Character sheaves on disconnected groups, IX

0.33 MB·English

by  G. Lusztig

CHARACTER SHEAVES ON DISCONNECTED GROUPS, IX
6
0
0 G. Lusztig
2

n
a
Introduction
J

0
Throughout this paper, G denotes a fixed, not necessarily connected, reductive
2
algebraic group over an algebraically closed field k. This paper is a part of a series

]
[L9] which attempts to develop a theory of character sheaves on G.
T
R One of the main constructions in [L3] (going back to [L14]) was a procedure
. which to any character sheaf on G0 associates a certain two-sided cell in an (ex-
h
t tended) Coxeter group. A variant of this construction (restricted to ”unipotent”
a
m character sheaves) was latergivenby Grojnowski [Gr]. Herewe givea construction
[ which generalizes that in [L3] (and takes into account the approach in [Gr]) which

to any (parabolic) character sheaf on Z associates a certain type of two-sided
1 J,D
v cell.
4
The paper is organized as follows. In Section 40 we study certain equivariant
0
5 sheaves on G0/U∗ × G0/U∗ (where U∗ is the unipotent radical of a Borel in
1 G0) under the convolution operation. Some results in this section are implicit in
0
6 [L14, Ch.1]. In Section 41 we study the character sheaves on Z∅,D (where D is a
0 connected component of G) by connecting them with sheaves on G0/U∗×G0/U∗.
/
h We use this study to attach a two-sided cell to any character sheaf on Z . (See
J,D
t
a 41.4.) In Section 42 we study the interaction between the duality operation d
m
(see 38.10, 38.11) and the functor f (see 36.4). The main result in this section
∅,I
:
v is Proposition 42.9 which contains [L3, III, Cor.15.8(b)] as a special case (with
Xi G = G0,v = 1).
r Notation We fix a 1-dimensional Q¯l-vector space V with a given isomorphism
a V⊗2 −→∼ Q¯ (1) (Tate twist of Q¯ ). For n ∈ N we set Q¯ (n/2) = V⊗n. For
l l l
n ∈ Z,n < 0 let Q¯ (n/2) be the dual space of Q¯ (−n/2). If X is an algebraic
l l
variety and A ∈ D(X),n ∈ Z we write A[[n/2]] instead of A[n](n/2). (When n is
even this agrees with the notation in [L9, II, p.73].)
Contents
40. Sheaves on G0/U∗ ×G0/U∗.
41. Character sheaves and two-sided cells.
Supported in part by the National Science Foundation.
Typeset by AMS-TEX
1
2 G. LUSZTIG
42. Duality and the functor f .
∅,I
40. Sheaves on G0/U∗ × G0/U∗
40.1. LetA = Z[v,v−1]. LetHˆ (resp. H)betheA-moduleconsistingofallformal
(resp. finite) linear combinations a T˜ 1 with a ∈ A. Note that
w∈W,λ∈s w,λ w λ w,λ
H is naturally an A-submodule oPf Hˆ with A-basis {T˜ 1 ;w ∈ W,λ ∈ s}. For
w λ
any n ∈ N∗, the A-submodule of H spanned by {T˜ 1 ;w ∈ W,λ ∈ s } may be
k w λ n
naturally identified with H (see 31.2(a)). There is a unique A-algebra structure
n
on Hˆ in which the product of two elements
h = w∈W,λ∈saw,λT˜w1λ, h′ = w′∈W,λ′∈sa′w′,λ′T˜w′1λ′
as abovPe is hh′ = y∈W,ν∈sby′,νT˜Py1ν where for any ν ∈ s,
w,w′∈Waw,w′P−1νa′w′,νT˜wT˜w′1ν = y∈Wby,νT˜y1ν
is cPomputed in the algebra structurePof Hn for any n such that ν ∈ sn. Thus
Hˆ becomes an associative algebra with 1; H is a subalgebra (without 1) and, for
n ∈ N∗, H is a subalgebra (with a different 1) with the algebra structure as in
k n
31.2.
