗}}(s)/Z_{G^{∗}}^{0}(s). Choose g_{x} ∈ G^{∗} such that g_{x}^{−1}F^{2}(g_{x}) = ẍ, and put s_{x} = g_{x}sg_{x}^{−1}. Then F^{2}(s_{x}) = s_{x}, and g_{x}T^{∗}g_{x}^{−1} = T_{x}^{∗} is a maximally split torus in Z_{G^{∗}}(s_{x}). We define W_{s_{x}}^{0} in a similar way as W_{s}^{0}. Under the isomorphism W_{s}^{0} → W_{s_{x}}^{0} induced by ad g_{x}, the action of ẋδ on W_{s}^{0} is transferred to the action of F^{2} on W_{s_{x}}^{0}. Hence each ẋδ-stable irreducible character E′ of W_{s}^{0} determines the F^{2}-stable irreducible character E′′ of W_{s_{x}}^{0}. Take an F^{2}-stable element ṡ_{x} such that π(ṡ_{x}) = s_{x}. We consider the irreducible character ρ_{ṡ_{x},E′′} of G̃^{F}^{2} as in 1.20, which we denote by ρ_{ṡ_{x},E′}, by abuse of the notation.

It is known by [S3, (4.4.2)] that there exists a natural bijection

(2.2.2) f : ∐_{E∈(\overline{Irr} W_{s}^{0})^{δ}} Ω̃_{s}(E)_{δ} ≃ ∐_{x∈(Ω_{s})_{δ}}(Irr W_{s}^{0})^{ẋδ}/Ω_{s}^{δ},

where in the right hand side, (Irr W_{s}^{0})^{ẋδ}/Ω_{s}^{δ} means the set of Ω_{s}^{δ}-orbits of ẋδ-stable irreducible characters of W_{s}^{0}. The bijection is described as follows. Take E in a δstable Ω_{s}-orbit in Irr W_{s}^{0}. For each ẏ ∈ Ω̃_{s}(E), there exists ẋ ∈ Ω_{s} and z ∈ Ω_{s} such that ẏ = z^{−1}ẋδ(z), where ẋ is one of the representatives of (Ω_{s})_{δ} chosen above. Then E_{x} = ^{z}E ∈ (Irr W_{s}^{0})^{ẋδ}. The correspondence (E, y) ↦ (x, E_{x}) gives rise to the required bijection f.

Under the above setting, we have

ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}} = ∑_{η∈Ω_{s}^{δ}(E)^{∧}} ρ_{η,y}. (2.2.3)

Let T_{s_{x},E_{x}} be the set of irreducible characters occurring in the restriction of ρ_{ṡ_{x},E_{x}} to G^{F^{2}}. We also denote by \overline{T}_{s,E} the set of ρ_{η,y} for (η, y) ∈ \overline{M}_{s,E}. Then (2.2.2) implies that

\overline{T}_{s,E} = ∐_{(x,E_{x})} T_{s_{x},E_{x}},

where (x, E_{x}) runs over all the pairs corresponding to (y, E) with y ∈ Ω̃_{s}(E)_{δ} under the map f.

Remark 2.3. In [S3, 4.5], the parameter set \overline{M}_{s,E} is defined as Ω_{s}^{δ}(E)^{∧} × Ω_{s}(E)_{δ}. Since Ω̃_{s}(E)_{δ} = Ω_{s}(E)_{δ}a, this set is in bijection with \overline{M}_{s,E} in this paper. However, the bijection depends on the choice of a ∈ Ω_{s}, and the definition of \overline{M}_{s,E} in this paper is more convenient for later applications.

2.4. We describe the decomposition of ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}} in (2.2.3) more precisely. It is known by [L2] that T_{s_{x},E_{x}} is in bijective correspondence with Ω_{s}^{δ}(E)^{∧}. This bijection is given as follows. The abelian group G̃^{F^{2}}/G^{F^{2}} acts transitively on T_{s_{x},E_{x}} by the conjugation action. Also its dual group (G̃^{F^{2}}/G^{F^{2}})^{∧} acts on Irr G̃^{F^{2}} by (θ, ρ̃) ↦ θ ⊗ ρ̃ for a linear character θ ∈ (G̃^{F^{2}}/G^{F^{2}})^{∧} and ρ̃ ∈ Irr G̃^{F^{2}}. Then for ρ_{0} ∈ T_{s_{x},E_{x}}, the stabilizer of ρ_{0} in G̃^{F^{2}}/G^{F^{2}} and the stabilizer of ρ_{ṡ_{x},E_{x}} in (G̃^{F^{2}}/G^{F^{2}})^{∧} are orthogonal to each other under the natural duality pairing G̃^{F^{2}}/G^{F^{2}} ×(G̃^{F^{2}}/G^{F^{2}})^{∧} → Q̄_{l} (cf. [L2, 9]). Let I(ρ_{ṡ_{x},E_{x}}) be the stabilizer of ρ_{ṡ_{x},E_{x}} in (G̃^{F^{2}}/G^{F^{2}})^{∧}. Then, under the choice of ρ_{0}, the set T_{s_{x},E_{x}} is in natural bijection with I(ρ_{ṡ_{x},E_{x}})^{∧}.

We show that I(ρ_{ṡ_{x},E_{x}}) is isomorphic to Ω_{s}^{δ}(E). First note that there exists a natural isomorphism

Z̃^{∗F^{2}} ≃ Hom(G̃^{F^{2}}/G^{F^{2}}, Q̄_{l}^{∗}) = (G̃^{F^{2}}/G^{F^{2}})^{∧}. (2.4.1)

If z is an element in Z̃^{∗F^{2}} corresponding to θ ∈ (G̃^{F^{2}}/G^{F^{2}})^{∧} under the above isomorphism, then θ maps E(G̃^{F^{2}}, {ṡ_{x}}) onto E(G̃^{F^{2}}, {zṡ_{x}}). Put

Z̃_{s_{x}}^{∗F^{2}} = {z ∈ Z̃^{∗F^{2}} | zṡ_{x} is conjugate to ṡ_{x} under G̃^{∗}},

which does not depend on the choice of ṡ_{x} for s_{x}. Then, under the identification in (2.4.1), I(ρ_{ṡ_{x},E_{x}}) is regarded as a subgroup of Z̃_{s_{x}}^{∗F^{2}}. Here we have a natural isomorphism

ω_{s_{x}} : Ω_{s}^{δ} = Ω_{s}^{xδ} ≃ Z_{G^{∗}}(s)^{ẍF^{2}}/Z_{G^{∗}}^{0}(s)^{ẍF^{2}} → Z̃_{s_{x}}^{∗F^{2}} (2.4.2)

defined as follows. For z̄ ∈ Ω_{s}^{xδ}, take a representative z ∈ Z_{G^{∗}}(s)^{ẍF^{2}} of z̄, and choose ż ∈ G̃^{∗ẍF^{2}} such that π(ż) = z. Then ^{g_{x}}(ṡ^{−1}żṡż^{−1}) ∈ Z̃_{s_{x}}^{∗F^{2}}, and the map z̄ ↦ ^{g_{x}}(ṡ^{−1}żṡż^{−1}) induces a well-defined isomorphism ω_{s_{x}} since π(ZG̃^{∗}(ṡ_{x})) = Z_{G^{∗}}^{0}(s_{x}). Now under the identification in (2.4.1), (2.4.2), we may see that I(ρ_{ṡ_{x},E_{x}}) is a subgroup of Ω_{s}^{δ}, and in fact, I(ρ_{ṡ_{x},E_{x}}) coincides with the stabilizer of E_{x} in Ω_{s}^{δ}. Thus we have I(ρ_{ṡ_{x},E_{x}}) = Ω_{s}^{δ}(E_{x}) = Ω_{s}^{δ}(E).

2.5. The bijection between T_{s_{x},E_{x}} and Ω_{s}^{δ}(E)^{∧} given in 2.4 depends on the choice of ρ_{0} ∈ T_{s_{x},E_{x}}. We have to choose a specific ρ_{0} for each T_{s_{x},E_{x}}. This problem is reduced to a certain special case, and is solved by the aid of generalized Gelfand-Graev characters.

Let 𝔤 be the Lie algebra of G with Frobenius map F. We have a bijection log : G_{uni} → 𝔤_{nil} by v ↦ v − 1, where G_{uni} (resp. 𝔤_{nil}) is the unipotent variety of G (resp. nilpotent variety of 𝔤). Let N be a nilpotent element in 𝔤^{F}. By Dynkin-Kostant theory, there exists a natural grading 𝔤 = ⨁_{i∈Z} 𝔤_{i} associated to N. Let 𝔲_{i} = ⨁_{j≥i} 𝔤_{j}. Then one can find an F -stable parabolic subgroup P = LU_{1} associated to N, where L is an F -stable Levi subgroup of P with Lie L = 𝔤_{0}, and U_{1} is the unipotent radical of P with Lie U_{1} = 𝔲_{1}. Moreover we have N ∈ 𝔤_{2}. Let k be an algebraic closure of F_{q}. It is known by Kawanaka (see [K1]), that there exists an F -stable subspace 𝔲 (𝔲_{1.5} in the notation of [S3]) of 𝔲_{1} containing 𝔲_{2} and an F -equivariant linear map λ : 𝔲 → k satisfying the following. There exists an F -stable connected unipotent subgroup U of U_{1} such that log(U) = 𝔲 and that the map λ ∘ log : U → k turns out to be an F -stable homomorphism of U. We define a linear character Λ_{N} of U^{F^{2}} by Λ_{N} = ψ_{2} ∘ λ ∘ log, where ψ_{2} : F_{q^{2}} → Q̄_{l}^{∗} is the additive character defined by ψ_{2} = ψ ∘ Tr _{F_{q^{2}}/F_{q}} for a non-trivial additive character ψ : F_{q} → Q̄_{l}^{∗}. The generalized Gelfand-Graev character Γ_{N} of G^{F^{2}} associated to N is defined as

Γ_{N} = Ind_{U^{F^{2}}}^{G^{F^{2}}} Λ_{N}.

The character Γ_{N} depends only on the G^{F^{2}}-conjugacy class of N.

We now consider the following special setting for the set \overline{M}_{s,E} determined by the pair (s, E).

(2.5.1) W_{s}^{0} is isomorphic to S_{b} × ⋯ × S_{b} (t-times) with b = n/t, and Ω_{s} ≃⟨w_{0}⟩, where w_{0} is an element of order t in W_{s} permuting the factors of W_{s}^{0} transitively. Moreover, E ∈ Irr W_{s}^{0} is of the form

E = E_{1} ⊠ E_{1} ⊠ ⋯ ⊠ E_{1} with E_{1} ∈ Irr S_{b}.

Then E is Ω_{s}-stable, and we have Ω_{s} = Ω_{s}(E).

Now it is known by Lusztig ([L1, 13.4], see [S3, 2.9] for a brief description) that there exists a map ρ ↦ O_{ρ} from Irr G̃^{F} to the set of nilpotent orbits in Lie G̃. We take N ∈ 𝔤^{F} a nilpotent element (in Jordan normal form) contained in O_{ρ} for ρ = ρ_{ṡ,E}. (To be more explicit, O_{ρ} is a nilpotent orbit corresponding to the partition of n dual to μ ∪ ⋯ ∪ μ (t-times), where μ is a partition of n/t corresponding to E_{1}). Then it is known that there exists a unique irreducible character ρ_{0} such that ρ_{0} occurs both in Γ_{N} and in ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}. By using this ρ_{0}, one obtains a bijection T_{s_{x},E_{x}} ↔ Ω_{s}^{δ}(E)^{∧} as in 2.4. This is the parametrization given in (2.2.3), where if (x, E_{x}) corresponds to (E, y) by (2.2.2), then ρ_{η,y} corresponds to η ∈ Ω_{s}^{δ}(E)^{∧}.

By the arguments in [S3, 4.5], the parametrization of T_{s_{x},E_{x}} in the general case is reduced to the case given in (2.5.1). Accordingly, ρ_{0} is determined for each T_{s_{x},E_{x}}. However, note that this parametrization still depends on the choice of a nilpotent element N in 𝔤. In what follows, we assume that

(2.5.2) Each nilpotent element N ∈ 𝔤^{F} is taken to be a Jordan normal form.

2.6 In order to apply the results in section 1, we need to know the condition when F (ρ) = \overline{ρ} for an irreducible character ρ of G^{F^{2}}. We return to the setting in 2.2, and further assume that F (s) = s^{−1}. Then F acts on W_{s}, preserving W_{s}^{0} and Ω_{s}. We denote this action by γ, so that γ^{2} = δ. Note that if ρ′ belongs to \overline{M}_{s,E}, then \overline{ρ}′ belongs to \overline{M}_{s^{−1},E} since E ∈ Irr W_{s}^{0} is self dual. Also F (ρ′) belongs to \overline{M}_{F (s),F (E)}. Hence if ρ as above belongs to \overline{M}_{s,E}, the Ω_{s}-orbit of E turns out to be γ-stable. It follows that γ leaves Ω̃_{s}(E) invariant, and induces an action on Ω̃_{s}(E)_{δ}. We denote by Ω̃_{s}(E)_{δ}^{γ} the set of γ-fixed points in Ω̃_{s}(E)_{δ}. γ acts also on the set Ω_{s}^{δ}(E)^{∧}. We denote by Ω_{s}^{δ}(E)_{−γ}^{∧} the set of η ∈ Ω_{s}^{δ}(E)^{∧} such that γ(η) = \overline{η}. We put, for E ∈ (\overline{Irr} W_{s}^{0})^{γ},

\overline{M}_{s,E}^{0} = Ω_{s}^{δ}(E)_{−γ}^{∧} × Ω̃_{s}(E)_{δ}^{γ}.

We have the following proposition.

Proposition 2.7. Let ρ_{η,y} be the irreducible character of G^{F^{2}} corresponding to (η, y) ∈ \overline{M}_{s,E}. Assume that F (s) = s^{−1}.

(i) If the Ω_{s}-orbit of E is not γ-stable, then F (ρ_{η,y}) ≠ \overline{ρ}_{η,y}.

(ii) Assume that the Ω_{s}-orbit of E is γ-stable. Then F (ρ_{η,y}) = \overline{ρ}_{η,y} if and only if (η, y) ∈ \overline{M}_{s,E}^{0}.

