Generic character sheaves on disconnected groups and character values
arXiv:0704.0999v1  [math.RT]  7 Apr 2007
GENERIC CHARACTER SHEAVES ON
DISCONNECTED GROUPS AND CHARACTER VALUES
G. Lusztig
Introduction
The theory of character sheaves [L3] on a reductive group G over an alge-
braically closed field and the theory of irreducible characters of G over a finite
field are two parallel theories; the first one is geometric (involving intersection
cohomology complexes on G), the second one involves functions on the group of
rational points of G. In the case where G is connected, a bridge between the two
theories was constructed in [L1] and strengthened in [L2], [S]. In this paper we
begin the construction of the analogous bridge in the general case, extending the
method of [L1]. Here we restrict ourselves to character sheaves which are ”generic”
(in particular their support is a full connected component of G) and show how such
character sheaves are related to characters of representations (see Theorem 1.2).
Contents
1. Statement of the Theorem.
2. Constructing representations of GF .
3. Proof of Theorem 1.2.
1. Statement of the Theorem
1.1. Let k be an algebraic closure of a finite field Fq. Let G be a reductive
algebraic group over k with identity component G0 such that G/G0 is cyclic,
generated by a fixed connected component D. We assume that G has a fixed
Fq-rational structure with Frobenius map F : G −→ G such that F (D) = D. Let
l be a prime number invertible in k; let Q̄l be an algebraic closure of the l-adic
numbers. All group representations are assumed to be finite dimensional over Q̄l.
We say ”local system” instead of ”Q̄l-local system”.
Let B be the variety of Borel subgroups of G0. Now F : G −→ G induces a
morphism B −→ B denoted again by F . We fix B∗ ∈ B and a maximal torus T of
B∗ such that F (B∗) = B∗, F (T ) = T . Let U∗ be the unipotent radical of B∗. Let
Supported in part by the National Science Foundation.
Typeset by AMS-TEX
http://arxiv.org/abs/0704.0999v1
2 G. LUSZTIG
NB∗ (resp. NT ) be the normalizer of B∗ (resp. T ) in G. Let T̃ = NT ∩NB∗, a
closed F -stable subgroup of G with identity component T . Let T̃D = T̃ ∩D.
Let N = NT ∩ G0. Let W = N /T be the Weyl group. Let D : T
−→ T ,
D : W
−→ W be the automorphisms induced by Ad(d) : N −→ N where d is any
element of T̃D. Now F : N −→ N induces an automorphism ofW denoted again by
F . For w ∈ W let [w] be the inverse image of w under the obvious map N −→ W
and let w be the automorphism Ad(x) : T −→ T for any x ∈ [w]. For w ∈ W
let Ow be the G
0-orbit in B × B (G0 acting by simultaneous conjugation on both
factors) that contains (B∗, xB∗x−1) for some/any x ∈ [w]. Define the ”length
function” l : W −→ N by l(w) = dimOw − dimB. For any y ∈ G
0 we define
k(y) ∈ N by y ∈ U∗k(y)U∗. For y ∈ G0, τ ∈ T̃ we have k(τyτ−1) = τk(y)τ−1
and F (k(y)) = k(F (y)). For x ∈ G0 we define Fx : G −→ G by Fx(g) = xF (g)x
this is the Frobenius map for an Fq-rational structure on G. (Indeed if y ∈ G
such that x = y−1F (y), then Ad(y) : G
−→ G carries Fx to F .) If w ∈W satisfies
D(w) = w and x ∈ [w] then T, T̃ are Fx-stable; thus Fx is the Frobenius map for
an Fq-rational structure on T̃ whose group of rational points is T̃
Fx . Since T̃FxD
is the set of rational points of T̃D (a homogeneous T -space under left translation)
for the rational structure defined by Fx : T̃D −→ T̃D, we have T̃
D 6= ∅.
Let Z∅ = {(B0, g) ∈ B ×D; gB0g
−1 = B0}. Let d ∈ T̃D. We set
Ż∅,d = {(h0U
∗, g) ∈ (G0/U∗)×D; h−10 gh0d
−1 ∈ B∗}.