Now in the definition of Hˆ given above, although T˜ 1 is defined, the elements
w λ
T˜ ,1 are not defined separately (as was the case in H ). To remedy this we set
w λ n
T˜ = T˜ 1 ∈ Hˆ (for w ∈ W) and 1 = T˜ 1 ∈ H (for λ ∈ s). Then T˜ 1
w λ∈s w λ λ 1 λ w λ
is thePproduct of T˜ ,1 in the algebra Hˆ. Note that T˜ is the unit element of Hˆ
w λ 1
and the following equalities hold in Hˆ:
1λ1λ = 1λ for λ ∈ s,1λ1λ′ = 0 for λ 6= λ′ in s;
T˜wT˜w′ = T˜ww′ for w,w′ ∈ W such that l(ww′) = l(w)+l(w′);
T˜ 1 = 1 T˜ for w ∈ W,λ ∈ s;
w λ wλ w
T˜2 = T˜ +(v −v−1) T˜ 1 for s ∈ I.
s 1 λ∈s;s∈Wλ s λ
By a standard argumenPt we see that
(a) H is exactly the A-algebra defined by the generators T˜ 1 (w ∈ W, λ ∈ s)
w l
and the relations:
(T˜w1λ)(T˜w′1λ′) = 0 if w,w′ ∈ W,λ,λ′ ∈ s,w′λ′ 6= λ,
(T˜w1w′λ′(T˜w′1λ′) = T˜ww′1λ′ if w,w′ ∈ W,λ,λ′ ∈ s, l(ww′) = l(w)+l(w′),
(T˜s1sλ′)(T˜s1λ′) = T˜11λ′ + (v − v−1)cT˜s1λ′ if s ∈ I, λ′ ∈ s where c = 1 for
s ∈ Wλ′ and c = 0 for s ∈/ Wλ′.
40.2. Let R,R+ be as in 28.3. The following result is well known:
(a) If w ∈ W,α ∈ R+ and s is as in 28.3 then we have l(ws ) > l(w) if and
α α
only if w(α) ∈ R+.
Let λ ∈ s. Let R ,R+,W ,H be as in 34.2. We write ∨ instead of ∨D (as
λ λ λ λ λ λ
in 34.4 with D = G0). We show:
(b) If w ∈ W then wW contains a unique element w of minimal length; it is
λ 1
characterized by the property w (R+) ⊂ R+.
1 λ
Let w be an element of minimal length in wW . Let α ∈ R+. Then l(w s ) ≥
1 λ λ 1 α
CHARACTER SHEAVES ON DISCONNECTED GROUPS, IX 3
l(w ). Since l(w s ) = l(w )+1 mod 2 we see that l(w s ) > l(w ). By (a) we
1 1 α 1 1 α 1
have w (α) ∈ R+. Thus, w (R+) ⊂ R+. Now let u ∈ W −{1}. We pick α ∈ R+
1 1 λ λ λ
such that u(α)−1 ∈ R+; then w u(α)−1 ∈ R+. If w u has minimal length in wW
λ 1 1 λ
then by an earlier part of the argument applied to w u instead of w we have
1 1
w u(α) ∈ R+, a contradiction. We see that w is the unique element of minimal
1 1
length in wW . It remains to show that if u ∈ W satisfies w u(R+) ⊂ R+ then
λ λ 1 λ
u = 1. If u 6= 1 then by an earlier part of the argument we have w u(α)−1 ∈ R+
1
for some α ∈ R+, a contradiction. This proves (b).
λ
We show:
(c) If s ∈ I and w ∈ W has minimal length in wW then either (i) sw has
λ
minimal length in swW or (ii) w−1sw ∈ W .
λ λ
There is a unique β ∈ R+ such that s(β)−1 ∈ R+. Assume that (i) does not
hold. By (b) there exists α ∈ R+ such that sw(α)−1 ∈ R+; moreover, w(α) ∈ R+.
λ
Hence w(α) = β. We have w−1(β) = α ∈ R hence w−1sw ∈ W and (ii) holds.
λ λ
This proves (c).
For z ∈ W let T˜λ,cλ ∈ H be as in 34.2 . Then cλ = pλ T˜λ where
λ z z λ z z′∈Wλ z′,z z′
pλ ∈ Z[v−1] are uniquely defined. P
z′,z
For any w ∈ W, λ ∈ s there is a unique element element of H which is equal
to c ∈ H (see 34.4) for any n such that λ ∈ s ; we denote this element again
w,λ n n
by c . We have
w,λ
cw,λ = w′∈Wπw′,w,λT˜w′1λ
where πw′P,w,λ ∈ Z[v−1] are uniquely defined. Note that
{c ;w ∈ W,λ ∈ s} is an A-basis of H.
w,λ
We show:
(d) Let w,w′ ∈ W. We write w = w z,w′ = w′z′ where w has minimal length
1 1 1
in wW , w′ has minimal length in w′W and z,z′ ∈ W . If w 6= w′ then
λ 1 λ λ 1 1
πw′,w,λ = 0. If w1 = w1′ then πw′,w,λ = pλz′,z.