The proposition will be proved in 2.11 after some preliminaries. First we note that

Lemma 2.8. For each N, we have F (Γ_{N}) = Γ_{N}, and \overline{Γ_{N}} = Γ_{N}.

Proof. The fact that F (Γ_{N}) = Γ_{N} can be checked directly for any N ∈ 𝔤^{F} since U^{F^{2}} is F -stable and Λ_{N} is also F -stable. On the other hand, it follows from the definition that we have \overline{Γ_{N}} = Γ_{−N}. So, in order to show the lemma, it is enough to see that N is conjugate to −N under G^{F^{2}}. Since N is given by a Jordan normal form, this is reduced to the case where N is regular nilpotent. Assume that N is a regular nilpotent element given in the Jordan normal form. There exists g = diag(a, −a, …, (−1)^{n−1}a) ∈ G̃ such that gNg^{−1} = −N. Then g ∈ G^{F^{2}} if and only if a ∈ F_{q^{2}} and (−1)^{k}a^{n} = 1 with k = [n/2]. We can set a = 1 if k is even, and set a = −1 if k is odd and n is odd. So, assume that n is even and k is odd, i.e., n = 2k. In this case, we may take a ∈ F_{q^{2}} such that a^{2} = −1. Thus we can always find g ∈ G^{F^{2}}, and the lemma follows.

As a corollary, we have

Corollary 2.9. Let ρ_{ṡ_{x},E_{x}} ∈ Irr G̃^{F^{2}} and ρ_{0} ∈ Irr G^{F^{2}} be as in 2.5. Assume that F (ρ_{ṡ_{x},E_{x}})|_{G^{F^{2}}} = \overline{ρ_{ṡ_{x},E_{x}}}|_{G^{F^{2}}}. Then we have F (ρ_{0}) = \overline{ρ_{0}}.

Proof. The parametrization of Irr G^{F^{2}} in terms of the set \overline{M}_{s,E} is reduced to the special case where \overline{M}_{s,E} is given by (2.5.1) through the steps (b) and (c) in [S3, 4.5]. Since the steps (b) and (c) are compatible with the F action and with taking duals, the assertion is reduced to the case of (2.5.1). In this case, we have F (\overline{Γ_{N}}) = Γ_{N} by Lemma 2.8. Note that the F -action and taking duals preserve the inner product. Since ρ_{0} is the unique irreducible character such that

⟨Γ_{N}, ρ_{0}⟩_{G^{F^{2}}} =⟨ρ_{s_{x},E_{x}}, ρ_{0}⟩_{G^{F^{2}}} = 1,

the corollary follows.

Lemma 2.10. Assume that the set \overline{M}_{s,E} satisfies the assumption of Proposition 2.7 (ii). Take y ∈ Ω̃_{s}(E)_{δ} and assume that (E, y) ↔ (x, E_{x}) under the map in (2.2.2). Then F (ρ_{ṡ_{x},E_{x}})|_{G^{F^{2}}} = \overline{ρ_{ṡ_{x},E_{x}}}|_{G^{F^{2}}} if and only if y ∈ Ω̃_{s}(E)_{δ}^{γ}.

Proof. We may choose ẏ ∈ Ω̃_{s}(E) as a representative of x ∈ (Ω_{s})_{δ}, so we may assume that E_{x} = E. Then T_{s_{x},E_{x}} = T_{s_{y},E} corresponds to the set Ω_{s}^{δ}(E)^{∧} × {y} under the correspondence T_{s,E} ↔ \overline{M}_{s,E} (cf. (2.2.3)). It is easy to see, for any pair (s_{1}, E_{1}), that F (ρ_{ṡ_{1},E_{1}}) = ρ_{F (ṡ_{1}),F (E_{1})}, where F (E_{1}) is the character of W_{F (s_{1})} corresponding to E_{1} under the isomorphism W_{s_{1}} ≃ W_{F (s_{1})}. On the other hand, we have \overline{ρ_{ṡ_{1},E_{1}}} = ρ_{ṡ_{1}^{−1},E_{1}} since W_{s_{1}} = W_{s_{1}^{−1}} and E_{1} is self dual. It follows that F (\overline{ρ_{ṡ_{y},E}}) = ρ_{F (ṡ_{y}^{−1}),F (E)}. By our assumption, F (s^{−1}) = s. Hence we have F (s_{y}^{−1}) ∈ T_{γ(y)}^{∗} and F (E) ∈ (Irr W_{s}^{0})^{γ(y)δ}. This implies that F (\overline{T_{s_{y},E}}) = T_{s_{γ(y)},F (E)} = T_{s_{γ(y)},^{u}E} for some u ∈ Ω_{s} since the Ω_{s}-orbit of E is F -stable. Since T_{s_{γ(y)},^{u}E} = T_{s_{y},E} if and only if γ(y) = y in Ω̃_{s}(E)_{δ}, the lemma is proved.

2.11. We shall prove Proposition 2.7. The assertion (i) follows from 2.6. We show (ii). Take (η, y) ∈ \overline{M}_{s,E}. If γ(y) ≠ y, then F (ρ_{η,y}) ≠ \overline{ρ_{η,y}} by Lemma 2.10. So, assume that y ∈ Ω̃_{s}(E)_{δ}^{γ}. Let ρ_{ṡ_{x},E_{x}} be the character of G̃^{F^{2}} containing ρ_{η,y}. Again by Lemma 2.10, we have F (ρ_{ṡ_{x},E_{x}}) = \overline{ρ_{ṡ_{x},E_{x}}}. Let ρ_{0} ∈ G^{F^{2}} be as in 2.5. Then by Corollary 2.9, we have F (ρ_{0}) = \overline{ρ}_{0}. If we write ρ_{η,y} = ^{g}ρ_{0} with g ∈ G̃^{F^{2}}, we have F (\overline{ρ}_{η,y}) = ^{F (g)}ρ_{0}. Now the action of F induces an action on (G̃^{F^{2}}/G^{F^{2}})^{∧} which is compatible with the natural pairing G̃^{F^{2}}/G^{F^{2}} × (G̃^{F^{2}}/G^{F^{2}})^{∧} → Q̄_{l}^{∗}. Then F stabilizes the subgroup I(ρ_{ṡ_{x},E_{x}}).

The arguments in 2.4 shows that the condition F (ρ_{η,y}) = \overline{ρ}_{η,y} is described by investigating the action of F on I(ρ_{ṡ_{x},E_{x}}). We follow the notation in 2.4. I(ρ_{ṡ_{x},E_{x}}) is regarded as a subgroup of Z̃^{∗F^{2}}. If we denote by ω̃_{s_{x}} the map Ω_{s}^{δ} → Z̃^{∗F^{2}} obtained as the composite of ω_{s_{x}} and the inclusion Z̃_{s_{x}}^{∗F^{2}} ↪ Z̃^{∗F^{2}}, then ω̃_{s_{x}}(Ω_{s}^{δ}(E_{x})) coincides with I(ρ_{ṡ_{x},E_{x}}). We note the following.

(2.11.1) Assume that x ∈ (Ω_{s})_{δ} is γ-stable. Then the following diagram commutes.

Ω_{s}^{δ} \underrightarrow{ω̃_{sx}} Z̃^{∗F^{2}}
−γ↓  ↓^{F =γ}
Ω_{s}^{δ} \underrightarrow{ω̃_{sx}} Z̃^{∗F^{2}},

where −γ : Ω_{s}^{δ} → Ω_{s}^{δ} is the map defined by z ↦ γ(z)^{−1}.

We show (2.11.1). Take z ∈ Ω_{s}^{δ}. Since F (s) = s^{−1}, we have
γ(ω̃_{s_{x}}(z)) = F (^{g_{x}}(ṡ^{−1}żṡż^{−1})) = ^{F (g_{x})}(ṡF (ż)ṡ^{−1}F (ż)^{−1}).

On the other hand, since x is γ-stable, ω̃_{s_{x}} coincides with ω̃_{s_{γ(x)}}, and we have

ω̃_{s_{x}}(−γ(z)) = ^{F (g_{x})}(ṡ^{−1}F (ż)^{−1}ṡF (ż)) = ^{F (g_{x})}(ṡF (ż)ṡ^{−1}F (ż)^{−1}).

since ṡ^{−1}F (ż)^{−1}ṡF (ż) is in the center Z̃^{∗} of G̃^{∗}. Hence (2.11.1) holds.

Now (2.11.1) shows that the F -action on I(ρ_{ṡ_{x},E_{x}}) is transferred to the −γ action on Ω_{s}^{δ}(E). Hence under the parametrization T_{s_{x},E_{x}} ↔ Ω_{s}^{δ}(E) × {y} given by ρ_{η,y} ↔ (η, y), we see that F (ρ_{η,y}) = \overline{ρ_{η,y}} if and only if η is −γ stable, i.e., η ∈ Ω_{s}^{δ}(E)_{−γ}^{∧}. This proves the proposition.

3. Almost characters of SL_{n}(F_{q^{2}})

3.1. We shall parametrize F^{2}-stable irreducible characters of G^{F^{2m}} for a sufficiently divisible integer m. Let s be an F^{2}-stable semisimple element in G^{∗}. We assume that m is large enough so that F^{2m} acts trivially on W_{s}^{0} and Ω_{s}. We denote by \overline{M}_{s,E}^{(m)} the set parametrizing irreducible characters of G^{F^{2m}} corresponding to \overline{M}_{s,E}

in the previous section. Hence, \overline{M}_{s,E}^{(m)} = Ω_{s}(E)^{∧} × Ω_{s}(E). Since s is F^{2}-stable, one can define a map δ = F^{2} : W_{s} → W_{s} as before. If the Ω_{s}-orbit of E is F^{2}-stable, then δ stabilizes Ω_{s}(E). For a pair (s, E) such that E ∈ (\overline{Irr} W_{s}^{0})^{δ}, we define a subset M_{s,E} of \overline{M}_{s,E}^{(m)} by

M_{s,E} = Ω_{s}(E)_{δ}^{∧} × Ω_{s}(E)^{δ},

where Ω_{s}(E)_{δ}^{∧} is the set of δ-stable irreducible characters in Ω_{s}(E)^{∧}. We denote by E(G^{F^{2m}}, {s})^{F^{2}} the subset of F^{2}-stable irreducible characters in E(G^{F^{2m}}, {s}). Then by [S3, (4.6.1)] it is known that under the parametrization in (2.2.1) for G^{F^{2m}}, we have

E(G^{F^{2m}}, \{s\})^{F^{2}} = ∐_{E∈(\overline{Irr} W_{s})^{δ}} M_{s,E}.

We denote by ρ_{η,z}^{(m)} the F^{2}-stable irreducible character of G^{F^{2m}} corresponding to (η, z) ∈ M_{s,E}.

In the case where F (s) = s^{−1}, one can define a map γ = F : W_{s} → W_{s} preserving Ω_{s} and W_{s}^{0}, and such that δ = γ^{2} as before. We denote by Ω_{s}(E)^{γ} the γ-fixed point subgroup of Ω_{s}(E), and by Ω_{s}(E)_{−γ}^{∧} the set of η ∈ Ω_{s}(E)^{∧} such that γ(η) = \overline{η}. Then we define a subset M_{s,E}^{0} of M_{s,E} by

M_{s,E}^{0} = Ω_{s}(E)_{−γ}^{∧} × Ω_{s}(E)^{γ}.

The following proposition can be proved in a similar way as in Proposition 2.7.

Proposition 3.2. Let ρ_{x}^{(m)} be an F^{2}-stable irreducible character of G^{F^{2m}} corresponding to x ∈ M_{s,E}. Assume that F (s) = s^{−1}.

(i) If the Ω_{s}-orbit of E is not F -stable, then F (ρ_{x}^{(m)}) ≠ \overline{ρ}_{x}^{(m)}. (ii) Assume that the Ω_{s}-orbit of E is F -stable. Then F (ρ_{x}^{(m)}) = \overline{ρ}_{x}^{(m)} if and only if

x ∈ M_{s,E}^{0}.

3.3. Following [S3, 4.6], we define almost characters of G^{F^{2}}. For a given Ω̃_{s}(E)_{δ}, we choose a = a_{E} ∈ Ω̃_{s}(E) and write it as Ω̃_{s}(E)_{δ} = Ω_{s}(E)_{δ}a_{E}. For x = (η, z) ∈ M_{s,E} and y = (η′, z′a) ∈ \overline{M}_{s,E}, we define a pairing \{x, y\} ∈ Q̄_{l}^{∗} by

{x, y} = |Ω_{s}(E)^{δ}|^{−1}η(z′)η′(z).

Then we define a class function R_{x} of G^{F^{2}} by

R_{x} = ∑_{y∈\overline{M}_{s,E}}\{x, y\}ρ_{y}. (3.3.1)

R_{x} are called almost characters of G^{F^{2}}. Note that the definition of the pairing { , } depends on the choice of a_{E} ∈ Ω̃_{s}(E)_{δ}. If a_{E} is replaced by a′ = b^{−1}a_{E} with b ∈ Ω_{s}(E)_{δ}, then R_{x} is replaced by η(b)R_{x}. Hence the almost character R_{x} is determined uniquely up to a root of unity multiple.

It is easy to see that (3.3.1) can be converted to the form

ρ_{y} = |Ω_{s}(E)^{δ}|^{−2} ∑_{x∈M_{s,E}}{x, y}^{−1}R_{x}. (3.3.2)

The following result describes the Shintani descent of irreducible characters of G^{F^{2m}}. Here we write the restriction of F^{2} on G^{F^{2m}} as δ instead of σ^{2}, in connection with the previous section.

Theorem 3.4 ([S3, Theorem 4.7]). Let ρ_{x}^{(m)} be an F^{2}-stable irreducible character of G^{F^{2m}} corresponding to x ∈ M_{s,E}, and choose an extension ρ̃_{x}^{(m)} to G^{F^{2m}}⟨δ⟩. Then

Sh_{F^{2m}/F^{2}}(ρ̃_{x}^{(m)}|_{G^{F^{2m}}δ}) = μ_{x}R_{x},

where μ_{x} is a root of unity depending on the extension ρ̃_{x}^{(m)} and on the choice of a_{E}.

Combining Theorem 3.4 with Proposition 3.2, we have the following refinement of Corollary 1.18.