Define a∅ : Ż∅,d −→ Z∅ by (h0U
∗, g) 7→ (h0B
∗h−10 , g). Now a∅ is a principal T -
bundle where T acts (freely) on Ż∅,d by t0 : (h0U
∗, g) 7→ (h0t
0 , g). Define p∅ :
Z∅ −→ D by (B0, g) 7→ g. We define b∅ : Ż∅,d −→ T by (h0U
∗, g) 7→ k(h−10 gh0d
Note that b∅ commutes with the T -actions where T acts on T by
(a) t0 : t 7→ t0tD(t
Let L be a local system of rank 1 on T such that
(i) L⊗n ∼= Q̄l for some n ≥ 1 invertible in k;
(ii) D∗L ∼= L;
From (i),(ii) we see (using [L3, 28.2(a)]) that L is equivariant for the T -action (a)
on T . Hence b∗
L is a T -equivariant local system on Ż∅,d. Since a∅ is a principal
T -bundle there is a well defined local system L̃∅ on Z∅ such that a
L̃∅ = b
Note that the isomorphism class of L̃∅ is independent of the choice of d. Assume
in addition that:
(iii) {w ∈W ;D(w) = w,w∗L ∼= L} = {1}.
We show:
(b) p∅!L̃∅ is an irreducible intersection cohomology complex on D.
We identify Z∅ with the variety X = {(g, xB
∗) ∈ G ×G0/B∗; x−1gx ∈ NB∗} (as
in [L3, I, 5.4] with P = B∗, L = T, S = T̃D) by (g, xB
∗) ↔ (xB∗x−1, g). Then L̃∅
becomes the local system Ē on X defined as in [L3, I, 5.6] in terms of the local
system E = j∗L on T̃D where j : T̃D −→ T is y 7→ d
−1y. (Note that E is equivariant
GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES3
for the conjugation action of T on T̃D.) In our case we have Ē = IC(X, Ē) since X
is smooth. Hence from [L3, I, 5.7] we see that p∅!Ē is an intersection cohomology
complex on D corresponding to a semisimple local system on an open dense subset
of D which, by the results in [L3, II, 7.10], is irreducible if and only if the following
condition is satisfied: if w ∈W,x ∈ [w] satisfy Ad(x)(T̃D) = T̃D and Ad(x)
∗E ∼= E ,
then w = 1. This is clearly equivalent to condition (iii). This proves (b).
From (b) and the definitions we see that p∅!L̃∅[dimD] is a character sheaf on D
in the sense of [L3, VI]. A character sheaf on D of this form is said to be generic.
We can state the following result.
Theorem 1.2. Let A be a generic character sheaf on D such that F ∗A ∼= A
where F : D −→ D is the restriction of F : G −→ G. Let ψ : F ∗A −→ A be an
isomorphism. Define χψ : D
F −→ Q̄l by g 7→
i∈Z(−1)
itr(ψ,Hig(A)) where H
the i-th cohomology sheaf and Hig is its stalk at g. There exists a G
F -module V
and a scalar λ ∈ Q̄∗l such that χψ(g) = λtr(g, V ) for all g ∈ D
The proof is given in §3. We now make some preliminary observations. In
the setup of 1.1 we have A = p∅!L̃∅[dimD] where L satisfies 1.1(i),(ii),(iii) and
F ∗(p∅!L̃∅) ∼= p∅!L̃∅. Hence we have p∅!F̃
∗L∅ ∼= p∅!L̃∅. By a computation in [L3,
IV, 21.18] we deduce that there exists w′ ∈ W such that D(w′) = w′, w′∗F ∗L ∼= L.
Setting w = F (w′) we see that
(a) D(w) = w, F ∗w∗L ∼= L.
1.3. Let w = (w1, w2, . . . , wr) be a sequence in W . Let lw = l(w1)+ l(w2)+ · · ·+
l(wr). Let
Zw = {(B0, B1, . . . , Br, g) ∈ B
r+1×D; gB0g
−1 = Br, (Bi−1, Bi) ∈ Owi(i ∈ [1, r])}.
This agrees with the definition in 1.1 when r = 0, that is w = ∅. Let d ∈ T̃D. We
define Żw,d as in 1.1 when r = 0 and by
Żw,d = {(h0U
∗, h1B
∗, . . . , hr−1B
∗, hrU
∗, g) ∈
(G0/U∗)× (G0/B∗)× . . .× (G0/B∗)× (G0/U∗)×D;
k(h−1i−1hi) ∈ [wi](i ∈ [1, r]), h
r gh0d
−1 ∈ U∗};
when r ≥ 1. Define aw : Żw,d −→ Zw as in 1.1 when r = 0 and by
∗, h1B
∗, . . . , hr−1B
∗, hrU
∗, g) 7→
∗h−10 , h1B
∗h−11 , . . . , hr−1B
∗hr−1, hrB
∗h−1r , g),
when r ≥ 1. Note that aw is a principal T -bundle where T acts (freely) on Żw,d
as in 1.1 when r = 0 and by
t0 : (h0U
∗, h1B
∗, . . . , hr−1B
∗, hrU
∗, g) 7→
∗, h1B
∗, . . . , hr−1B
∗, hrdt
−1U∗, g)
4 G. LUSZTIG
when r ≥ 1. Define pw : Zw −→ D by (B0, B1, . . . , Br, g) 7→ g.