From the definitions we see that if wλ 6= w′λ then πw′,w,λ = 0. Thus we may
assume that wλ = w′λ. We choose a sequence s ,s ,...,s in I such that wλ =
1 2 r
w′λ = s s ...s λ 6= s ...s λ 6= ··· =6 s λ 6= λ.
r r−1 1 r−1 1 1
Weshow thatfork ∈ [0,r],s s ...s hasminimallengthins s ...s W .
k k−1 1 k k−1 1 λ
We argue by induction. For k = 0 the result is obvious. Assume now that k ∈
[1,r]. Since s ...s has minimal length in s ...s W and s s ...s λ 6=
k−1 1 k−1 1 λ k k−1 1
s ...s λweseefrom(c)thats s ...s hasminimallengthins s ...s W
k−1 1 k k−1 1 k k−1 1 λ
as required.
In particular, s s ...s has minimal length in s s ...s W . Since wλ =
r r−1 1 r r−1 1 λ
s s ...s λ we have w = s s ...s h h where h ∈ ∨ , h ∈ W . Then
r r−1 1 r r−1 1 1 2 1 λ 2 λ
both w and s s ...s h have minimal length in s s ...s h W = wW =
1 r r−1 1 1 r r−1 1 1 λ λ
w W ; using (b) we deduce that s s ...s h = w . Hence s ...s w =
1 λ r r−1 1 1 1 1 r
s ...s w z = h z. Similarly, s ...s w′ = h′z′ where h′ ∈ ∨ .
1 r 1 1 1 r 1 1 λ
Fromtheresults in34.7-34.10wesee that πw′,w,λ = pλs1...srw′,s1...srw = pλh′1z′,h1z.
Using h ,h′ ∈ ∨ and the definitions (34.2) we see that pλ = 0 if h 6= h′
1 1 λ h′1z′,h1z 1 1
4 G. LUSZTIG
and pλ = pλ if h = h′.
h′1z′,h1z z′,z 1 1
It remains to show that we have w = w′ if and only if h = h′. We have
1 1 1 1
s s ...s = h−1w and similarly s s ...s = (h′)−1w′. Hence h−1w =
r r−1 1 1 1 r r−1 1 1 1 1 1
(h′)−1w′. We see that w = w′ if and only if h = h′. This proves (d).
1 1 1 1 1 1
For w′ ≤ w in W, λ ∈ s and i ∈ Z we define Ni,w′,w,λ ∈ Z by
(e) πw′,w,λ = vl(w′)−l(w) i∈ZNi,w′,w,λvi, that is,
pλz′,z = vl(w′)−l(w) i∈ZNPi,w′,w,λvi if w′Wλ = wWλ and z,z′ are as in (d),
Ni,w′,w,λ = 0 if w′PWλ 6= wWλ.
Note that Ni,w′,w,λ is 0 unless i is even.
40.3. Let B∗ ∈ B. Let U∗ = UB∗ and let T be a maximal torus of B∗. Let
r = dimT. Let WT = NG0T/T. We identify T = T,WT = W as in 28.5. For any
w ∈ W we denote by w˙ a representative of w in NG0T.
Let C = G0/U∗ ×G0/U∗. We have a partition C = ∪ C where
w∈W w
C = {(hU∗,h′U∗) ∈ C;h−1h′ ∈ B∗w˙B∗}.
w
For w ∈ W let d = dimC and let
w w
C¯ = {(hU∗,h′U∗) ∈ C;h−1h′ ∈ B∗w˙B∗}
w
(closure in G0). Now C¯ is an irreducible variety and we have a partition C¯ =
w w
∪w′;w′≤wCw′ with Cw smooth, open dense in C¯w.