Corollary 3.5. Let M_{s,E} be such that F (s) = s^{−1} and that the Ω_{s}-orbit of E is F -stable. Then

m_{2}(R_{x}) = \begin{cases}ζ_{x} & if x ∈ M_{s,E}^{0}, \\ 0 & otherwise, \end{cases}

where ζ_{x} is a certain root of unity.

The following result describes the action of twisting operators on almost characters. In the special case where F^{2} acts trivially on the center, this was proved by Bonnafé [B, Théorème 5.5.4]. We note that this result can also be derived from the property of character sheaves, by making use of Lusztig’s conjecture for SL_{n}, which will be discussed in [S4].

Theorem 3.6. For any x = (η, z) ∈ M_{s,E}, we have

t_{1}^{∗}(R_{x}) = η(z)^{−1}R_{x}.

The theorem will be proved in 3.15 after some preliminaries. First we recall some general properties of twisting operators.

Lemma 3.7. Let Γ be a connected algebraic group defined over F_{q} with Frobenius map F, and H a connected F -stable subgroup of Γ. Then the twisting operator t_{1}^{∗} commutes with the induction Ind_{H^{F}}^{Γ^{F}}.

Proof. It is clear that t_{1}^{∗} commutes with the restriction functor Res_{H^{F}}^{Γ^{F}}. Moreover t_{1}^{∗} is an isometry with respect to the inner product on C(H^{F}/∼) and C(Γ^{F}/∼). The lemma follows from these two facts.

3.8 Let Γ be as in the lemma. For each integer m > 0, we consider the group Γ^{F^{m}}, and its semidirect product Γ̃^{F^{m}} = Γ^{F^{m}}⟨σ⟩, where σ is the restriction of F on Γ^{F^{m}}, and ⟨σ⟩ is the cyclic group of order m with generator σ. Then the twisting operator t_{1}^{∗} : C(Γ^{F}/∼) → C(Γ^{F}/∼) can be lifted to the operator

τ_{1}^{∗−1} : C(Γ^{F^{m}}σ/∼) → C(Γ^{F^{m}}σ/∼)

in the following way. We define a map τ_{1} : Γ^{F^{m}}σ/∼→ Γ^{F^{m}}σ/∼ by τ_{1}(xσ) = (xσ)^{1−m}, and define τ_{1}^{∗} by its transpose. It is shown in [S1, Lemma 4.2] that, under the condition that m is sufficiently divisible, τ_{1}^{∗} is an isomorphism and satisfies the following commutative diagram.

C(Γ^{F^{m}}σ/∼) ⟶^{τ_{1}^{∗}} C(Γ^{F^{m}}σ/∼)
Sh_{F^{m}/F} ↓  ↓^{Sh_{F^{m}/F}}
C(Γ^{F}/∼) ⟵^{t_{1}^{∗}} C(Γ^{F}/∼).

We have the following result.

Theorem 3.9 ([S1, Theorem 4.7]). Let ρ̃ be an extension of an F -stable irreducible character of Γ^{F^{m}} to Γ̃^{F^{m}}. Then for an appropriate choice of (sufficiently divisible) m, there exists a root of unity λ such that

τ_{1}^{∗}(ρ̃|_{Γ^{F^{m}}σ}) = λ(ρ̃|_{Γ^{F^{m}}σ}).

The following related result seems to be worth mentioning though it is not used later. In [S1], under some condition on p, the notion of almost characters was established for any connected algebraic group Γ. Then in view of (3.8.1) together with Theorem 3.9, we have

Corollary 3.10. For each almost character R_{x} of Γ^{F}, there exists a root of unity λ_{x} such that

t_{1}^{∗}(R_{x}) = λ_{x}R_{x}.

The following result was proved by Digne and Michel, which holds for any connected reductive groups.

Proposition 3.11 ([DM]). Let H be an F -stable Levi subgroup of a parabolic subgroup of a connected reductive group Γ. Then the Lusztig induction R_{H}^{Γ} : C(H^{F}/ ∼) → C(Γ^{F}/∼) commutes with the twisting operator t_{1}^{∗}.

3.12. We now return to our original setting, and consider G = SL_{n}. The modified generalized Gelfand-Graev characters were introduced by Kawanaka (see [K1]), which is a refinement of generalized Gelfand-Graev characters. The modified generalized Gelfand-Graev characters are used in [S3] to parametrize irreducible characters of SL_{n}. Here we discuss the action of twisting operators on the modified generalized Gelfand-Graev characters. We follow the notation in 2.5 (but replacing F^{2} by F).

By [S3, 2.6], we may choose 𝔲 so that 𝔲 is L-stable. Let A_{λ} = Z_{L}(λ)/Z_{L}^{0}(λ). Then by [S3, 2.7], we have

A_{λ} ≃ A_{G}(N) = Z_{G}(N)/Z_{G}^{0}(N).

In particular, A_{λ} is an abelian group. F acts naturally on A_{λ}, and we consider the quotient group (A_{λ})_{F} of A_{λ}. Put

\overline{M} = (A_{λ})_{F} × (A_{λ}^{F})^{∧}.

For each pair (c, ξ) ∈ \overline{M} one can define a modified generalized Gelfand-Graev character Γ_{c,ξ} as follows. For c ∈ A_{λ}, we choose a representative ċ ∈ Z_{L}(λ). Then we find α_{c} ∈ L such that α_{c}^{−1}F (α_{c}) = ċ. We define a linear map λ_{c} : 𝔲 → k by λ_{c} = λ ∘ Ad α_{c}^{−1}, and define a linear character Λ_{c} = ψ ∘ λ_{c} ∘ log on U^{F}. Since Z_{L}(λ_{c})^{F} = Z_{L}(Λ_{c})^{F}, the linear character Λ_{c} can be extended to a linear character on Z_{L}(λ_{c})^{F} U^{F} trivial on Z_{L}^{0}(λ_{c})^{F}, which we denote also by Λ_{c}. On the other hand, since A_{λ} is abelian, we can define a linear character ξ^{♮} of Z_{L}(λ_{c})^{F} trivial on Z_{L}^{0}(λ_{c})^{F} by

ξ^{♮} : Z_{L}(λ_{c})^{F} → (Z_{L}(λ_{c})/Z_{L}^{0}(λ_{c}))^{F} ≃ A_{λ}^{ċF} = A_{λ}^{F} → Q̄_{l}^{∗},

where the last step is given by ξ : A_{λ}^{F} → Q̄_{l}^{∗}. We denote by the same symbol ξ^{♮} the lift of ξ^{♮} to Z_{L}(λ_{c})^{F} U^{F} under the homomorphism Z_{L}(λ_{c})^{F} U^{F} → Z_{L}(λ_{c})^{F}. Under these setting we define Γ_{c,ξ} by

Γ_{c,ξ} = Ind_{Z_{L}(λ_{c})^{F} U^{F}}^{G^{F}} (ξ^{♮} ⊗ Λ_{c}).

3.13. We choose m large enough so that F^{m} acts trivially on A_{λ}. Replacing F by F^{m}, we have a modified generalized Gelfand-Graev character Γ_{(c,ξ)}^{(m)} on G^{F^{m}}. Now the parameter set \overline{M} is replaced by A_{λ}×(A_{λ})^{∧}. We denote by M the subset of A_{λ}×(A_{λ})^{∧} defined by

M = A_{λ}^{F} × (A_{λ})_{F}^{∧},

where (A_{λ})_{F}^{∧} is the set of F -stable irreducible characters of A_{λ}. Following [S3, 1.8], we construct, for each (c, ξ) ∈ M, an F -stable modified generalized Gelfand-Graev character Γ_{c,ξ}^{(m)}, and its extension to G^{F^{m}}⟨σ⟩, where σ = F |_{G^{F^{m}}}. For c ∈ A_{λ}^{F}, we choose ċ ∈ L^{F}. We construct the linear character Λ_{c}^{(m)} of U^{F^{m}} as in 3.12, i.e., we choose β_{c} ∈ L such that β_{c}^{−1}F^{m}(β_{c}) = ċ, and define λ_{c} by λ_{c} = λ ∘ Ad β_{c}^{−1}, and put Λ_{c}^{(m)} = ψ_{m} ∘ λ_{c} ∘ log, where ψ_{m} = ψ ∘ Tr _{F_{q^{m}}/F_{q}}. Put ĉ = β_{c}F (β_{c}^{−1}) ∈ L^{F^{m}}. Then Λ_{c}^{(m)} turns out to be ĉF -stable.

On the other hand, it can be checked that ĉF acts on Z_{L}(λ_{c}) commuting with F^{m}, and that under the isomorphism

ad β_{c}^{−1} : Z_{L}(λ_{c})^{F^{m}}/Z_{L}^{0}(λ_{c})^{F^{m}} ≃ Z_{L}(λ)^{ċF^{m}}/Z_{L}^{0}(λ)^{ċF^{m}} ≃ A_{λ},

the action of ĉF on Z_{L}(λ_{c})^{F^{m}} is transferred to the action of F on A_{λ}. Hence if we take ξ ∈ (A_{λ})_{F}^{∧}, it produces a ĉF -stable linear character ξ^{♮} on Z_{L}(λ_{c})^{F^{m}}. It follows that ξ^{♮} ⊗ Λ_{c}^{(m)} is ĉF -stable for (c, ξ) ∈ M, and we conclude that Γ_{c,ξ}^{(m)} is F -stable.

Put ĉ_{0} = (ĉσ)^{m} ∈ L^{F^{m}}. We note that ĉ_{0} ∈ Z_{L}(λ_{c})^{F^{m}} = Z_{L}(Λ_{c}^{(m)})^{F^{m}}. In fact, since Λ_{c}^{(m)} is ĉF -stable, it is stable by (ĉσ)^{m} = ĉ_{0}. We also note that

β_{c}^{−1}ĉ_{0}β_{c} ≡ ċ^{−1} (mod Z_{L}^{0}(λ)^{ċF^{m}})

since (ĉσ)^{m} = β_{c}F^{m}(β_{c}^{−1}) and ċ = β_{c}^{−1}F^{m}(β_{c}). In particular, we have

ξ^{♮}(ĉ_{0}) = ξ(c^{−1}). (3.13.1)

Put M_{c} = Z_{L}(λ_{c})^{F^{m}} and M_{c}^{0} = Z_{L}^{0}(λ_{c})^{F^{m}}. We consider a subgroup M_{c}U^{F^{m}}⟨ĉσ⟩ of G^{F^{m}}⟨σ⟩ generated by M_{c}U^{F^{m}} and ĉσ. Since ξ^{♮} ∈ M_{c}^{∧} is ĉF -stable, and (ĉσ)^{m} = ĉ_{0} ∈ M_{c}, ξ^{♮} may be extended to a linear character ξ̃^{♮} of M_{c}⟨ĉσ⟩ in m distinct ways. The extension ξ̃^{♮} is determined by the value ξ̃^{♮}(ĉσ) = μ_{c,ξ}, where μ_{c,ξ} is any m-th root of ξ^{♮}(ĉ_{0}).

We fix an extension ξ̃^{♮} of ξ^{♮} to M_{c}⟨ĉσ⟩. Since M_{c}U^{F^{m}}⟨ĉσ⟩ is the semidirect product of M_{c}⟨ĉσ⟩ with U^{F^{m}}, ξ̃^{♮} may be regarded as a character of M_{c}U^{F^{m}}⟨ĉσ⟩. On the other hand, since Λ̃_{c}^{(m)} is ĉσ-stable, it can be extended to a linear character on M_{c}U^{F^{m}}⟨ĉσ⟩ by Λ̃_{c}^{(m)}(ĉσ) = 1. Thus we have a character ξ̃^{♮} ⊗ Λ̃_{c}^{(m)} of M_{c}U^{F^{m}}⟨ĉσ⟩ which is an extension of ξ^{♮} ⊗ Λ_{c}^{(m)} on M_{c}U^{F^{m}}. We put

Γ̃_{c,ξ}^{(m)} = Ind_{M_{c}U^{F^{m}}⟨ĉσ⟩}^{G^{F^{m}}⟨σ⟩}(ξ̃^{♮} ⊗ Λ̃_{c}^{(m)}).

Then Γ̃_{c,ξ}^{(m)} gives rise to an extension of Γ_{c,ξ}^{(m)} to G^{F^{m}}⟨σ⟩. Note that μ_{c,ξ}^{−1}Γ̃_{c,ξ}^{(m)}|_{G^{F^{m}}σ} depends only on the choice of (c, ξ).

Now we have the following result.

Proposition 3.14. Let the notations be as above. We have

τ_{1}^{∗}(Γ̃_{c,ξ}^{(m)}|_{G^{F^{m}}σ}) = ξ(c)(Γ̃_{c,ξ}^{(m)}|_{G^{F^{m}}σ})

for an appropriate choice of (sufficiently divisible) m.

Proof. The following proof is an analogy of the argument in [S1, Corollary 5.10]. Put H = LU. Then H is an F -stable connected subgroup of G. For each (c, ξ) ∈ M, we denote by θ the linear character ξ^{♮} ⊗ Λ_{c}^{(m)} of M_{c}U^{F^{m}}, and by θ̃ its extension ξ̃^{♮} ⊗ Λ̃_{c}^{(m)} to M_{c}U^{F^{m}}⟨ĉσ⟩. We put H̃^{F^{m}} = H^{F^{m}}⟨σ⟩, V = M_{c}U^{F^{m}}, and Ṽ = M_{c}U^{F^{m}}⟨ĉσ⟩. We consider the induced characters

ρ_{c,ξ} = Ind_{V}^{H^{F^{m}}} θ, ρ̃_{c,ξ} = IndH̃^{F^{m}} ; Ṽ θ̃.

Then ρ_{c,ξ} is an F -stable character of H^{F^{m}}, and ρ̃_{c,ξ} is an extension of ρ_{c,ξ} to H̃^{F^{m}}. Moreover, ρ_{c,ξ} is irreducible by [S3, Lemma 1.7]. Note that

Γ̃_{c,ξ}^{(m)}|_{G^{F^{m}}σ} = Ind_{H^{F^{m}}σ}^{G^{F^{m}}σ}(ρ̃_{c,ξ}|_{H^{F^{m}}σ}).