In the remainder of this subsection we assume that w1w2 . . . wr = 1; this holds
automatically when r = 0. We define bw : Żw,d −→ T as in 1.1 when r = 0 and by
∗, h1B
∗, . . . , hr−1B
∗, hrU
∗, g) 7→ k(h−10 h1)k(h
1 h2) . . . k(h
r−1hr)
when r ≥ 1. Note that bw commutes with the T -actions where T acts on T as in
1.1(a).
Let L be a local system of rank 1 on T such that 1.1(i),(ii) hold. As in 1.1, L
is equivariant for the T -action 1.1(a) on T . Hence b∗
L is a T -equivariant local
system on Żw,d. Since aw is a principal T -bundle there is a well defined local
system L̃w on Zw such that a
L̃w = b
Lemma 1.4. Assume that w1w2 . . . wr = 1 and that L (as in 1.3) satisfies
(i) α̌∗L 6∼= Q̄l for any coroot α̌ : k
∗ −→ T .
Then pw!L̃w[lw](lw/2) ∼= p∅!L̃∅. (Note that lw is even.)
Assume first that for some i ∈ [1, r] we have wi = w
i where w
i in W
satisfy l(w′iw
i ) = l(w
i) + l(w
i ). Let
′ = (w1, w2, . . . , wi−1, w
i , wi+1, . . . , wn).
The map (B0, B1, . . . , Br+1, g) 7→ (B0, B1, Bi−1, Bi+1, . . . , Br+1, g) defines an iso-
morphism Zw′ −→ Zw compatible with the maps pw′ , pw and with the local systems
L̃w′ , L̃w. Since lw′ = lw we have
(a) pw!L̃w[lw](lw/2) ∼= pw′!L̃w′ [lw′ ](lw′/2).
Using (a) repeatedly we can assume that l(wi) = 1 for all i ∈ [1, r]. We will prove
the result in this case by induction on r. Note that r is even. When r = 0 the
result is obvious. We now assume that r ≥ 2. Since w1w2 . . . wr = 1, we can
find j ∈ [1, r − 1] such that l(w1w2 . . . wj) = j, l(w1w2 . . . wj+1) = j − 1. We can
find a sequence w′ = (w′1, w
2, . . . , w
r) in W such that l(w
i) = 1 for all i ∈ [1, r],
2 . . .w
j = w1w2 . . . wj , w
j = w
j+1, w
i = wi for i ∈ [j + 1, r]. Let
u = (w1w2 . . . wj , wj+1, . . . , wr) = (w
2 . . . w
j , w
j+1, . . . , w
Using (a) repeatedly we see that
pw!L̃w[lw](lw/2) ∼= pu!L̃u[lu](lu/2) ∼= pw′!L̃w′ [lw′ ](lw′/2).
Replacing w by w′ we see that we may assume in addition that wj = wj+1
for some j ∈ [1, r − 1]. We have a partition Zw = Z
∪ Z ′′
where Z ′
(resp.
) is defined by the condition Bj−1 = Bj+1 (resp. Bj−1 6= Bj+1). Let w
(w1, w, . . . , wj−1, wj+2, . . . , wr), w
′′ = (w1, w, . . . , wj−1, wj+1, . . . , wr). Define c :
−→ Zw′ by
(B0, B1, . . . , Br, g) 7→ (B0, B1, . . . , Bj−1, Bj+2, . . . , Br, g).
GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES5
This is an affine line bundle and L̃w|Z′
= c∗L̃w′ . Let p
be the restriction of pw
to Z ′
. We have p′
= pw′c. Since the induction hypothesis applies to w
′ we have
w!(L̃w|Z′w)[lw](lw/2) = pw′!c!c
∗L̃w′ [lw](lw/2)
= pw′!L̃w′ [−2](−1)[lw](lw/2) = pw′!L̃w′ [lw′ ](lw′/2) = p∅!L̃∅.(b)
Define e : Z ′′
−→ Zw′′ by
(B0, B1, . . . , Br, g) 7→ (B0, B1, . . . , Bj−1, Bj+1, . . . , Br, g).
Let p′′
be the restriction of pw to Z
. We have p′′
= pw′′e. We show that
w!(L̃w|Z′′w) = 0. It is enough to show that
(c) pw′′!e!(L̃w|Z′′
) = 0.