Define γ : B∗w˙B∗ −→ T by γ (g) = t where g ∈ U∗w˙tU∗ with t ∈ T. Define
w˙ w˙
ψ : C −→ T by ψ(hU∗,h′U∗) = γ (h−1h′).
w w˙
For L ∈ s we set L = ψ∗L, a local system on C . (Using 28.1(c) we see
w w
that the isomorphism class of ψ∗L is independent of the choice of w˙.) Let L♯ =
w
IC(C¯ ,L ) ∈ D(C¯ ).
w w w
40.4. For w ∈ W,L ∈ s let L = j L , L♯ = ¯j L♯ where j : C −→ C, ¯j :
w w! w w w! w w w w
C¯ −→ C are the inclusions. Let Cˆ be the full subcategory of D(C) whose objects
w
are the simple perverse sheaves on C which are equivariant for the G0 × T × T
action
(a) (x,t,t′) : (hU∗,h′U∗) 7→ (xhtnU∗,xh′t′nU∗)
on C (for some n ∈ N∗) or equivalently, are isomorphic to L♯ [d ] for some L ∈ s
k w w
and some w ∈ W. Let Dcs(C) be the subcategory of D(C) whose objects are
those K ∈ D(C) such that for any j, any simple subquotient of pHjK is in Cˆ.
If w,L are as above then L ∈ Dcs(C). Indeed this constructible sheaf is
w
equivariant for the action (a) (for some n) hence so is each pHj(L ).
w
We have a diagram C ×C ←−r (G0/U∗)3 −→s C where
r(h U∗,h U∗,h U∗) = ((h U∗,h U∗),(h U∗,h U∗)),
1 2 3 1 2 2 3
s(h U∗,h U∗,h U∗) = (h U∗,h U∗).
1 2 3 1 3
We define a bi-functor D(C)×D(C) −→ D(C) by A,A′ 7→ A∗A′ = s r∗(A⊠A′).
!
The ”product” A∗A′ is associative in an obvious sense. We show:
(b) A,A′ 7→ A∗A′ restricts to a bi-functor Dcs(C)×Dcs(C) −→ Dcs(C).
Let A,A′ ∈ Dcs(C). To show that A∗A′ ∈ Dcs(C) we may assume that A,A′ ∈ Cˆ.
CHARACTER SHEAVES ON DISCONNECTED GROUPS, IX 5
Then each pHj(A∗A′) is equivariant for the action (a) (for some n). This proves
(b).
40.5. For w′ ≤ w in W, λ ∈ s, L ∈ λ and i ∈ Z we show:
(a) Hi(L♯w)|Cw′ ∼= (Lw′(−i/2))⊕Ni,w′,w,λ.
(Both sides are 0 unless i is even.)
Let
C˜ = {(h,h′) ∈ G0 ×G0;h−1h′ ∈ B∗w˙B∗}×k∗,
w
C¯˜ = {(h,h′) ∈ G0 ×G0;h−1h′ ∈ B∗w˙B∗}×k∗.
w
Now C¯˜w is an irreducible variety and we have a partition C¯˜w = ∪w′;w′≤wC˜w′ with
C˜ smooth, open dense in C¯˜ . Define d¯: C¯˜ −→ C¯ , d : C˜ −→ C¯ by (h,h′,z) 7→
w w w w w w
(hU∗,h′U∗). Let L˜ = d∗L , a local system on C˜ . Let L˜♯ = IC(C¯˜ ,L˜ ). Since
w w w w w w
d,d¯are principal U∗ ×k∗-bundles it is enough to show
(b) Hi(L˜♯w)|C˜w′ ∼= (L˜w′(−i/2))⊕Ni,w′,w,λ.
(Both sides are 0 unless i is even.)
We choose κ ∈ Hom(T,k∗),E ∈ s(k∗) such that L ∼= κ∗E, see 28.1(c).
Now B∗ acts on (B∗w˙B∗) × k∗ and on (B∗w˙B∗) × k∗ by t u : (g,z) 7→
1
(g(t u)−1,κ(t )z) where t ∈ T, u ∈ U∗. Let P¯κ = ((B∗w˙B∗) × k∗)/B∗,
1 1 1 w
PPκ = ((B∗w˙B∗) × k∗)/B∗. Now Pκ is a smooth open dense subvariety of
w w
the irreducible variety P¯kw and P¯κw = ∪w′;w′≤wPκw′ is a partition. The mor-
phism (B∗w˙B∗) × k∗ −→ k∗ given by (g,z) 7→ κ(γ (g))z factors through a mor-
w˙
phism φ : Pκ −→ k∗. Let Eκ = φ∗E, a local system of rank 1 on Pκ. Let
w w w
Eκ♯ = IC(P¯κ,Eκ) ∈ D(P¯κ). From [L14, 1.24] we see that
w w w w
(c) Hi(Ewκ♯)|Pκw′ ∼= (Ewκ′(−i/2))⊕Ni,w′,w,λ.