In order to prove the proposition, we have only to show the following formula since τ_{1}^{∗} commutes with the induction Ind_{H^{F^{m}}σ}^{G^{F^{m}}σ} by Lemma 3.7 and (3.8.1).

τ_{1}^{∗}(ρ̃_{c,ξ}|_{H^{F^{m}}σ}) = ξ(c)(ρ̃_{c,ξ}|_{H^{F^{m}}σ}). (3.14.1)

We show (3.14.1). We choose m so that m is a multiple of some fixed integer A, where A is divisible by |A_{λ}|p, and that m − 1 is prime to the order of H̃^{F^{m}}. The existence of such m is shown in [S1, Lemma 4.8]. Then the map f : H̃^{F^{m}} → H̃^{F^{m}}, g ↦ g^{1−m} is a bijection, and τ_{1} is obtained by restricting f to H^{F^{m}}σ. Since f stabilizes the conjugacy classes, it induces an isomorphism f^{∗} : C(H̃^{F^{m}}/∼) → C(H̃^{F^{m}}/∼). The map f^{∗} stabilizes the space C(Ṽ/∼), and we denote by f ∗ ; Ṽ the restriction of f^{∗} on Ṽ. We note that

(3.14.2) f ∗ ; Ṽ(θ̃) is a linear character of Ṽ such that f ∗ ; Ṽ(θ̃)|_{V} = θ.

In fact, since f induces a homomorphism on H̃^{F^{m}} modulo the commutator subgroup, f ∗ ; Ṽ maps linear characters to linear characters. We show that the restriction of f ∗ ; Ṽ(θ̃) on V coincides with θ. Since λ ∘ log : U → k is a homomorphism of algebraic groups and m is divisble by p, λ(g^{m}) = 0 for g ∈ U. This implies that Λ_{c}^{(m)}(g^{1−m}) = Λ_{c}^{(m)}(g) for g ∈ U^{F^{m}}. On the other hand, since m is divisible by |A_{λ}|, ξ^{♮}(g^{1−m}) = ξ^{♮}(g) for g ∈ M_{c}. It follows that θ(g^{1−m}) = θ(g) for any g ∈ V = M_{c}U^{F^{m}}, and the claim follows.

Now it is easy to see that f^{∗} commutes with the induction

IndH̃^{F^{m}} ; Ṽ : C(Ṽ/∼) → C(H̃^{F^{m}}/∼).

Thus f^{∗}(ρ̃_{c,ξ}) is also an extension of ρ_{c,ξ} to H̃^{F^{m}}. Now the extension of θ to θ̃ is characterized by the value θ̃(ĉσ), and it determines the extension ρ̃_{c,ξ}. Since f(ĉσ) = (ĉσ)^{1−m} = ĉσ · ĉ_{0}^{−1}, we see that

f ∗ ; Ṽ(ξ̃^{♮} ⊗ Λ̃_{c}^{(m)})(ĉσ) = ξ^{♮}(ĉ_{0}^{−1}) · ξ̃^{♮} ⊗ Λ̃_{c}^{(m)}(ĉσ)
= ξ(c) · ξ̃^{♮} ⊗ Λ̃_{c}^{(m)}(ĉσ)

by (3.13.1). This proves (3.14.1), and so the proposition follows.

3.15. We are in a position to prove Theorem 3.6. We apply the previous results to our situation by replacing F by F^{2}. By [S3, 4.5], the parametrization of irreducible characters, ρ_{η′,z′} ↔ (η′, z′) ∈ \overline{M}_{s,E} are divided into three steps. Accordingly, the parametrization of almost characters R_{η,z} ↔ (η, z) ∈ M_{s,E} are divided similarly. The cases (b) and (c) in [loc. cit.] are reduced to the case (a) via Harish-Chandra induction and Lusztig induction. Since the twisting operator commutes with Lusztig induction by Proposition 3.11, the proof of Theorem 3.6 is reduced to the case (a), i.e., the case where (s, E) satisfies the condition (2.5.1).

So, assume that (s, E) is as above. In this case, irreducible characters belonging to \overline{M}_{s,E} and almost characters belonging to M_{s,E} are characterized by modified generalized Gelfand-Graev characters as follows. Let ρ_{ṡ,E} be an irreducible character of G̃^{F^{2}} for some ṡ ∈ G̃^{∗} such that π(ṡ) = s, and N be a nilpotent element such that the nilpotent orbit O_{N} containing N coincides with the orbit O associated to ρ_{ṡ,E} (see e.g., [S3, 2.9]). Let A_{λ} be the finite group given in 3.12. For a certain quotient group \overline{A}_{λ} of A_{λ} with F^{2}-action, we put

M_{s,N} = \overline{A}_{λ}^{δ} × (\overline{A}_{λ})_{δ}^{∧},

where δ is the action of F^{2} on \overline{A}_{λ} and on (\overline{A}_{λ})^{∧} as before. Let (T_{s,E}^{(m)})^{F^{2}} be the set of F^{2}-stable irreducible characters of G^{F^{2m}} belonging to M_{s,E}. Then there exists a parametrization M_{s,N} ↔ (T_{s,E}^{(m)})^{F^{2}} via (c, ξ) ↔ ρ_{c,ξ}^{(m)} satisfying the following properties. Put M̃_{s,N} = A_{λ}^{δ} × (\overline{A}_{λ})_{δ}^{∧}. Since M̃_{s,N} is a subset of A_{λ}^{δ} × (A_{λ})_{δ}^{∧}, one can define an F^{2}- stable character Γ_{c,ξ}^{(m)} of G^{F^{2m}} for each pair (c, ξ) ∈ M_{s,N} (see 3.13). Let φ : M̃_{s,N} → M_{s,N} be the natural projection. Then for (c, ξ) ∈ M̃_{s,N} and (c′, ξ′) ∈ M_{s,N}, we have the following (cf. [S3, Corollary 2.21]).

(3.15.1) ⟨Γ_{c,ξ}^{(m)}, ρ_{c′,ξ′}^{(m)}⟩ = \begin{cases}1 & if φ(c, ξ) = (c′, ξ′), \\ 0 & otherwise. \end{cases}

Assume that φ(c, ξ) = (c′, ξ′), i.e., ξ = ξ′ and c′ is the image of c under the map A_{λ} → \overline{A}_{λ}. Let Γ̃_{c,ξ}^{(m)} and ρ̃_{c′,ξ′}^{(m)} be extensions of Γ_{c,ξ}^{(m)} and ρ_{c′,ξ′}^{(m)} to G^{F^{2m}}⟨δ⟩, respectively. Now by Proposition 3.14, Γ̃_{c,ξ}^{(m)}|_{G^{F^{2m}}δ} is an eigenfunction for τ_{1}^{∗} with eigenvalue ξ(c). Since ρ_{c′,ξ′}^{(m)} occurs in the decomposition of Γ_{c,ξ}^{(m)} with multiplicity 1 by (3.15.1), by applying Theorem 3.9 we see that

τ_{1}^{∗}(ρ̃_{c′,ξ′}^{(m)}|_{G^{F^{2m}}δ}) = ξ′(c′)(ρ̃_{c′,ξ′}^{(m)}|_{G^{F^{2m}}δ}). (3.15.2)

(Note that ξ(c) = ξ′(c′)). The set (T_{s,E}^{(m)})^{F^{2}} is also parametrized by the set M_{s,E} via ρ_{η,z}^{(m)} ↔ (η, z) ∈ M_{s,E}. Here we need the following fact.

Lemma 3.16. The bijection M_{s,N} ↔ M_{s,E}, (c′, ξ′) ↔ (η, z) parametrizing the set (T_{s,E}^{(m)})^{F^{2}} satisfies the property that ξ′(c′) = η(z).

Assuming Lemma 3.16, we continue the proof of the theorem. By the lemma, we have

τ_{1}^{∗}(ρ̃_{η,z}^{(m)}|_{G^{F^{2m}}δ}) = η(z)(ρ̃_{η,z}^{(m)}|_{G^{F^{2m}}δ}).

Now the theorem follows from Theorem 3.4, in view of the commutativity of t_{1}^{∗−1} and τ_{1}^{∗} given in (3.8.1). This completes the proof of Theorem 3.6 modulo Lemma 3.16.

3.17. We shall prove Lemma 3.16. Here we use the results from [S3]. See 2.20, Corollary 2.21 and 4.11 in [S3] for details. M_{s,E} is a subset of \overline{M}_{s,E}^{(m)} = Ω_{s}^{∧} × Ω_{s}, and M_{s,N} is a subset of \overline{M}_{s,N}^{(m)} = \overline{A}_{λ}×\overline{A}_{λ}^{∧}, where \overline{M}_{s,N}^{(m)} is a set parametrizing T_{s,E}^{(m)}. Hence it is enough to show a similar fact for \overline{M}_{s,E} and \overline{M}_{s,N} with respect to G^{F}, assuming that F acts trivially on Ω_{s} and on \overline{A}_{λ}. Thus we have \overline{M}_{s,E} = Ω_{s}^{∧} × Ω_{s}, \overline{M}_{s,N} = \overline{A}_{λ} × \overline{A}_{λ}^{∧}, and there exists a bijection \overline{M}_{s,E} ≃ \overline{M}_{s,N} through bijections Ω_{s}^{∧} ≃ \overline{A}_{λ}, Ω_{s} ≃ \overline{A}_{λ}^{∧}. Now for x ∈ Ω_{s}, we have an irreducible character ρ_{ṡ_{x},E} of G̃^{F}, and it is decomposed in terms of the parameter set \overline{M}_{s,N},

ρ_{ṡ_{x},E}|_{G^{F}} = ∑_{c∈\overline{A}_{λ}} ρ_{c,ξ_{x}}. (3.17.1)

(Compare this with (2.2.3). Here E_{x} = E by our assumption that Ω_{s}(E) = Ω_{s}). ξ_{x} ∈ \overline{A}_{λ}^{∧} is uniquely determined by x ∈ Ω_{s}. The map h : Ω_{s} → \overline{A}_{λ}^{∧}, x ↦ ξ_{x}, is described as follows (cf. [S3, 2.13]). There exists an F -stable Levi subgroup M̃ of a parabolic subgroup of G̃ (depending on the pair (s, E)) such that ZL̃(λ) ⊂ M̃, and that ṡ_{x} is contained in the center Z(M̃^{∗}) of the dual group M̃^{∗}. We choose m large enough so that ṡ_{x} ∈ Z(M̃^{∗})^{F^{m}}, and let θ̂′_{x} be the corresponding linear character of M̃^{F^{m}}. Then the restriction θ̂_{x} of θ̂′_{x} to ZL̃(λ)^{F^{m}} is F -stable, and we put θ_{x} = Sh_{F^{m}/F} (θ̂_{x}), which is a linear character of ZL̃(λ)^{F}. Then the restriction of θ_{x} to Z_{L}^{0}(λ)^{F} is trivial, and the restriction of θ_{x} to Z_{L}(λ)^{F} induces a character of A_{λ}^{F} = A_{λ} (see Lemma 2.18 and Theorem 2.16 in [S3]). This character factors through \overline{A}_{λ}, which gives ξ_{x} ∈ \overline{A}_{λ}^{∧}. The map h gives a bijection Ω_{s} ≃ \overline{A}_{λ}^{∧}. Now by our assumption, s is F -stable, and F acts trivially on Ω_{s}. This implies, by [S3, Lemma 2.18] that ξ_{1} (the case where x = 1) is the trivial character of \overline{A}_{λ}. Let us write ṡ_{x} = ṡz_{x} with z_{x} ∈ Z̃^{∗}. Since ṡ_{x} is xF -stable, we have

ṡ^{−1}ẋṡẋ^{−1} = z_{x}F (z_{x})^{−1}, (3.17.2)

where ẋ ∈ N_{G^{∗}}(T^{∗}) is a representative of x ∈ Ω_{s} ⊂ W_{s}. We may assume that z_{x} ∈ Z̃^{∗F^{m}}. Let ψ̂′_{x} be the linear character of G̃^{∗F^{m}} corresponding to z_{x}. Then it follows from the above discussion that the restriction ψ̂_{x} of ψ̂′_{x} to ZL̃(λ)^{F^{m}} is F -stable, and we obtain a linear character ψ_{x} = Sh_{F^{m}/F} (ψ̂_{x}) of ZL̃(λ)^{F}. Since ξ_{1} = 1, we see that ξ_{x} is obtained from the restriction of ψ_{x} to Z_{L}(λ)^{F}.

Next, we shall describe the bijection between \overline{A}_{λ} and Ω_{s}^{∧}. There exists a surjective homomorphism f_{1} : G̃^{F} /G^{F} → \overline{A}_{λ} defined as follows (cf [S3, 2.19]). For g ∈ G̃^{F}, we can write g = g_{1}z, with g_{1} ∈ G, z ∈ Z̃. Then g_{1}^{−1}F (g_{1}) ∈ Z, and it determines an element in A_{λ} = Z_{L}(λ)/Z_{L}^{0}(λ), and so an element in \overline{A}_{λ}, which is unique up to F -conjugacy. Since F acts trivially on \overline{A}_{λ}, this gives a well-defined map f_{1} : G̃^{F} /G^{F} → \overline{A}_{λ}. On the other hand, we construct f_{2} : G̃^{F} /G^{F} → Ω_{s}^{∧} as follows. From (3.17.2), we have ṡ^{−1}ẋṡẋ^{−1} ∈ Z̃^{∗F} (we may choose ẋ ∈ N_{G^{∗}}(T^{∗})^{F}), and this defines a well-defined injective homomorphism f_{2}^{∗} : Ω_{s} → Z̃^{∗F} , x ↦ ṡ^{−1}ẋṡẋ^{−1}. Since Z̃^{∗F} ≃ (G̃^{F} /G^{F})^{∧}, we have a surjective map f_{2} as the transpose of f_{2}^{∗}. Then Ker f_{1} = Ker f_{2}, and these maps induce the bijection \overline{A}_{λ} ≃ Ω_{s}^{∧}.