Hence it is enough to show that e!(L̃w|Z′′
) = 0. It is also enough to show that, if
E is a fibre of e, then Hic(E, L̃w|E) = 0 for any i. As in the proof of [L3, VI, 28.10]
we may identify E = k∗ in such a way that L̃w|E becomes α̌
∗(L) for some coroot
α̌ : k∗ −→ T . We then use that Hic(k
∗, α̌∗L) = 0 which follows from α̌∗L 6∼= Q̄l.
Using (c) and the exact triangle
(pw′′!e!(L̃w|Z′′
), pw!L̃w, p
w!(L̃w|Z′w))
we see that
pw!L̃w[lw](lw/2) = p
w!(L̃w|Z′w)[lw])(lw/2) = p∅!L̃∅
(the last equality follows from (b)). The lemma is proved.
Lemma 1.5. Assume that L (as in 1.3) satisfies 1.1(iii). Then L satisfies 1.4(i).
Let RL be the set of roots α : T −→ k
∗ such that the corresponding coroot α̌
satisfies α̌∗L ∼= Q̄l. Let WL be the subgroup of W generated by the reflections
with respect to the various α ∈ RL. Since D
∗L ∼= L we have D(WL) = WL.
Assume that 1.4(i) does not hold. Then RL 6= ∅ and WL 6= {1}. By [DL, 5.17]
the fixed point set of D : WL −→ WL is 6= {1}. Let w ∈ WL − {1} be such that
D(d)w = w. Since w ∈WL we have w
∗L ∼= L (see [L3, VI, 28.3(b)]). Thus 1.1(iii)
does not hold. The lemma is proved.
2. Constructing representations of GF
2.1. In this section we construct some representations of GF using the method of
[DL]. See [M],[DM] for other results in this direction.
Let L be a local system of rank 1 on T such that 1.1(i) holds. For any t ∈ T let
Lt be the stalk of L at t. Assume that we are given w ∈W and x ∈ [w] such that
6 G. LUSZTIG
(i) F ∗xL
∼= L;
(Fx : T −→ T as in 1.1). Let φ : F
xL −→ L be the unique isomorphism of
local systems on T which induces the identity map on L1. For t ∈ T , φ induces
an isomorphism LFx(t)
−→ Lt. When t ∈ T
Fx this is an automorphism of the
1-dimensional vector space Lt given by multiplication by θ(t) ∈ Q̄
l . It is well
known that t 7→ θ(t) is a group homomorphism TFx −→ Q̄∗l .
Following [DL] we define
Y = {hU∗ ∈ G0/U∗; h−1F (h) ∈ U∗xU∗}.
For (g, t) ∈ G0F × TFx we define eg,t : Y −→ Y by hU
∗ 7→ ght−1U∗. Note
that (g, t) 7→ eg,t is an action of G
0F × TFx on Y . Hence G0F × TFx acts on
Hic(Y ) := H
c(Y, Q̄l) by (g, τ) 7→ e
g−1,τ−1
. We set
Hic(Y )θ = {ξ ∈ H
c(Y ); e
1,t−1ξ = θ(t)
−1ξ for all t ∈ TFx};
this is a G0F × TFx-stable subspace of Hic(Y ).
For g ∈ G0F we define ǫg : H
c(Y )θ −→ H
c(Y )θ by ǫg(ξ) = e
g−1,1
. This makes
Hic(Y )θ into a G
0F -module.
We can find an integer r ≥ 1 such that
F r(x) = x, xF (x) . . . F r−1(x) = 1.
Indeed we first find an integer r1 ≥ 1 such that F
r1(x) = x and then we find
an integer r2 ≥ 1 such that (xF (x) . . .F
r1−1(x))r2 = 1. Then r = r1r2 has the
required properties. Then hU∗ 7→ F r(h)U∗ is a well defined map Y −→ Y denoted
again by F r. Also,
F r = F rx : G −→ G.
(We have F rx (g) = (xF (x) . . . F
r−1(x))F r(g)(xF (x) . . .F r−1(x))−1 = F r(g).) Hence
F r acts trivially on TFx . We see that F r : Y −→ Y commutes with eg,t : Y −→ Y
for any (g, t) ∈ G0F × TFx . Hence (F r)∗ : Hic(Y ) −→ H
c(Y ) leaves stable the
subspace Hic(Y )θ. Note that:
for any i, all eigenvalues of (F r)∗ : Hic(Y ) −→ H
c(Y ) are of the form root of 1
times qnr/2 where n ∈ Z.