(Both sides are 0 unless i is even.)
We can find n ∈ N∗ such that E ∈ s (k∗). Define c¯: C¯˜ −→ P¯ , c : C¯˜ −→ P¯
k n w w w w
by (h,h′,z) 7→ B∗−orbit of (h−1h′,zn). Now c¯,c are locally trivial fibrations with
smooth fibres of pure dimension. Hence (b) follows from (c) provided that we can
show that c∗Ewκ′ = L˜w′ for w′ ≤ w. We may assume that w = w′. We have a
commutative diagram
Pκ ←−−c−− C˜ ×k∗ −−−d−→ C
w w w
φ φ′ κψ
  
ky∗ ←−c−′−− k∗ ×y k∗ −−−d−′→ ky∗
with φ,ψ,c,d as above, φ′(h,h′,z) = (κ(γ (h−1h′)),z), c′(z′,z) = z′zn, d′(z′,z) =
w˙
z′. Using this and the definitions we have L˜ = φ′∗d′∗E, c∗E = φ′∗c′∗E. It
w w
remains to show that d′∗E = c′∗E. This expresses the fact that E is equivariant
for the k∗-action z : z 7→ znz on k∗ which follows from E ∈ s (k∗). This proves
1 1 n
(b) hence (a).
6 G. LUSZTIG
40.6. Let w,w′ ∈ W, L,L′ ∈ s. We set L = L ∗L′ ∈ Dcs(C). Let
w w′
X = {(h U∗,h U∗,h U∗) ∈ (G0/U∗)3;h−1h ∈ B∗w˙B∗,h−1h ∈ B∗w˙′B∗},
1 2 3 1 2 2 3
X¯ = {(h U∗,h B∗,h U∗) ∈ G0/U∗ ×G0/B∗ ×G0/U∗;
1 2 3
h−1h ∈ B∗w˙B∗,h−1h ∈ B∗w˙′B∗}.
1 2 2 3
We have a commutative diagram with a cartesian square
X −−−f−→ X¯ −−−σ¯−→ C
τ τ¯
 
 f′ 
T ×y T −−−−→ Ty
where f is given by (h U∗,h U∗,h U∗) 7→ (h U∗,h B∗,h U∗),
1 2 3 1 2 3
f′ is (t,t′) 7→ Ad(w˙′)−1(t)t′,
τ is (h U∗,h U∗,h U∗) 7→ (t,t′) with h−1h ∈ U∗w˙tU∗,h−1h ∈ U∗w˙′t′U∗,
1 2 3 1 2 2 3
τ¯ is (h U∗,h B∗,h U∗) 7→ Ad(w˙′)−1(t)t′ with t,t′ as in the definition of τ,
1 2 3
σ¯ is (h U∗,h B∗,h U∗) 7→ (h U∗,h U∗).
1 2 3 1 3
From the definitions we have L = σ¯ f τ∗(L ⊠ L′). Using the diagram above, we
! !
have L = σ¯ τ¯∗f′(L⊠L′). From the definitions we see that either (i) or (ii) below
! !
holds:
(i) L ∼=6 (Ad(w˙′)−1)∗L′ and f′(L⊠L′) = 0;
!
(ii) L ∼= (Ad(w˙′)−1)∗L′ and L⊠L′ = f′∗L′.
If (i) holds then K = 0. If (ii) holds then, as in 32.16, we have
f′(L⊠L′) = f′f′∗L′ = L′ ⊗f′Q¯ ≎ {L′ ⊗He(f′Q¯ )[−e],e ∈ Z},
! ! ! l ! l
r
L′ ⊗He(f′Q¯ )[−e] ≎ {L′(r−e),...,L′(r−e),( copies)}.
! l (cid:18)2r−e(cid:19)
Setting L¯ = σ¯ τ¯∗(L′), it follows that
!
r
L ≎ {L¯(r−e)[−e],...,L¯(r−e)[−e],( copies),e ∈ Z}.
(cid:18)2r−e(cid:19)
We now consider L¯ for certain choices of w,w′.
If w,w′ satisfy l(ww′) = l(w)+l(w′) then σ¯ restricts to an isomorphism X¯ −→
Cww′ and L¯ = L′ww′.