Now the parametrization is given as follows. There exists a unique ρ_{0} ∈ Irr G^{F} such that ρ_{0} occurs in ρ_{ṡ_{x},E}|_{G^{F}} and in Γ_{N}. In our parametrization, then ρ_{0} = ρ_{1,ξ_{x}} = ρ_{1,x} ((1, ξ_{x}) ∈ \overline{M}_{s,N}, (1, x) ∈ \overline{M}_{s,E}). Then any ρ contained in ρ_{ṡ_{x},E}|_{G^{F}} is obtained as ^{g}ρ_{0} with g ∈ G̃^{F} /G^{F}. We then have ρ = ρ_{c,ξ_{x}} = ρ_{η,x} with c = f_{1}(g) and η = f_{2}(g). Thus, in order to prove the lemma, it is enough to show

f_{1}^{∗} ∘ h = f_{2}^{∗}, (3.17.3)

where f_{1}^{∗} : \overline{A}_{λ}^{∧} → Z̃^{∗F} is the transpose of f_{1}.

Take x ∈ Ω_{s}, and let θ be the linear character of G̃^{F} corresponding to f_{2}^{∗}(x) ∈ Z̃^{∗F}. For g = g_{1}z ∈ G̃^{F}, put y = g_{1}^{−1}F (g_{1}) ∈ Z ⊂ Z̃^{F}. Take ŷ ∈ Z̃^{F^{m}} such that N_{F^{m}/F} (ŷ) = y. Then (3.17.3) is deduced from the formula

θ(g) = ψ̂′_{x}(ŷ) (3.17.4)

for g ∈ G̃^{F} /G^{F}. We show (3.17.4). Since G̃^{F} /G^{F} ≃ T̃^{F} /T^{F}, we may assume that g ∈ T̃^{F} and g_{1} ∈ T. Take α ∈ F̄_{q}^{∗} such that det g = α^{n}. Then z = Diag(α, …, α) and g_{1} = z^{−1}g. Since f_{2}^{∗}(g) ∈ Z̃^{∗F}, by considering the restriction of θ on T̃^{F}, we see that there exists a homomorphism ω : F_{q}^{∗} → Q̄_{l}^{∗} such that

θ(g) = ω(deg g) = ω(α^{n}). (3.17.5)

Let θ̂ be a character of G̃^{F^{m}} such that θ = Sh_{F^{m}/F} (θ̂). It is checked that for any ĝ ∈ T̃^{F^{m}}, one can write θ̂(ĝ) = ω̂(det ĝ), where ω̂ is a character of F_{q^{m}}^{∗} such that Sh_{F^{m}/F} (ω̂) = ω. Then by (3.17.2), we have

θ̂ = ψ̂′_{x}F (ψ̂′_{x})^{−1}. (3.17.6)

On the other hand, y ∈ Z^{F} is expressed as

y = g_{1}^{−1}F (g_{1}) = zF (z^{−1}) = Diag(α^{1−q}, …, α^{1−q}).

Take β ∈ F̄_{q}^{∗} such that α = β^{q^{m}−1}, and put α̂ = β^{q−1}. Then ŷ = Diag(α̂^{1−q}, …, α̂^{1−q}). By making use of (3.17.6), we have

ψ̂′_{x}(ŷ) = θ̂(Diag(α̂, …, α̂)) = ω̂(α̂^{n}) = ω(α^{n}).

Comparing this with (3.17.5), we see that (3.17.4) holds. Thus Lemma 3.16 is proved.

4. Determination of m_{2}(ρ_{ṡ,E}|_{G^{F^{2}}})

4.1. Assume that s is F^{2}-stable, and the pair (s, T^{∗}) is given as in Section 2. Let ρ_{ṡ,E} be an irreducible character of G̃^{F^{2}} as in 1.20. In this section, we shall compute the value m_{2}(ρ_{ṡ,E}|_{G^{F^{2}}}). Now ρ_{ṡ,E} is given as

ρ_{ṡ,E} = εG̃^{∗}ε_{ZG̃^{∗}(ṡ)}|Wṡ|^{−1} ∑_{w∈Wṡ} Tr (wδ, Ẽ)RT̃_{w}^{∗}(ṡ), (4.1.1)

where ε_{H} = (−1)^{F_{q^{2}}−rank(H)} for any reductive group H, and Ẽ is a certain extension of E ∈ (Irr Wṡ)^{δ} to Wṡ⟨δ⟩. Therefore we compute the value m_{2}(RT̃_{w}^{∗}(ṡ)) for each w ∈ Wṡ.

We consider the isomorphism Z̃^{∗F^{2}} ≃ (G̃^{F^{2}}/G^{F^{2}})^{∧} as in (2.4.1), and a similar one by replacing F^{2} by F. By the property of the dual torus, we have the following commutative diagram.

\begin{aligned}Z̃^{∗F^{2}} & ⟶^{∼} & (G̃^{F^{2}}/G^{F^{2}})^{∧} \\ ^{N_{F^{2}/F}} ↓ & & ↓^{Res} \\ Z̃^{∗F} & ⟶^{∼} & (G̃^{F} /G^{F})^{∧}, \end{aligned} (4.1.2)

where Res is the restriction of the character of G̃^{F^{2}}/G^{F^{2}} on G̃^{F} /G^{F}, and N_{F^{2}/F} is the norm map z → zF (z). The norm map is also described as in 1.1. By using this, it is easy to see that Ker N_{F^{2}/F} coincides with the subset {z^{−1}F (z) | z ∈ Z̃^{∗F^{2}}}, and so Z̃^{∗F} can be identified with (Z̃^{∗F^{2}})_{F} via the map zF (z) ↔ z for z ∈ (Z̃^{∗F^{2}})_{F}.

First we note the following general fact.

Lemma 4.2. Let χ be a class function of G̃^{F^{2}}. Then

m_{2}(χ|_{G^{F^{2}}}) = ∑_{θ∈(G̃^{F} /G^{F})^{∧}} m_{2}(χ ⊗ θ̃),

where θ̃ is a character of G̃^{F^{2}}/G^{F^{2}}, regarded as a linear character of G̃^{F^{2}}, which is an extension of θ via the inclusion G̃^{F} /G^{F} ↪ G̃^{F^{2}}/G^{F^{2}}.

Proof. By the Frobenius reciprocity, we have

⟨χ|_{G^{F^{2}}}, Ind_{G^{F}}^{G^{F^{2}}} 1⟩ =⟨χ, IndG̃^{F^{2}} ; G^{F} 1⟩

=⟨χ, IndG̃^{F^{2}} ; G̃^{F}(IndG̃^{F} ; G^{F} 1)⟩

=⟨χ, ∑_{θ∈(G̃^{F} /G^{F})^{∧}} IndG̃^{F^{2}} ; G̃^{F} θ⟩ .

But for any linear character θ̃ of G̃^{F^{2}} such that θ̃|G̃^{F} = θ, we have

⟨χ, IndG̃^{F^{2}} ; G̃^{F} θ⟩ =⟨χ ⊗ θ̃^{−1}, IndG̃^{F^{2}} ; G̃^{F} 1⟩ = m_{2}(χ ⊗ θ̃^{−1}).

27

Thus the lemma is proved.

By applying the above formula to the class function RT̃_{w}^{∗}(ṡ) of G̃^{F^{2}}, Lemma 4.3. We have

m_{2}(RT̃_{w}^{∗}(ṡ)|_{G^{F^{2}}}) = ∑_{z∈(Z̃^{∗F^{2}})_{F}} m_{2}(RT̃_{w}^{∗}(ṡż)), (4.3.1)

where ż is a representative of z in Z̃^{∗F^{2}}. Proof. By 4.1, (Z̃^{∗F^{2}})_{F} is isomorphic to (G̃^{F} /G^{F})^{∧}. We denote by θ the character of G̃^{F} /G^{F} corresponding to z ∈ (Z̃^{∗F^{2}})_{F}. Then by (4.1.2), the representative ż ∈ Z̃^{∗F^{2}} corresponds to a linear character θ̃ of G̃^{F^{2}}/G^{F^{2}}, which is an extension of θ. Now it is known that RT̃_{w}^{∗}(ṡ) ⊗ θ̃ = RT̃_{w}^{∗}(ṡż). Hence the lemma follows from Lemma 4.2.

We have the following proposition. Proposition 4.4. Let s be an element in T_{w}^{∗} such that F^{2}(s) = s.

(i) Assume that the G^{∗F^{2}}-orbit of s does not contain s′ such that F (s′) = s′^{−1}. Then m_{2}(RT̃_{w}^{∗}(ṡ)|_{G^{F^{2}}}) = 0 for any ṡ ∈ T̃_{w}^{∗} such that π(ṡ) = s.

(ii) Assume that F (s) = s^{−1}. Then there exists ṡ ∈ T̃_{w}^{∗} such that π(ṡ) = s and that F (ṡ) = ṡ^{−1}, and we have

m_{2}(RT̃_{w}^{∗}(ṡ)|_{G^{F^{2}}}) = ∑_{x∈Ω_{s}^{−γ}} m_{2}(RT̃_{w}^{∗}(ṡz_{x})), (4.4.1)

where Ω_{s}^{−γ} = {x ∈ Ω_{s} | γ(x) = x^{−1}} is the subgroup of Ω_{s}^{δ}, and z_{x} ∈ Z̃^{∗F^{2}} is a representative of an element in (Z̃^{∗F^{2}})_{F} such that z_{x}F (z_{x}) = ω_{s}(x) under the

map ω_{s} : Ω_{s}^{δ} → Z̃∗F^{2} ; ṡ ⊂ Z̃^{∗F^{2}} (see 2.4).

Proof. First we show (i). It is known that RT̃_{w}^{∗}(ṡ)|_{G^{F^{2}}} coincides with the Deligne-Lusztig character R_{T_{w}^{∗}}(s) of G^{F^{2}}. Then by [L3, 2.7 (a)], we have m_{2}(R_{T_{w}^{∗}}(s)) = 0. (Note that in [loc. cit.], it is assumed that the center of G is connected. However, the above fact holds without this assumption, by 2.3 and 2.6 (b) in [loc. cit.]). Thus (i) holds.

Next we show (ii). Assume that F (s) = s^{−1}. Take ṡ_{1} ∈ T̃^{∗} such that π(ṡ_{1}) = s. Then there exists some z ∈ Z̃^{∗} such that F (ṡ_{1}) = ṡ_{1}^{−1}z for some z ∈ Z̃^{∗}. Take z_{1} ∈ Z̃^{∗} ≃ G_{m} such that z = z_{1}F (z_{1}) = z_{1}^{q+1}, and put ṡ = ṡ_{1}z_{1}. Then π(ṡ) = s and F (ṡ) = ṡ^{−1} as asserted.

Take ṡ as above, and consider the formula (4.3.1). Again by [L3, Lemma 2.8], we may only consider, in the sum of the right hand side of (4.3.1), z ∈ (Z̃^{∗F^{2}})_{F} such that F (ṡż) is conjugate to (ṡż)^{−1} in G̃^{∗}. Here we note that (4.4.2) F (ṡż) is conjugate to (ṡż)^{−1} if and only if there exists x ∈ Ω_{s}^{−γ} such that żF (ż) = ω_{s}(x).

We show (4.4.2). Assume that x ∈ Ω_{s}^{−γ}, and let ẋ be an element in G̃^{∗} such that π(ẋ) is a representative of x. Then by (2.11.1), ω_{s}(x) = ṡ^{−1}ẋṡẋ^{−1} ∈ Z̃^{∗F^{2}} is γ-stable, i.e., ω_{s}(x) ∈ Z̃^{∗F}. Hence by (4.1.2) there exists ż ∈ Z̃^{∗F^{2}} such that ω_{s}(x) = żF (ż). It follows that ṡż = ẋṡẋ^{−1}F (ż)^{−1}, and we have F (ṡż) = ẋ^{−1}(ṡż)^{−1}ẋ. This shows that F (ṡż) is conjugate to (ṡż)^{−1} in G̃^{∗}. Conversely, assume that F (ṡż) is conjugate to (ṡż)^{−1} in G̃^{∗}. Then there exists ẋ ∈ G̃^{∗} such that F (ṡż) = ẋ^{−1}(ṡż)^{−1}ẋ. Clearly π(ẋ) ∈ Z_{G^{∗}}(s), and its image in Ω_{s} determines an element x ∈ Ω_{s}. Since ṡ(żF (ż)) = ẋṡẋ^{−1}, we have żF (ż) ∈ Z̃_{s}^{∗F^{2}}. Moreover, żF (ż) is γ-stable. Hence by (2.4.2) and (2.11.1), we see that x ∈ Ω_{s}^{−γ}. This proves (4.4.2).

Since ω_{s}(x) ∈ Z̃^{∗F}, ż ∈ Z̃^{∗F^{2}} such that żF (ż) = ω_{s}(x) has a unique image on (Z̃^{∗F^{2}})_{F} ≃ Z̃^{∗F} by (4.1.2). We choose z_{x} from such ż for each x. Then the formula (4.4.1) is immediate from (4.4.2).

By using Lusztig’s formula in [L3], we shall compute the right hand side of (4.4.1) explicitly. We show

Lemma 4.5. Under the notation in Proposition 4.4 (ii), we have

m_{2}(RT̃_{w}^{∗}(ṡz_{x})) = ♯{u ∈ Wṡ | w = u(^{xγ}u)}.

Proof. Take ẋ ∈ NG̃^{∗}(T̃^{∗}) whose image in W is a representative of x ∈ Ω_{s}^{−γ}. We note that one can choose ẋ such that F (ẋ) = ẋ^{−1}. In fact, take any x′ ∈ NG̃^{∗}(T̃^{∗}) in the inverse image of x. Since γ(x) = x^{−1}, we have x′F (x′) = t ∈ T̃^{∗}. We can find t_{1} ∈ T̃^{∗} such that t_{1}^{−1}F^{2}(t_{1}) = t. Then ẋ = t_{1}x′F (t_{1})^{−1} satisfies the required condition.

Take g ∈ G̃^{∗} such that g^{−1}F (g) = ẋ. Put s′ = ^{g}(ṡz_{x}), T̃′ = ^{g}T̃^{∗} and W′ = NG̃^{∗}(T̃′)/T̃′. Then F (s′) = s′^{−1}, F (T̃′) = T̃′, and s′ ∈ T̃′. Moreover, we have g^{−1}F^{2}(g) = ẋF (ẋ) = 1 and so g ∈ G̃^{∗F^{2}}. We have an isomorphism f : W \underrightarrow{∼} W′ via ad g, and we see that the pair (s′, T̃′_{f(w)}) is G̃^{∗F^{2}}-conjugate to the pair (ṡz_{x}, T̃_{w}^{∗}), where T̃′_{f(w)} is an F^{2}-stable maximal torus obtained from T̃′ by twisting by f(w) ∈ W′.