(See [L1, 6.1(e)] and the references there.)
Replacing r by an integer multiple we may therefore assume that r satisfies in
addition the following condition:
(a) for any i, all eigenvalues of (F r)∗ : Hic(Y ) −→ H
c(Y ) are of the form q
where n ∈ Z.
2.2. We preserve the setup of 2.1 and assume in addition that L satisfies 1.4(i).
Let i0 = 2dimU
∗ − l(w). Note that
(a) Hic(Y )θ = 0 for i 6= i0; if i = i0 then all eigenvalues of (F
r)∗ : Hic(Y )θ −→
Hic(Y )θ are of the form q
ir/2.
For the first statement in (a) see [DL, 9.9] and the remarks in the proof of [L1,
8.15]. The second statement in (a) is deduced from 2.1(a) as in the proof of [L1,
6.6(c)].
GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES7
2.3. We preserve the setup of 2.1 and assume in addition that L satisfies 1.1(ii)
and that w ∈W satisfies D(w) = w. From the definitions we see that D : T −→ T
commutes with Fx : T −→ T hence D restricts to an automorphism of T
Fx and
(a) θ(D(t)) = θ(t) for any t ∈ TFx .
We show:
(b) there exists a homomorphism θ̃ : T̃Fx −→ Q̄∗l such that θ̃|TFx = θ.
Let d ∈ T̃FxD . Let n = |G/G
0| = |T̃Fx/TFx |. Then t0 := d
n ∈ TFx . Let c ∈ Q̄∗l
be such that cn = θ(t0). For any t ∈ T
Fx and j ∈ Z we set θ̃(djt) = cjθ(t).
This is well defined: if djt = dj
t′ with j, j′ ∈ Z, t, t′ ∈ TFx then j′ = j + nj0,
j0 ∈ Z and t
′ = t
0 t so that θ(t
′) = cnj0θ(t) and cjθ(t) = cj
θ(t′). We show that
if j, j′ ∈ Z, t, t′ ∈ TFx then θ̃(djtdj
t′) = θ̃(djt)θ̃(dj
t′) that is cj+j
θ(D−j
(t)t′) =
cjθ(t)cj
θ(t′); this follows from (a). This proves (b).
Let Γ = {(g, τ) ∈ GF×T̃Fx ; gτ−1 ∈ G0}, a subgroup of GF×T̃Fx . For (g, τ) ∈ Γ
we define eg,τ : Y −→ Y by hU
∗ 7→ ghτ−1U∗. To see that this is well defined we
assume that h ∈ G0 satisfies h−1F (h) ∈ U∗xU∗ and (g, τ) ∈ Γ; we compute
(ghτ−1)−1F (ghτ−1) = τh−1g−1gF (h)F (τ−1)
= τh−1F (h)F (τ−1) ∈ τU∗xU∗F (τ−1) = U∗τxF (τ−1)U∗ = U∗xU∗,
since τxF (τ−1) = x (that is Fx(τ) = τ). Note that (g, τ) 7→ eg,τ is an action
of Γ on Y (extending the action of G0F × TFx). Hence Γ acts on Hic(Y ) by
(g, τ) 7→ e∗
g−1,τ−1
. Note that Hic(Y )θ is a Γ-stable subspace of H
c(Y ). This follows
from the identity
eg−1,τ−1e1,t−1 = e1,τ−1t−1τeg−1,τ−1
for g ∈ GF , τ ∈ T̃Fx , t ∈ TFx together with the identity θ(t) = θ(τ−1tτ) which is
a consequence of (a).
For g ∈ GF we define ǫg : H
c(Y )θ −→ H
c(Y )θ by
ǫg(ξ) = θ̃(τ)e
g−1,τ−1ξ
for any ξ ∈ Hic(Y )θ and any τ ∈ T̃
Fx such that gτ−1 ∈ G0. Assume that τ ′ ∈ T̃Fx
is another element such that gτ ′−1 ∈ G0. Then τ ′ = τt with t ∈ TFx and
θ̃(τ ′)e∗g−1,τ ′−1ξ = θ̃(τ)θ(t)e
g−1,τ−1e
1,t−1ξ = θ̃(τ)e
g−1,τ−1ξ
so that ǫg is well defined. For g, g
′ in GF we choose τ, τ ′ in T̃Fx such that gτ−1 ∈
G0, g′τ ′−1 ∈ G0; we have
ǫgǫg′ξ = θ̃(τ
′)θ̃(τ)e∗g−1,τ−1e
g′−1,τ ′−1ξ = θ̃(ττ
′)e∗(gg′)−1,(ττ ′)−1ξ = ǫgg′ξ.