Now assume that α,αˇ,s are as in 28.3 and that w = w′ = s ∈ I. We have
α α
L¯ ≎ {j L¯ ;u ∈ W}
u! u
where j : C −→ C is the inclusion and L¯ = j∗L¯. Let X¯ = σ¯−1(C ). Then
u u u u u u
L¯ = σ¯ τ¯∗(L′) where σ¯ : X¯ −→ C , τ¯ : X¯ −→ T are the restrictions of σ¯,τ¯.
u u! u u u u u u
CHARACTER SHEAVES ON DISCONNECTED GROUPS, IX 7
If u ∈/ {1,s } then X¯ = ∅ and L¯ = 0. If u = 1 then σ¯ : X¯ −→ C is an affine
α u u u u u
line bundle and τ¯∗(L′) = σ¯∗L′ ; hence σ¯ τ¯∗(L′) = σ¯ σ¯∗L′ = L′ [[−1]]. If u = s
u u u u! u u! u u u α
then σ¯ : X¯ −→ C is a principal k∗-bundle and either (iii) or (iv) below holds:
u u u
(iii) αˇ∗L′ 6∼= Q¯ and σ¯ τ¯∗(L′) = 0,
l u! u
(iv) αˇ∗L′ ∼= Q¯ and τ¯∗(L′) = σ¯∗L′ .
l u u u
If (iv) holds then, as in case (ii) above, we have
σ¯ τ¯∗(L′) = σ¯ σ¯∗L′ = L′ ⊗σ¯ Q¯ ≎ {L′ ⊗He(σ¯ Q¯ )[−e],e ∈ Z},
u! u u! u u u u! l u u! l
L′ ⊗He(σ¯ Q¯ )[−e] ≎ {L′ (1−e),...,L′ (1−e),( 1 copies)}.
u u! l u u 2−e
(cid:0) (cid:1)
40.7. In this subsection we assume that k is an algebraic closure of a finite field.
Now the A-module K(C) is defined as in 36.8 (the character sheaves on C are
taken to be the objects in Cˆ).
For (w,λ) ∈ W×s, let [w;λ] be the basis element of K(C) given by L♯ [[d /2]];
w w
we choose L ∈ λ and we regard L ,L♯ as mixed complexes on C whose restriction
w w
to C is pure of weight 0; then gr(L ),gr(L♯ ) are defined in K(C) as in 36.8. We
w w w
denote these elements of K(C) by [w;λ]′,[w;λ]′♯ respectively. From 40.5(a) we see
that
(a) (−v)dw[w;λ] = [w;λ]′♯ = w′∈W i∈2ZNi,w′,w,λvi[w′;λ]′ in K(C).
where Ni,w′,w,λ is as in 40.2(e).P P
Let r,s be as in 40.4. By 40.4(b), s r∗ : D(C × C) −→ D(C) restricts to a
!
functor Dcs(C × C) −→ Dcs(C) where the character sheaves on C × C are by
definition complexes of the form A⊠A′ with A ∈ Cˆ,A′ ∈ Cˆ. Hence the A-linear
map gr(s r∗) : K(C ×C) −→ K(C) or equivalently K(C)⊗ K(C) −→ K(C) is well
! A
defined. (We have canonically K(C×C) = K(C)⊗ K(C).) We write ξ∗ξ′ instead
A
of gr(s r∗)(ξ⊠ξ′) where ξ,ξ′ ∈ K(C). Note that ξ,ξ′ 7→ ξ∗ξ′ defines an associative
!
A-algebra structure on K(C).
Let w,w′ ∈ W, λ,λ′ ∈ s. From 40.6 we see that:
if w′λ′ 6= λ then [w;λ]′ ∗[w′;λ′]′ = 0 in K(C);
if w′λ′ = λ and l(ww′) = l(w)+l(w′) then [w;λ]′∗[w′,λ′]′ = (v2−1)r[ww′;λ′]′
in K(C);
if s ∈ I and sλ′ = λ then [s;λ]′∗[s,λ′]′ = (v2 −1)r(v2[1;λ′]′ +(v2 −1)c[s;λ′]′)
where c = 1 for s ∈ Wλ′ and c = 0 for s ∈/ Wλ′.
Using this and (a), 40.1(a), 40.2(e), we see that
(b) the unique A-linear isomorphism ω : K(C) −→ H (H as in 40.1) given by
[w,λ]′ 7→ vl(w)T˜ 1 for w ∈ W, λ ∈ s, satisfies ω([w,λ]) = (−v)−dwvl(w)c for
w λ w,λ
w ∈ W, λ ∈ s and ω(x∗x′) = (v2 −1)rω(x)ω(x′) for any x,x′ ∈ K(C).