It follows that

RT̃_{w}^{∗}(ṡz_{x}) = RT̃′_{f(w)}(s′). (4.5.1)

If we put W′_{s′} = \{w′ ∈ W′ | w′(s′) = s′\}, f induces an isomorphism Wṡ \underrightarrow{∼} W′_{s′}. Now it is known by [L3, Lemma 2.8, (b)] that

m_{2}(RT̃′_{f(w)}(s′)) = ♯{y ∈ W′_{s′} | f(w) = yF (y)}. (4.5.2)

Hence, by (4.5.1) and (4.5.2), we have

m_{2}(RT̃_{w}(ṡz_{x})) = ♯{u ∈ Wṡ | f(w) = f(u)F (f(u))}
= ♯{u ∈ Wṡ | w = u(^{xγ}u)},

since F ∘ f = f ∘ ẋF. This proves the lemma.

We are in a position to determine m_{2}(ρ_{ṡ,E}|_{G^{F^{2}}}).

Theorem 4.6. Let ρ_{ṡ,E} be an irreducible character of G̃^{F^{2}} as before, and put s = π(ṡ).

(i) If s is not G^{∗F^{2}}-conjugate to s′ such that F (s′) = s′^{−1}, then m_{2}(ρ_{ṡ,E}|_{G^{F^{2}}}) = 0. (ii) Assume that F (s) = s^{−1}. Then we have

m_{2}(ρ_{ṡ,E}|_{G^{F^{2}}}) = \begin{cases}|Ω_{s}^{−γ}(E)| & if there exists x ∈ Ω_{s}^{−γ} such that ^{xγ}E = E , \\ 0 & otherwise, \end{cases}

where Ω_{s}^{−γ}(E) is the stabilizer of E in Ω_{s}^{−γ}. Proof. ρ_{ṡ,E} is given as in (4.1.1). Thus (i) is immediate from Proposition 4.4 (i). We show (ii). So, assume that F (s) = s^{−1}. Since R_{T̃_{w}^{∗}(ṡ)}|_{G^{F^{2}}} does not depend on the choice of a representative ṡ of s, we may assume that ṡ satisfies the property that F (ṡ) = ṡ^{−1}. Then ε_{G̃∗}ε_{ZG̃^{∗}(s)} = 1 by [L3, 1.5 (b)]. Hence by (4.4.1) together with Lemma 4.5, we have

m_{2}(ρ_{ṡ,E}|_{G^{F^{2}}}) = |Wṡ|^{−1} ∑_{w∈Wṡ} T r(wδ, Ẽ)m_{2}(RT̃_{w}^{∗}(ṡ)|_{G^{F^{2}}})

= |Wṡ|^{−1} ∑_{w∈Wṡ} Tr (wδ, Ẽ) ∑_{x∈Ω_{s}^{−γ}} m_{2}(RT̃_{w}^{∗}(ṡz_{x}))

= |Wṡ|^{−1} ∑_{x∈Ω_{s}^{−γ}} ∑_{w∈Wṡ} Tr (wδ, Ẽ)♯{u ∈ Wṡ | w = u(^{xγ}u)}.

Now by Lemma 2.11 in [L3] (see also the formula in the proof of Proposition 2.13 there), one can write

∑_{E′∈(Wṡ)_{xγ}^{∧}} Tr (wδ, Ẽ′) = ♯{u ∈ Wṡ | w = u(^{xγ}u)},

where (Wṡ)_{xγ}^{∧} is the set of xγ-stable characters of Wṡ, and the extension Ẽ′ of E′ is chosen to be realized over Q. (Note that (xγ)^{2} = δ since x ∈ Ω_{s}^{−γ}). It follows that

m_{2}(ρ_{ṡ,E}|_{G^{F^{2}}}) = ∑_{x∈Ω_{s}^{−γ}} ∑_{E′∈(Wṡ)_{xγ}^{∧}} |Wṡ|^{−1} ∑_{w∈Wṡ} Tr (wδ, Ẽ) Tr (wδ, Ẽ′).

But since

|Wṡ|^{−1} ∑_{w∈Wṡ} Tr (wδ, Ẽ) Tr (wδ, Ẽ′) = \begin{cases}1 & if Ẽ = Ẽ′, \\ 0 & if E ≠ E′, \end{cases}

(here the extension Ẽ is chosen to be over Q, see [L1, 3.2]), we have

m_{2}(ρ_{ṡ,E}|_{G^{F^{2}}}) = ♯{x ∈ Ω_{s}^{−γ} | ^{xγ}E = E }.

If there exists x_{1} such that ^{x_{1}γ}E = E, then {x ∈ Ω_{s}^{−γ} | ^{xγ}E = E} = Ω_{s}^{−γ}(E)x_{1}. Thus the theorem is proved.

We shall apply the formula in the theorem to the case ρ_{ṡ_{x},E_{x}}. First we note that Lemma 4.7. Assume that F (s) = s^{−1}. Let s_{x} be an element corresponding to x ∈ (Ω_{s})_{δ}. Then the G^{∗F^{2}}-class of s_{x} contains an element s′ such that F (s′) = s′^{−1} if and only if there exists u ∈ Ω_{s} such that uγ(u) gives a representative of x in Ω_{s}.

Proof. Let ẍ ∈ Z_{G^{∗}}(s) be an element such that its image to Ω_{s} gives a representative of x. Let s′ be an element contained in the G^{∗F^{2}}-class of s_{x}. Then s′ can be obtained as s′ = ^{g}s for some g ∈ G^{∗} such that g^{−1}F^{2}(g) = ẍ. It is easy to see that F (s′) = s′^{−1} if and only if g^{−1}F (g) ∈ Z_{G^{∗}}(s). Assume that g^{−1}F (g) ∈ Z_{G^{∗}}(s). Then we have uγ(u) ∈ Ω_{s} gives a representative of x if we put u the image of g^{−1}F (g) in Ω_{s}. Conversely, assume that there exists u ∈ Ω_{s} as in the lemma, and let u̇ its representative in Z_{G^{∗}}(s). Then s′ = ^{g}s for g ∈ G^{∗} such that g^{−1}F (g) = u̇ satisfies the property F (s′) = s′^{−1}.

4.8. We prepare a notation. Let s be a semisimple element such that F (s) = s^{−1}, and E ∈ Irr W_{s}^{0} such that the Ω_{s}-orbit of E is γ-stable. We define a subset Ω̃_{s}(E)_{δ}^{+} (resp. Ω_{s}(E)_{δ}^{+}) of Ω̃_{s}(E)_{δ} (resp. of Ω_{s}(E)_{δ}) by

Ω̃_{s}(E)_{δ}^{+} = the image of {uγ(u) | u ∈ Ω_{s},^{uγ}E = E} into Ω̃_{s}(E)_{δ},
Ω_{s}(E)_{δ}^{+} = the image of {vγ(v) | v ∈ Ω_{s}(E)} into Ω_{s}(E)_{δ}.

Then we can see that there exists a_{E} ∈ Ω_{s} such that

Ω̃_{s}(E)_{δ}^{+} = Ω_{s}(E)_{δ}^{+}a_{E}. (4.8.1)

In fact, since the Ω_{s}-orbit of E is γ-stable, there exists b ∈ Ω_{s} such that ^{bγ}E = E. Then a_{E} = bγ(b) is contained in Ω̃_{s}(E), and we have Ω̃_{s}(E) = Ω_{s}(E)a_{E}. (4.8.1) follows from this.

As a corollary to Theorem 4.6, we have the following.

Corollary 4.9. Assume that s is semisimple in G^{∗} such that F (s) = s^{−1}, and that E ∈ (\overline{Irr} W_{s}^{0})^{δ}. Take y ∈ Ω̃_{s}(E)_{δ} and let (E, y) ↔ (x, E_{x}) be as in (2.2.2). Then we have

(i) If the Ω_{s}-orbit of E is not γ-stable, then m_{2}(ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}) = 0. (ii) Assume that the Ω_{s}-orbit of E is γ-stable. Then

m_{2}(ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}) = \begin{cases}|Ω_{s}^{−γ}(E)| & if y ∈ Ω̃_{s}(E)_{δ}^{+}, \\ 0 & otherwise. \end{cases}

Proof. Assume that m_{2}(ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}) ≠ 0. Then s_{x} is G^{∗F^{2}}-conjugate to some s′ such that F (s′) = s′^{−1} by Theorem 4.6. Since x and y are in the same class in (Ω_{s})_{δ}, there exists u ∈ Ω_{s} such that the image of uF (u) to (Ω_{s})_{δ} coincides with y, by Lemma 4.7. Let u̇ ∈ Z_{G^{∗}}(s) be a representative of u. Then there exists g ∈ G^{∗} such that g^{−1}F (g) = u̇. We see that g^{−1}F^{2}(g) = u̇F (u̇) gives a representative of y as its image on Ω_{s}, which we denote by ẏ ∈ Ω_{s}. Then as discussed in (2.2.2), there exists z ∈ Ω_{s} such that ẏ = z^{−1}ẋδ(z), and we have E_{x} = ^{z}E ∈ (Irr W_{s}^{0})^{ẋδ}. Put g_{1} = gż^{−1}, where ż ∈ Z_{G^{∗}}(s) is a representative of z ∈ Ω_{s}. Then g_{1}^{−1}F^{2}(g_{1}) gives ẋ as its image to Ω_{s}, and so we may assume that s_{x} = ^{g_{1}}s.

Now ad g_{1} gives an isomorphism W_{s}^{0} → W_{s_{x}}^{0}, and by this isomorphism, E_{x} ∈ (Irr W_{s}^{0})^{ẋδ} is sent to E′′ ∈ (Irr W_{s_{x}}^{0})^{F^{2}}. ρ_{ṡ_{x},E_{x}} is defined as ρ_{ṡ_{x},E′′} (cf. 2.2). Here ad g_{1} : W_{s}^{0} → W_{s_{x}}^{0} is factored as

W_{s}^{0} ⟶^{ad ż^{−1}} W_{s}^{0} ⟶^{ad g} W_{s_{x}}^{0}.

In the first isomorphism, E_{x} ∈ (Irr W_{s}^{0})^{ẋδ} is sent to E ∈ (Irr W_{s}^{0})^{ẏδ}. Moreover, in the second isomorphism, the actions F, F^{2} on W_{s_{x}}^{0} is transfered to the actions uF, yF^{2} on W_{s}^{0}. Then by Theorem 4.6, applied to ρ_{ṡ,E} ∈ Irr G^{(u̇F)^{2}} with (u̇F)(s) = s^{−1}, m_{2}(ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}) ≠ 0 is equivalent to the condition that there exists h ∈ Ω_{s}^{−uγ} = Ω_{s}^{−γ} such that ^{huγ}E = E. In particular, the Ω_{s}-orbit of E is γ-stable. Since huγ(hu) = uγ(u) = y, this implies that y ∈ Ω̃_{s}(E)_{δ}^{+}. Moreover in that case m_{2}(ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}) is given by the formula as claimed in the corollary. Conversely, assume that y ∈ Ω̃_{s}(E)_{δ}^{+}. Then there exists u̇ ∈ Z_{G^{∗}}(s) such that the image of u̇F (u̇) to Ω_{s} gives a representative of y. Take g ∈ G^{∗} such that g^{−1}F (g) = u̇. Then a similar argument as above implies, by Theorem 4.6, that m_{2}(ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}) ≠ 0. This proves the corollary.

5. Determination of m_{2}(ρ) for ρ ∈ Irr SL_{n}(F_{q^{2}})

5.1. In this section, we shall determine m_{2}(ρ) for all irreducible characters of G^{F^{2}}. Our strategy is to compute m_{2} for almost characters of G^{F^{2}} first, and then derive the formula for m_{2}(ρ) from it.

First we prepare some notation. Let s be an F^{2}-stable semisimple element in G^{∗}, and E an δ-stable irreducible character of W_{s}^{0}. We recall two sets \overline{M}_{s,E} = Ω_{s}^{δ}(E)^{∧} × Ω̃_{s}(E)_{δ} and M_{s,E} = Ω_{s}(E)_{δ}^{∧} × Ω_{s}(E)^{δ}. Assuming that F (s) = s^{−1} and that the Ω_{s}-orbit of E is F -stable, we define subsets Ω_{s}^{δ}(E)_{−}^{∧} ⊂ Ω_{s}^{δ}(E)_{−γ}^{∧} ⊂ Ω_{s}^{δ}(E)^{∧} by

Ω_{s}^{δ}(E)_{−γ}^{∧} = {θ ∈ Ω_{s}^{δ}(E)^{∧} | γ(θ) = θ^{−1}},
Ω_{s}^{δ}(E)_{−}^{∧} = {θ^{−1}γ(θ) | θ ∈ Ω_{s}^{δ}(E)^{∧}}.

We also consider subsets Ω̃_{s}(E)_{δ}^{+} ⊂ Ω̃_{s}(E)_{δ}^{γ} ⊂ Ω̃_{s}(E)_{δ}, where

Ω̃_{s}(E)_{δ}^{γ} = {u ∈ Ω̃_{s}(E)_{δ} | γ(u) = u},

and Ω̃_{s}(E)_{δ}^{+} is defined as in 4.8. We define subsets Ω_{s}(E)_{δ}^{+} ⊂ Ω_{s}(E)_{δ}^{γ} ⊂ Ω_{s}(E)_{δ} in a similar way as above.

Put

|Ω_{s}(E)^{δ}| = t, |Ω_{s}(E)^{γ}| = d, |Ω_{s}(E)^{−γ}| = d′.

Then we see easily that

|Ω_{s}^{δ}(E)^{∧}| = |Ω_{s}(E)_{δ}| = t,
|Ω_{s}^{δ}(E)_{γ}^{∧}| = |Ω_{s}(E)_{δ}^{γ}| = d,
|Ω_{s}^{δ}(E)_{−γ}^{∧}| = |Ω_{s}(E)_{δ}^{−γ}| = d′.