We see that
8 G. LUSZTIG
g 7→ ǫg defines a G
F -module structure on Hic(Y )θ extending the G
0F -module
structure in 2.1.
(Note that this extension depends on the choice of θ̃.) We show:
(c) If (g, τ) ∈ Γ then F reg,τ : Y −→ Y is the Frobenius map of an Fq-rational
structure on Y .
Since eg,t is a part of a Γ-action, it has finite order. Since F
r = F rx : G −→ G (see
2.1), we see that F r : Y −→ Y commutes with eg,τ : Y −→ Y . Hence (c) holds.
2.4. We preserve the setup of 2.3 and assume in addition that L satisfies 1.3(i).
Let i0 = 2dimU
∗ − l(w). Using 2.2(a), 2.3(c) and Grothendieck’s trace formula
we see that for (g, d) ∈ Γ we have
(−1)l(w)θ̃(d)qi0r/2tr(ǫg, H
c (Y )θ)
= θ̃(d)
(−1)itr((F r)∗ǫg, H
c(Y )θ) =
(−1)itr((F r)∗e∗g−1,d−1 , H
c(Y )θ)
(−1)i|TFx |−1
t∈TFx
tr((F r)∗e∗g−1,d−1e
1,t−1 , H
c(Y ))θ(t)
= |TFx |−1
t∈TFx
(−1)itr((F r)∗e∗g−1,(dt)−1 , H
c(Y ))θ(t)
= |TFx |−1
t∈TFx
g−1,(dt)−1 |θ(t)
= |TFx |−1
t∈TFx
|{hU∗ ∈ (G0/U∗); h−1F (h) ∈ U∗xU∗, h−1g−1F r(h)dt ∈ U∗}|θ(t).
3. Proof of Theorem 1.2
3.1. Let A, ψ, χψ be as in 1.2. Let L, w be as in the end of 1.2. Let x ∈ [w]. From
1.2(a) we see that 2.1(i) holds. Let r ≥ 1 be as in 2.1. Let
w = (w, F (w), . . . , F r−1(w)).
By the choice of r we have wF (w) . . . F r−1(w) = 1. Define a morphism F̃ : Zw −→
Zw by
F̃ (B0, B1, . . . , Br, g) = (F (g
−1Br−1g), F (B0), F (B1), . . . , F (Br−1), F (g)).
We show:
(a) Let g ∈ DF and let F̃g : p
(g) −→ p−1
(g) be the restriction of F̃ : Zw −→ Zw.
Then F̃g is the Frobenius map of an Fq-rational structure on p
It is enough to note that the map Br+1 −→ Br+1 given by
(B0, B1, . . . , Br) 7→ (F (g
−1Br−1g), F (B0), F (B1), . . . , F (Br−1))
GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES9
is the composition of the map
F ′ : (B0, B1, . . . , Br) 7→ (F (B0), F (B1), . . . , F (Br))
(the Frobenius map of an Fq-rational structure on B
r+1) with the automorphism
(B0, B1, . . . , Br) 7→ (g
−1Br−1g, B0, B1, . . . , Br−1)
of Br+1 which commutes with F ′ and has finite order (since g has finite order in
Let d ∈ T̃FxD . Define a morphism F̃
′ : Żw,d −→ Żw,d by
F̃ ′(h0U
∗, h1B
∗, . . . , hr−1B
∗, hrU
∗, g) = (h′0U
∗, h′1B
∗, . . . , h′r−1B
∗, h′rU
∗, F (g))
where
h′0 = F (g
−1hr−1k(h
r−1hr))x
−1d, h′r = F (hr−1k(h
r−1hr)x
h′i = F (hi−1) for i ∈ [1, r − 1].
This is well defined since
(F (hr−1k(h
r−1hr)x
−1)−1F (g)F (g−1hr−1k(h
r−1hr))x
−1)dd−1 = 1.
We show that the T -action on Żw,d (see 1.3) satisfies F̃
′(t0x̃) = Fx(t0)F̃
′(x̃) for
t0 ∈ T, x̃ ∈ Żw,d. Let (hi) be as above. We must show:
F (g−1hr−1k(h
r−1hrdt
−1))x−1d = F (g−1hr−1k(h
r−1hr))x
−1dxF (t−10 )x
F (hr−1k(h
r−1hrdt
−1)x−1 = F (hr−1k(h
r−1hr)x
−1dxF (t0)
−1x−1d−1,
which follow from F (d) = x−1dx. Note that
(b) awF̃
′ = F̃ aw : Żw,d −→ Zw.