40.8. For w,w′ ∈ W and λ,λ′ ∈ s we have
cw,λcw′,λ′ = y∈W,ν∈sγyw,,νλ;w′,λ′cy,λ
in the algebra HP. Here γw,λ;w′,λ′ ∈ A. We have:
y,ν
(a) γw,λ;w′,λ′ ∈ N[v,v−1].
y,ν
By the arguments in 34.4-34.10 (with D = G0) this is reduced to the analogous
8 G. LUSZTIG
(well known) statement for the structure constants of the algebra HD with its
λ
basis (cλ) (see 34.2).
w
40.9. For any J ⊂ I let H be the A-submodule of H spanned by {c ;w ∈
J w,λ
W ,λ ∈ s} or equivalently by {T˜ 1 ;w ∈ W ,λ ∈ s}. From the definitions we
J w λ J
seethatHJ isasubalgebraofH. ForanyJ ⊂ I,J′ ⊂ Iwedefinearelation(cid:22)J,J′ on
W×sasfollows. Wesaythat(y,ν) (cid:22)J,J′ (w,λ)ifthereexistw1 ∈ WJ,w2 ∈ WJ′,
λ ,λ ∈ s such that in the expansion (in the algebra H):
1 2
cw1,λ1cw,λcw2,λ2 = y′∈W,ν′∈say′,ν′cy′,ν′
(with ay′,ν′ ∈ A) we hPave ay,ν 6= 0.
Using the associativity of the product in H, the fact that HJ,HJ′ are sub-
algebras of H and 40.8(a), we see that (cid:22)J,J′ is transitive. Using the formula
c c c = c we see that it is reflexive. Thus, it is a preorder. Let
1,wλ w,λ 1,λ w,λ
∼J,J′ be the equivalence relation attached to (cid:22)J,J′; thus, (y,ν) ∼J,J′ (w,λ) if
(y,ν) (cid:22)J,J′ (w,λ) and (w,λ) (cid:22)J,J′ (y,ν). The equivalence classes for ∼J,J′ are
called (J,J′)-two-sided cells. The (I,I)-two sided cells in W × s are also called
two-sided cells.
40.10. Let w,w′,w′′ ∈ W, L,L′,L′′ ∈ s. We set K = L ∗L′ ♯ ∗L′′ ∈ Dcs(C).
w w′ w′′
Let
X = {(h U∗,h U∗,h U∗,h U∗) ∈ (G0/U∗)4;
1 2 3 4
h−1h ∈ B∗w˙B∗,h−1h ∈ B∗w˙′B∗,h−1h ∈ B∗w˙′′B∗},
1 2 2 3 3 4
an irreducible variety. Let X be the smooth open dense subset of X defined by
0
the condition h−1h ∈ B∗w˙′B∗. Define σ : X −→ C by
2 3
(h U∗,h U∗,h U∗,h U∗) 7→ (h U∗,h U∗).
1 2 3 4 1 4
Define τ : X −→ T ×T ×T by
0
(h U∗,h U∗,h U∗,h U∗) 7→ (t,t′,t′′)
1 2 3 4
with h−1h ∈ U∗w˙tU∗,h−1h ∈ U∗w˙′t′U∗,h−1h ∈ U∗w˙′′t′′U∗.
1 2 2 3 3 4
Let F = τ∗(L⊠L′⊠L′′), a local system on X . Then F♯ := IC(X,F) ∈ D(X) is
0
defined and we have K = σ F♯.
!
Let X¯ (resp. X¯ ) be the the variety of all (h U∗,h B∗,h B∗,h U∗) ∈ G0/U∗×
0 1 2 3 4
G0/B∗×G0/B∗×G0/U∗ that satisfythe sameequationsasthose defining X (resp.