Since Ω_{s}(E)^{δ} is a cyclic group, Ω_{s}(E)^{δ} is written as a product of Ω_{s}(E)^{γ} and Ω_{s}(E)^{−γ}. If t = |Ω_{s}(E)^{δ}| is even, Ω_{s}(E)^{δ} contains a unique element of order 2. In that case Ω_{s}(E)^{γ} and Ω_{s}(E)^{−γ} have a non-trivial intersection, and so t = dd′/2. If t is odd, then Ω_{s}(E)^{δ} = Ω_{s}(E)^{γ} × Ω_{s}(E)^{−γ}, and so t = dd′.

There is a surjective homomorphism Ω_{s}^{δ}(E)^{∧} → Ω_{s}^{δ}(E)_{−}^{∧} given by θ → θ^{−1}γ(θ), whose kernel is given by Ω_{s}^{δ}(E)_{γ}^{∧}. It follows that Ω_{s}^{δ}(E)_{−}^{∧} is a subgroup of Ω_{s}^{δ}(E)_{−γ}^{∧} of order t/d. Hence we have

[Ω_{s}^{δ}(E)_{−γ}^{∧} : Ω_{s}^{δ}(E)_{−}^{∧}] = \begin{cases}1 & if t = dd′, \\ 2 & if t = dd′/2. \end{cases} (5.1.1)

Similarly, we have a surjective homomorphism Ω_{s}(E)_{δ} → Ω_{s}(E)_{δ}^{+} given by z ↦ zγ(z) with kernel Ω_{s}(E)_{δ}^{−γ}. It follows that Ω_{s}(E)_{δ}^{+} is a subgroup of Ω_{s}(E)_{δ}^{γ} of degree t/d′. Hence we have

[Ω_{s}(E)_{δ}^{γ} : Ω_{s}(E)_{δ}^{+}] = \begin{cases}1 & if t = dd′, \\ 2 & if t = dd′/2. \end{cases} (5.1.2)

The following result describes the values of m_{2} for almost characters of G^{F^{2}}.

Theorem 5.2. Assume that s is a semisimple element in G^{∗F^{2}}, and E is an irreducible character of W_{s}^{0} such that Ω_{s}-orbit of E is δ-stable. Let R_{η,z} be an almost character associated to (η, z) ∈ M_{s,E}. Then

(i) Assume that s is not G^{∗}-conjugate to an element s′ such that F (s′) = s′^{−1}. Then m_{2}(R_{η,z}) = 0.

(ii) Assume that F (s) = s^{−1}. If the Ω_{s}-orbit of E is not γ-stable, then m_{2}(R_{η,z}) = 0. (iii) Assume that F (s) = s^{−1}, and that the Ω_{s}-orbit of E is γ-stable.

(a) Assume that |Ω_{s}(E)^{δ}| is odd. Then we have

m_{2}(R_{η,z}) = \begin{cases}1 & if η ∈ Ω_{s}(E)_{−γ}^{∧} and z ∈ Ω_{s}(E)^{γ}, \\ 0 & otherwise. \end{cases}

(b) Assume that |Ω_{s}(E)^{δ}| is even. Then we have

m_{2}(R_{η,z}) = \begin{cases}1 & if η ∈ Ω_{s}(E)_{−γ}^{∧} and z ∈ Ω_{s}(E)^{+}, \\ ε & if η ∈ Ω_{s}(E)_{−γ}^{∧} and z ∈ Ω_{s}(E)^{γ} − Ω_{s}(E)^{+}, \\ 0 & otherwise, \end{cases}

where ε = c_{2}(ρ_{1,z′′_{0}}) = ±1 for any z′′_{0} ∈ Ω̃_{s}(E)_{δ}^{γ} − Ω̃_{s}(E)_{δ}^{+}.

Proof. We show (i). By Theorem 4.6 (i), m_{2}(ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}) = 0 for any x ∈ (Ω_{s})_{δ}. It follows that m_{2}(ρ_{η′,z′}) = 0 for any (η′, z′) ∈ \overline{M}_{s,E}. Hence m_{2}(R_{η,z}) = 0 for any (η, z) ∈ M_{s,E}.

A similar proof works for the assertion (ii) since m_{2}(ρ_{ṡ_{x},E_{x}}) = 0 for any x ∈ (Ω_{s})_{δ} by Corollary 4.9 (i).

We show (iii) by computing m_{2}(R_{η,z}) for (η, z) ∈ M_{s,E}. By Theorem 3.6, we have m_{2}(R_{η,z}) = η(z)^{−1}m_{2}(t_{1}^{∗−1}R_{η,z}). By definition of R_{η,z}, together with Corollary 1.11, applied to the case where r = 2, we have

m_{2}(R_{η,z}) = η(z)^{−1} ∑_{(η′,z′a)∈\overline{M}_{s,E}}\{(η, z), (η′, z′)\}c_{2}(ρ_{η′,z′a}). (5.2.1)

with a = a_{E}. Moreover by Corollary 1.16 (applied to the case where m = 1, see also [K2, Theorem 2.1.3]), together with Proposition 2.7, we have

(5.2.2) c_{2}(ρ_{η′,z′a}) = \begin{cases}±1 & if (η′, z′) ∈ Ω_{s}^{δ}(E)_{−γ}^{∧} × Ω_{s}(E)_{δ}^{γ}, \\ 0 & otherwise. \end{cases}

We note the following.

(5.2.3) Assume that z′ ∈ Ω_{s}(E)_{δ}^{+}. Then c_{2}(ρ_{η′,z′a}) = 1 for any η′ ∈ Ω_{s}^{δ}(E)_{−γ}^{∧}.

We show (5.2.3). Let (x, E_{x}) corresponding to (E, z′) via f in (2.2.2). Then we have m_{2}(ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}) = |Ω_{s}(E)^{−γ}| by Corollary 4.9. Since the twisting operator t_{1}^{∗} acts trivially on ρ_{ṡ_{x},E_{x}}, we have c_{2}(ρ_{ṡ_{x}.E_{x}}|_{G^{F^{2}}}) = |Ω_{s}(E)^{−γ}| by Corollary 1.11. On the other hand, ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}} can be decomposed as in (2.2.3). Hence by (5.2.2), we have

c_{2}(ρ_{ṡ_{x},E_{x}}|_{G^{F^{2}}}) = ∑_{η′∈Ω_{s}^{δ}(E)_{−γ}^{∧}} c_{2}(ρ_{η′,z′a}).

Since |Ω_{s}(E)^{−γ}| = |Ω_{s}^{δ}(E)_{−γ}^{∧}| = d′, we can conclude that c_{2}(ρ_{η′,z′a}) = 1, and (5.2.3) follows.

We now compute m_{2}(R_{η,z}). In view of (5.2.2), the formula (5.2.1) can be written as

m_{2}(R_{η,z}) = η(z)^{−1}|Ω_{s}^{δ}(E)|^{−1} ∑_{(η′,z′)∈Ω_{s}^{δ}(E)_{−γ}^{∧}×Ω_{s}(E)_{δ}^{γ}} η(z′)η′(z)c_{2}(ρ_{η′,z′a}). (5.2.4)

First consider the case where t is odd, i.e., the case where t = dd′. Then by (5.1.2), we have Ω_{s}(E)_{δ}^{γ} = Ω_{s}(E)_{δ}^{+}. It follows by (5.2.3) that

m_{2}(R_{η,z}) = η(z)^{−1}|Ω_{s}^{δ}(E)|^{−1} ∑_{(η′,z′)∈Ω_{s}^{δ}(E)_{−γ}^{∧}×Ω_{s}(E)_{δ}^{γ}} η(z′)η′(z).

This implies that m_{2}(R_{η,z}) = 0 unless η is trivial on Ω_{s}(E)_{δ}^{γ} and z ∈ Ω_{s}^{δ}(E) is such that η′(z) = 1 for any η′ ∈ Ω_{s}^{δ}(E)_{−γ}^{∧}. But since Ω_{s}^{δ}(E)_{−γ}^{∧} = Ω_{s}^{δ}(E)_{−}^{∧}, the condition for z is equivalent to the condition that z ∈ Ω_{s}(E)^{γ}. Similarly, since Ω_{s}(E)_{δ}^{γ} = Ω_{s}(E)_{δ}^{+}, the condition for η is equivalent to the condition that η ∈ Ω_{s}(E)_{−γ}^{∧}. Now assume that η ∈ Ω_{s}^{δ}(E)_{−γ}^{∧} and z ∈ Ω_{s}(E)^{γ}. Since |Ω_{s}^{δ}(E)| = |Ω_{s}^{δ}(E)_{−γ}^{∧}| × |Ω_{s}(E)_{δ}^{γ}|, and η(z) = 1, (5.2.4) implies that m_{2}(R_{η,z}) = 1. This proves (a) of (iii).

Next we consider the case where t is even, i.e., the case where t = dd′/2. In this case, Ω_{s}^{δ}(E)_{−}^{∧} is an index two subgroup of Ω_{s}^{δ}(E)_{−γ}^{∧}, and Ω_{s}(E)_{δ}^{+} is an index two subgroup of Ω_{s}(E)_{δ}^{γ}. We fix η′_{0} ∈ Ω_{s}^{δ}(E)_{−γ}^{∧} − Ω_{s}^{δ}(E)_{−}^{∧} and z′_{0} ∈ Ω_{s}(E)_{δ}^{γ} − Ω_{s}(E)_{δ}^{+}. Then by using (5.2.3), (5.2.4) can be written as

m_{2}(R_{η,z}) = η(z)^{−1}|Ω_{s}^{δ}(E)|^{−1} ∑_{(η′,z′)∈Ω_{s}^{δ}(E)_{−}^{∧}×Ω_{s}(E)_{δ}^{+}} η(z′)η′(z)A_{η′,z′}, (5.2.5)

where

A_{η′,z′} = 1 + η′_{0}(z) + η(z′_{0})c_{2}(ρ_{η′,z′z′_{0}a}) + η(z′_{0})η′_{0}(z)c_{2}(ρ_{η′η′_{0},z′z′_{0}a}).

It is known by Corollary 3.5 that

(5.2.6) |m_{2}(R_{η,z})| = \begin{cases}1 & if (η, z) ∈ Ω_{s}(E)_{−γ}^{∧} × Ω_{s}(E)^{γ}, \\ 0 & otherwise. \end{cases}

We now assume that m_{2}(R_{η,z}) ≠ 0. Hence η ∈ Ω_{s}(E)_{−γ}^{∧} and z ∈ Ω_{s}(E)^{γ}. We note that η ∈ Ω_{s}(E)^{∧} is contained in Ω_{s}(E)_{−γ}^{∧} if and only if η is trivial on Ω_{s}(E)_{δ}^{+}. Similarly, η′ ∈ Ω_{s}^{δ}(E)^{∧} is contained in Ω_{s}^{δ}(E)_{−}^{∧} if and only if η′ is trivial on Ω_{s}(E)^{γ}. In particular, we have η(z′) = η′(z) = 1 for any (η′, z′) ∈ Ω_{s}^{δ}(E)_{−}^{∧} × Ω_{s}(E)_{δ}^{+} in the sum in (5.2.5). Since η(z), η′_{0}(z), η(z′_{0}) take values ±1, we see that m_{2}(R_{η,z}) ∈ Q. This implies that m_{2}(R_{η,z}) = ±1 by (5.2.6).

We shall consider the two cases, whether z ∈ Ω_{s}(E)^{γ} is contained in ∈ Ω_{s}(E)^{+} or not. First assume that z ∈ Ω_{s}(E)^{+}. Then η(z) = 1, η′_{0}(z) = 1 and η(z′_{0}) = ±1. Since |Ω_{s}^{δ}(E)_{−}^{∧}| × |Ω_{s}(E)_{δ}^{+}| = t/2, it follows from (5.2.5) that

m_{2}(R_{η,z})t = t + ∑_{(η′,z′)} η(z′_{0})(c_{2}(ρ_{η′,z′z′_{0}a}) + c_{2}(ρ_{η′e′_{0},z′z′_{0}a})).

Let C be the sum part of this formula. Then we have −t ≤ C ≤ t. Since m_{2}(R_{η,z}) = ±1, this forces that C = 0, and we have m_{2}(R_{η,z}) = 1.

Next assume that z ∈ Ω_{s}(E)^{γ}−Ω_{s}(E)^{+}. Since z^{−1}z′_{0} ∈ Ω_{s}(E)_{δ}^{+}, we have η(z^{−1}z′_{0}) = 1. Moreover η′_{0}(z) = −1. Hence by (5.2.5), we can write

m_{2}(R_{η,z})t = ∑_{(η′,z′)}(c_{2}(ρ_{η′,z′z′_{0}a}) − c_{2}(ρ_{η′η′_{0},z′z′_{0}a})).

But since

∑_{(η′,z′)} |c_{2}(ρ_{η′,z′z′_{0}a}) − c_{2}(ρ_{η′η′_{0},z′z′_{0}a})| ≤ t = |m_{2}(R_{η,z})t|,

we see that c_{2}(ρ_{η′,z′z′_{0}a}) = −c_{2}(ρ_{η′η′_{0},z′z′_{0}a}) has a common value for any (η′, z′), which coincides with c_{2}(ρ_{1,z′_{0}a}) = −c_{2}(ρ_{η′_{0},z′_{0}a}). This implies that m_{2}(R_{η,z}) = c_{2}(ρ_{1,z′_{0}a}) = ε. By putting z′′_{0} = z′_{0}a, we obtain the theorem.

We can now easily translate Theorem 5.2 to the form on m_{2}(ρ) for irreducible characters ρ.

Theorem 5.3. Assume that s is a semisimple element in G^{∗F^{2}}, and that E ∈ Irr W_{s}^{0} is such that the Ω_{s}-orbit of E is δ-stable. Let ρ_{η′,ζ′′} be an irreducible character of G^{F^{2}} associated to (η′, z′′) ∈ \overline{M}_{s,E}. Then

(i) Assume that s is not G^{∗}-conjugate to an element s′ such that F (s′) = s′^{−1}. Then m_{2}(ρ_{η′,z′′}) = 0.

(ii) Assume that F (s) = s^{−1}. If the Ω_{s}-orbit of E is not γ-stable, then m_{2}(ρ_{η′,z′′}) = 0. (iii) Assume that F (s) = s^{−1} and that the Ω_{s}-orbit of E is γ-stable.