We show:
(c) |a−1
(y)F̃
| = |TFx | for any y ∈ ZF̃
Since a−1
(y) is a homogeneous T -space this follows from Lang’s theorem applied
to (T, Fx).
We have
(d) pwF̃ = Fpw : Zw −→ D.
3.2. We show:
(a) bwF̃
′ = Fxbw : Żw,d −→ T .
Let (h0, h1, . . . , hr, g) ∈ (G
0)r+1 ×D be such that
∗, h1B
∗, . . . , hr−1B
∗, hrU
∗, g) ∈ Żw,d.
10 G. LUSZTIG
Let (h′1, h
2, . . . , h
r) be as in 3.1. We set
µ = k(h−10 h1)k(h
1 h2) . . . k(h
r−1hr) ∈ T,
µ′ = k(h−10 h1)k(h
1 h2) . . . k(h
r−2hr−1) ∈ B
∗F r−1(x)−1B∗
µ̃ = k(h′0
−1h′1)k(h
−1h′2) . . . k(h
−1h′r) ∈ T
so that µ = µ′k(h−1r−1hr) and
µ̃ = k(d−1xF (k(h−1r−1hr)
−1h−1r−1gh0))
× k(F (h−10 h1)) . . . k(F (h
r−3hr−2))k(F (h
r−2hr−1k(h
r−1hr))x
= d−1xF (k(h−1r−1hr)
−1)F (d)k(F (d−1)F (h−1r−1gh0))F (µ
′)F (k(h−1r−1hr))x
= d−1xF (d)F (µ)x−1 = xF (µ)x−1 = Fx(µ),
as required.
3.3. Let φ : F ∗xL
−→ L, θ : TFx −→ Q̄∗l be as in 2.1. We shall denote by ? the
various isomorphisms induced by φ such as:
(a) F̃ ′∗b∗
L = b∗
F ∗xL
−→ b∗
L (see 3.2(a)),
(b) F̃ ′∗a∗
−→ a∗
L̃w (coming from (a)),
(c) a∗
F̃ ∗L̃w
−→ a∗
L̃w (see (b) and 3.1(b)),
(d) F̃ ∗L̃w
−→ L̃w (coming from (c)),
(e) pw!F̃
−→ pw!L̃w (coming from (d)),
(f) F ∗pw!L̃w
−→ pw!L̃w (coming from (e) and 3.1(d)).
(g) F ∗(pw!L̃w[lw])
−→ pw!L̃w[lw] (coming from (f)).
3.4. For any g ∈ DF we compute
(−1)itr(?,Hig(pw!L̃w)) =
(−1)itr(?, Hic(p
(g), L̃w))
(g);F̃ (y)=y
tr(?, (L̃w)y)
where Hi is the i-th cohomology sheaf. (The last two sums are equal by the
Grothendieck trace formula applied in the context of 3.1(a).) Using 3.1(c) we see
that the last sum equals
|TFx |−1
(g))F̃
tr(?, (a∗
L̃w)ỹ) = |T
Fx |−1
(g))F̃
tr(?, (b∗
Lw)ỹ)
= |TFx |−1
(g))F̃
tr(?, (Lw)bw(ỹ)).
GENERIC CHARACTER SHEAVES ON DISCONNECTED GROUPS AND CHARACTER VALUES11
Now a−1
(g))F̃
can be identified with the set of all
∗, h1B
∗, . . . , hr−1B
∗, hrU
∗) ∈ (G0/U∗)×(G0/B∗)× . . .×(G0/B∗)×(G0/U∗)
such that
(a) k(h−1i−1hi) ∈ F
i−1(x)T for i ∈ [1, r],
(b) h−1r gh0d
−1 ∈ U∗,
(c) h0U
∗ = F (g−1hr−1k(h
r−1hr))x
−1dU∗,
(d) hiB
∗ = F (hi−1)B
∗ for i ∈ [1, r − 1].
(We then have automatically hrU
∗ = F (hr−1k(h
r−1hr)x
−1U∗.) If h0U
∗ is given,
then (d) determines successively h2B
∗, . . . hr−1B
∗ in a unique way and (b) deter-
mines hrU
∗ in a unique way. We see that the equations (a)-(d) are equivalent to
the following equations for h0U
h−10 F (h0) ∈ B
∗xB∗, F r−1(h0)
−1gh0d
−1 ∈ B∗F r−1(x)B∗,
F r(h0)
−1gh0d
−1U∗ = k(F r(h0)
−1gF (h0)F (d
−1))x−1U∗
(if r ≥ 2) and
h−10 gh0d
−1 ∈ B∗xB∗, F (h0)
−1gh0d
−1U∗ = k(F (h0)
−1gF (h0)F (d
−1))x−1U∗
(if r = 1). In both cases these equations are equivalent to
(e) h−10 F (h0) ∈ U
∗txF (t)−1U∗, F r(h0)
−1gh0d
−1 ∈ F r(t)U∗
for some t ∈ T . We then have F r−1(h0)
−1gh0d
−1 ∈ U∗F r−1(t)F r−1(x)U∗. For
∗, t as in (e) we compute
k(h−10 F (h0))k(F (h0)
−1F 2(h0)) . . . k(F
r−2(h0)
−1F r−1(h0))k(F
r−1(h0)
−1gh0d
= (txF (t)−1)(F (t)F (x)F 2(t−1)) . . . (F r−2(t)F r−2(x)F r−1(t)−1)(F r−1(t)F r−1(x))
= txF (x) . . . F r−1(x) = t.
By 3.2(a) the result of the last computation is necessarily in TFx . Thus Fx(t) = t.
Hence F r(t) = t and the equations (e) become
(f) h−10 F (h0) ∈ U
∗xU∗, F r(h0)
−1gh0d
−1 ∈ TFxU∗.
We see that
(−1)itr(?,Hig(pw!L̃w)) = |T
Fx |−1
t∈TFx
at = |T
Fx |−1
t′∈TFx
where
at = |{hU
∗ ∈ (G0/U∗); h−1F (h) ∈ U∗xU∗, dh−1g−1F r(h)t ∈ U∗}|θ(t),
12 G. LUSZTIG
a′t′ = |{hU
∗ ∈ (G0/U∗); h−1F (h) ∈ U∗xU∗, h−1g−1F r(h)dt′ ∈ U∗}|θ(dt′d−1).
Comparing with the last formula in 2.4 and using θ(dt′d−1) = θ(t′) for t′ ∈ TFx
we obtain (with i0 as in 2.4):
(−1)itr(?,Hig(pw!L̃w)) = (−1)
l(w)θ̃(d)qi0r/2tr(ǫg, H
c (Y )θ).
Let us choose an isomorphism pw!L̃w[lw] ∼= p∅!L̃∅. (This exists by 1.4; note that
1.4(i) holds by 1.5.) Via this isomorphism, the isomorphism 3.3(g) corresponds to
an isomorphism F ∗(p∅!L̃∅) −→ p∅!L̃∅ that is to an isomorphism ψ
′ : F ∗A
−→ A so
that ∑
(−1)itr(?,Hig(pw!L̃w)) =
(−1)itr(ψ′,Hig(A))
for any g ∈ DF . (We use that lw is even.) Since A is irreducible, we must have
ψ = λ′ψ′ for some λ′ ∈ Q̄∗l . It follows that
(−1)itr(ψ,Hig(A)) = λ
′(−1)l(w)θ̃(d)qi0r/2tr(ǫg, H
c (Y )θ)
for any g ∈ DF . Thus Theorem 1.2 holds with V being the GF -module Hi0c (Y )θ,
which is irreducible (even as a G0F -module) if G0 has connected centre, but is not
necessarily irreducible in general.
References
[DL] P.Deligne and G.Lusztig, Representations of reductive groups over finite fields, Ann.Math.
103 (1976), 103-161.
[DM] F.Digne and J.Michel, Groupes réductifs non connexes, Ann.Sci. École Norm.Sup.
27 (1994), 345-406.
[L1] G.Lusztig, Green functions and character sheaves, Ann.Math. 131 (1990), 355-408.
[L2] G. Lusztig, Remarks on computing irreducible characters, J.Amer.Math.Soc. 5 (1992),
971-986.
[L3] G.Lusztig, Character sheaves on disconnected groups,I, Represent. Th. (electronic) 7
(2003), 374-403; II 8 (2004), 72-124; III 8 (2004), 125-144; IV 8 (2004), 145-178; Er-
rata 8 (2004), 179-179; V 8 (2004), 346-376; VI 8 (2004), 377-413; VII 9 (2005), 209-266;
VIII 10 (2006), 314-352; IX 10 (2006), 353-379.
[M] G.Malle, Generalized Deligne-Lusztig characters, J.Algebra 159 (1993), 64-97.
[S] T.Shoji, Character sheaves and almost characters of reductive groups, Adv.in Math. 111
(1995), 244-313; II 111 (1995), 314-354.
Department of Mathematics, M.I.T., Cambridge, MA 02139