(a) Assume that |Ω_{s}(E)^{δ}| is odd. Then we have

m_{2}(ρ_{η′,z′′}) = \begin{cases}1 & if η′ ∈ Ω_{s}^{δ}(E)_{−γ}^{∧} and z′′ ∈ Ω̃_{s}(E)_{δ}^{γ}, \\ 0 & otherwise. \end{cases}

(b) Assume that |Ω_{s}(E)^{δ}| is even. Then we have

m_{2}(ρ_{η′,z′′}) = \begin{cases}1 + ε & if η′ ∈ Ω_{s}^{δ}(E)_{−}^{∧} and z′′ ∈ Ω̃_{s}(E)_{δ}^{+} \\ 1 − ε & if η′ ∈ Ω_{s}^{δ}(E)_{−γ}^{∧} − Ω_{s}^{δ}(E)_{−}^{∧} and z′′ ∈ Ω̃_{s}(E)_{δ}^{+} \\ 0 & otherwise, \end{cases}

where ε = c_{2}(ρ_{1,z′′_{0}}) = ±1 for any z′′_{0} ∈ Ω̃_{s}(E)_{δ}^{γ} − Ω̃_{s}(E)_{δ}^{+}.

Proof. The assertion (i) and (ii) are already shown in the proof of Theorem 5.2. We show (iii). First assume that |Ω_{s}(E)^{δ}| is odd. Then (3.3.2) implies, in view of Theorem 5.2, that

m_{2}(ρ_{η′,z′a}) = |Ω_{s}(E)^{δ}|^{−1} ∑_{(η,z)∈Ω_{s}(E)_{−γ}^{∧}×Ω_{s}(E)^{γ}} η(z′)^{−1}η′(z)^{−1}.

It follows that m_{2}(ρ_{η′,z′a}) = 0 unless η′ is trivial on Ω_{s}(E)^{γ}, and η(z′) = 1 for any η ∈ Ω_{s}(E)_{−γ}^{∧}, and in which case m_{2}(ρ_{η′,z′a}) = 1. But this condition is equivalent to the condition that η′ ∈ Ω_{s}^{δ}(E)_{−γ}^{∧} and z′ ∈ Ω_{s}(E)_{δ}^{+}. By replacing z′a by z′′, we obtain (a).

Next assume that |Ω_{s}(E)^{δ}| is even. Let us fix z_{0} ∈ Ω_{s}(E)^{γ} − Ω_{s}(E)^{+}. Again by Theorem 5.2, we have

m_{2}(ρ_{η′,z′a}) = |Ω_{s}(E)^{δ}|^{−1} ∑_{(η,z)∈Ω_{s}(E)_{−γ}^{∧}×Ω_{s}(E)^{+}} η(z′)^{−1}η′(z)^{−1}(1 + η′(z_{0})^{−1}ε).

It follows that m_{2}(ρ_{η′,z′a}) = 0 unless η′ is trivial on Ω_{s}^{δ}(E)^{+} and η(z′) = 1 for any η ∈ Ω_{s}(E)_{−γ}^{∧}, and in which case m_{2}(ρ_{η′,z′a}) = 1 + η′(z_{0})^{−1}ε. The condition for z′ is the same as before, and η′ is trivial on Ω_{s}^{δ}(E)^{+} if and only if η′ ∈ Ω_{s}^{δ}(E)_{−γ}^{∧}. Moreover,

η′(z_{0}) = \begin{cases}1 & if η′ ∈ Ω_{s}^{δ}(E)_{−}^{∧}, \\ −1 & if η′ ∈ Ω_{s}^{δ}(E)_{−γ}^{∧} − Ω_{s}^{δ}(E)_{−}^{∧}. \end{cases}

Hence (b) holds, and the theorem is proved.

Remark 5.4. In [L3], Lusztig gave a uniform description of m_{2}(ρ) for any irreducible character ρ of G^{F^{2}} in the case where G is a connected reductive group with connected center. He expects that his formulation will be extended also to the disconnected center case. We shall compare our results with Lusztig’s conjectural description. Take (η, z) ∈ \overline{M}_{s,E}. For z ∈ Ω̃_{s}(E)_{δ}, take a representative ż ∈ Ω_{s}(E) of z, and put

\sqrt{z} = the image of \{y ∈ Ω_{s} | ^{yγ}E = E, (yγ)^{2} = żδ\} into Ω_{s}(E)_{δ}.

Then Ω_{s}(E)^{δ} acts on \sqrt{z} by the F -twisted conjugation. We denote by \sqrt{z} the corresponding permutation representation also. Let [η : \sqrt{z}] the multiplicity of η in this permutation representation. Now assume that F (s) = s^{−1} and that the Ω_{s}-orbit of E is F -stable. Then we have the following.

(5.4.1) Assume that (η, z) ∈ Ω_{s}^{δ}(E)_{−}^{∧} × Ω̃_{s}(E)_{δ}^{+}. Then we have

[η : \sqrt{z}] = \begin{cases}1 & if |Ω_{s}(E)^{δ}| is odd, \\ 2 & if |Ω_{s}(E)^{δ}| is even. \end{cases}

If (η, z) ∉ Ω_{s}^{δ}(E)_{−}^{∧} × Ω̃_{s}(E)_{δ}^{+}, then we have [η : \sqrt{z}] = 0.

In fact, in our setting, Ω̃_{s}(E)_{δ}^{+} is the set of z ∈ Ω̃_{s}(E)_{δ} such that \sqrt{z} ≠ ∅. Hence if z ∉ Ω̃_{s}(E)_{δ}^{+}, then \sqrt{z} = ∅, and so [η : \sqrt{z}] = 0. If z ∈ Ω̃_{s}(E)_{δ}^{+}, then \sqrt{z} = Ω_{s}(E)_{δ}^{−γ}y for some y ∈ \sqrt{z}. Let χ be the character of the representation \sqrt{z}. Then χ(u) = |Ω_{s}(E)_{δ}^{−γ}| if u ∈ Ω_{s}(E)^{γ} and χ(u) = 0 otherwise. It follows that

[η : \sqrt{z}] = \begin{cases}|Ω_{s}(E)_{δ}^{−γ}||Ω_{s}(E)^{γ}|/|Ω_{s}(E)^{δ}| & if η ∈ Ω_{s}^{δ}(E)_{−}^{∧}, \\ 0 & otherwise. \end{cases}

(5.4.1) follows from this.

In the connected center case, [η : \sqrt{z}] gives the value m_{2}(ρ). In our situation, by comparing with Theorem 5.3, we see that [η : \sqrt{z}] coincides with m_{2}(ρ_{η,z}) if and only if ε = 1. As an example of SL_{2} shows (see 5.7), Lusztig’s formula does not hold in general, as far as we use the parametrization of Irr G^{F^{2}} based on the choice of N ∈ 𝔤^{F} as in (2.5.2), i.e., N is given by Jordan normal form. However, as the referee pointed out, this discrepancy can be removed (at least for (s, E) treated in Lemma 5.6), by choosing N ∈ 𝔤^{F^{2}} such that F (N) = −N instead of (2.5.2) (see the discussion in Lemma 5.6).

5.5. We shall determine the value ε explicitly in certain special cases. We assume that (s, E) satisfies the property in (2.5.1). In that case, we have a bijection \overline{M}_{s,E} ≃ \overline{M}_{s,N} for a nilpotent element N ∈ 𝔤^{F} (cf. [S3, 4.5]). We have Ω_{s}(E) = Ω_{s}, and \overline{M}_{s,E} = (Ω_{s}^{δ})^{∧} × (Ω̃_{s})_{δ}. Moreover, \overline{M}_{s,N} = (\overline{A}_{λ})_{δ} × (\overline{A}_{λ}^{δ})^{∧}, and the above bijection is given through bijections (Ω_{s}^{δ})^{∧} ≃ (\overline{A}_{λ})_{δ} and (Ω̃_{s})_{δ} ≃ (\overline{A}_{λ}^{δ})^{∧}. (Compare with 3.7). These bijections can be explicitly described in a similar way as in 3.7. Now there exists g ∈ G such that ^{g}N = −N. Take α ∈ G such that α^{−1}F (α) = g, and put N′ = ^{α}N. Then we have F (N′) = −N′. Since N′ ∈ 𝔤^{F^{2}}, one can write N′ = N_{c_{1}} for a c_{1} ∈ (A_{λ})_{δ}. By the isomorphism (\overline{A}_{λ})_{δ} ≃ (Ω_{s}^{δ})^{∧}, c_{1} determines a unique element η_{1} ∈ (Ω_{s}^{δ})^{∧}. The following result was inspired by the comment of the referee. We are very grateful to the referee for this.

Lemma 5.6. Assume that (s, E) is as in (2.5.1), and that we are in a setting in (iii),

(b) in Theorem 5.3. Let η_{1} ∈ (Ω_{s}^{δ})^{∧} be as in 5.5. Then we have η_{1} ∈ (Ω_{s}^{δ})−γ^{∧}, and

ε = \begin{cases}1 & if η_{1} ∈ (Ω_{s}^{δ})_{−}^{∧}, \\ −1 & if η_{1} ∈ (Ω_{s}^{δ})_{−γ}^{∧} − (Ω_{s}^{δ})_{−}^{∧}. \end{cases}

Proof. We consider the generalized Gelfand-Graev character Γ_{N′} = Ind_{U^{F^{2}}}^{G^{F^{2}}}(Λ_{c_{1}}). Since F (N′) = −N′, we see that Λ_{c_{1}} is a linear character of U^{F^{2}} such that F (Λ_{c_{1}}) = \overline{Λ}_{c_{1}}. Then there exists a basis of the representation Γ_{N′} such that the corresponding matrix R(g) satisfies the property (*) that R(F (g)) coincides with the complex conjugate of R(g) for all g ∈ G^{F^{2}}. Let ρ be an irreducible character of G^{F^{2}} such that F (ρ) = \overline{ρ}. Then the representation obtained as the ρ-isotypic part of Γ_{N′} also satisfies this property. It follows that if ρ has multiplicity one in Γ_{N′}, then ρ is afforded by a matrix representation satisfying the property (*) as above. Now it is known by [KM], for any ρ ∈ Irr G^{F^{2}}, that c_{2}(ρ) = 1 if and only if ρ is afforded by a representation satisfying (*). Summing up the above argument, we see that c_{2}(ρ) = 1 if ⟨Γ_{N′}, ρ⟩ = 1.

Now by our parametrization, ρ_{c_{1},ξ_{x}} ((c_{1}, ξ_{x}) ∈ \overline{M}_{s,N}) is the unique character of G^{F^{2}} appearing in Γ_{N′} and in ρ_{ṡ_{x},E} with multiplicity one. By \overline{M}_{s,N} ≃ \overline{M}_{s,E}, (c_{1}, ξ_{x}) ↔ (η_{1}, x), this implies that c_{2}(ρ_{η_{1},x}) = 1 for any x ∈ (Ω̃_{s})_{δ}. We use the notation in the proof of Theorem 5.2. By (5.2.2), we know that c_{2}(ρ_{η′,z′a}) = 0 unless η′ ∈ (Ω_{s}^{δ})_{−γ}^{∧} for (η′, z′a) ∈ \overline{M}_{s,E}. It follows that η_{1} ∈ (Ω_{s}^{δ})_{−γ}^{∧}. In the last part of the proof of Theorem 5.2, we know that c_{2}(ρ_{η′,z′z′_{0}a}) = −c_{2}(ρ_{η′η′_{0},z′z′_{0}a}) has a common value for any (η′, z′) ∈ (Ω_{s}^{δ})_{−}^{∧} × (Ω_{s})_{δ}^{+}, which coincides with ε. So if η_{1} ∈ (Ω_{s}^{δ})_{−}^{∧} (resp. η_{1} ∉ (Ω_{s}^{δ})_{−}^{∧}), then η_{1} = η′ (resp. η_{1} = η′η′_{0}) for some η′, and we have ε = 1 (resp. ε = −1), respectively. This proves the lemma.

5.7. We give an example in the case where G = SL_{2} and G̃ = GL_{2}. Assume that p is odd, and let ṡ = Diag(1, −1) be a regular semisimple element in G̃^{∗} and put s = π(ṡ). Then we have W_{s} = Ω_{s} ≃ Z/2Z, on which F acts trivially. We consider the pair (s, E), where E is the trivial character of W_{s}^{0} = {1}. Then (s, E) satisfies the property (2.5.1), and we have \overline{M}_{s,E} = Ω_{s}^{∧} × Ω_{s}. Since F (s) = s = s^{−1}, and |Ω_{s}| = 2, we see that the pair (s, E) falls in the class (iii), (b) in Theorem 5.2. We have (Ω_{s})_{−γ}^{∧} = Ω_{s}^{∧}, and (Ω_{s})_{−}^{∧} = {1}.

On the other hand for ρ = ρ_{ṡ,E} ∈ Irr G̃^{F^{2}}, O_{ρ} = O_{N}, where N = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) is a

regular nilpotent element in 𝔤^{F}. We have A_{λ} ≃ Z/2Z on which F acts trivially. Put t = Diag(α, α^{−1}) ∈ G, then ^{t}N = α^{2}N. We choose α ∈ \overline{F}_{q} such that α^{2} = −1. Take g ∈ G such that g^{−1}F (g) = t. Then N′ = ^{g}N satisfies the relation F (N′) = −N′. We can choose g = Diag(β, β^{−1}) such that β^{q−1} = α. Then g^{−1}F^{2}(g) = Diag(γ, γ^{−1}), where

γ = β^{q^{2}−1} = α^{q+1} = \begin{cases}1 & if q ≡ 3 (mod 4), \\ −1 & if q ≡ 1 (mod 4). \end{cases}

It follows that N′ = N_{c_{1}} with c_{1} ∈ A_{λ} ≃ {±1}, where c_{1} = 1 (resp. c_{1} = −1) if q ≡ 3 (mod 4) (resp. q ≡ 1 (mod 4)), respectively. Thus under the isomorphism A_{λ} ≃ Ω_{s}^{∧}, c_{1} corresponds to η_{1} ∈ Ω_{s}^{∧} = {±1}. In view of Lemma 5.6, we conclude that

(5.7.1) ε = \begin{cases}1 & if q ≡ 3 (mod 4), \\ −1 & if q ≡ 1 (mod 4). \end{cases}

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