X ). Note that X¯ is irreducible and X¯ is an open dense smooth subset of X¯. We
0 0
have a cartesian diagram
X −−−f−→ X¯ −−−σ¯−→ C
x x
X −−−f0−→ X¯
0 0
τ τ¯
 
 f′ 
T ×Ty ×T −−−−→ Ty
CHARACTER SHEAVES ON DISCONNECTED GROUPS, IX 9
where X −→ X,X¯ −→ X¯ are the obvious imbeddings,
0 0
f,f are given by (h U∗,h U∗,h U∗,h U∗) 7→ (h U∗,h B∗,h B∗,h U∗),
0 1 2 3 4 1 2 3 4
f′ is (t,t′,t′′) 7→ Ad(w˙′w˙′′)−1(t)Ad(w˙′′)−1(t′)t′′,
τ¯ is (h U∗,h B∗,h B∗,h U∗) 7→ Ad(w˙′w˙′′)−1(t)Ad(w˙′′)−1(t′)t′′ with t,t′,t′′ as
1 2 3 4
in the definition of τ,
σ¯ is (h U∗,h B∗,h B∗,h U∗) 7→ (h U∗,h U∗).
1 2 3 4 1 4
Assume that L ∼= (Ad(w˙′)−1)∗L′ and L′ ∼= (Ad(w˙′′)−1)∗L′′. Then L⊠L′ ⊠L′′ =
f′∗L′′. We have F = τ∗f′∗L′′ = f∗τ¯∗L′′. Since f is a principal T × T-bundle
0
and X = f−1(X¯ ) it follows that F♯ = f∗IC(X¯,τ¯∗L′′). Note that f Q¯ ≎
0 0 ! l
{He(f Q¯ )[−e],2r ≤ e ≤ 4r},
! l
2r
He(f Q¯ ) ≎ {Q¯ (2r−e),...,Q¯ (2r−e),( copies)}.
! l l l
(cid:18)4r−e(cid:19)
Hence setting K¯ = σ¯ (IC(X¯,τ¯∗L′′)) we have
!
K = σ f∗IC(X¯,τ¯∗L′′) = σ¯ f f∗IC(X¯,τ¯∗L′′) = σ¯ (IC(X¯,τ¯∗L′′)⊗f Q¯ ),
! ! ! ! ! l
2r
(a) K ≎ {K¯(2r−e)[−e],...,K¯(2r−e)[−e],( copies),2r ≤ e ≤ 4r}.
(cid:18)4r−e(cid:19)
We now show:
(b) if A ∈ Cˆ is such that A ⊣ K¯, then A ⊣ K.
We may regard L,L′,L′′ as mixed local systems (with respect to a rational struc-
ture over a sufficiently large finite subfield of k) which are pure of weight 0. Then
K,K¯ are naturally mixed complexes and (a) is compatible with the mixed struc-
tures. For any mixed perverse sheaf P, let P be the subquotient of P of pure
h
weight h. We can find h ∈ Z such that A ⊣ pHj(K¯) for some j ∈ Z; moreover we
h
mayassumethathismaximumpossible. NotethatA ⊣ pHj+4r(K¯[−4r](−2r))
h+2r
and A 6⊣ pHj′(K¯[−e](2r−e)) for 2r ≤ e < 4r and any j′; hence from (a) we
h+2r
see that A ⊣ pHj+4r(K) . In particular, A ⊣ K, and (b) is proved.
h+2r
40.11. Let w,w′L,L′,X,X¯,τ be as in 40.6. We set L = L♯ ∗L′ ♯ ∈ Dcs(C). Let
w w′
A = L′w′′′♯[dw′′]. We show:
(a) If A ⊣ L then [w′′,λ′′] appears with non-zero coefficient in the expansion of
the product [w,λ]∗[w′,λ′] in terms of the basis ([y,ν]) of K(C).
Let
X = {(h U∗,h U∗,h U∗) ∈ (G0/U∗)3;h−1h ∈ B∗w˙B∗,h−1h ∈ B∗w˙′B∗},
1 2 3 1 2 2 3
X¯ = {(h U∗,h B∗,h U∗) ∈ G0/U∗ ×G0/B∗ ×G0/U∗;
1 2 3
h−1h ∈ B∗w˙B∗,h−1h ∈ B∗w˙′B∗}.
1 2 2 3
10 G. LUSZTIG
Note that X (resp. X¯) is naturally an open dense subset of X (resp. X¯). Define
σ′ : X −→ C by (h U∗,h U∗,h U∗) 7→ (h U∗,h U∗). Define σ¯′ : X¯ −→ C by
1 2 3 1 3
(h U∗,h B∗,h U∗) 7→ (h U∗,h U∗). Let F = τ∗(L ⊠L′), a local system on X.
1 2 3 1 3
Then F♯ := IC(X,F) ∈ D(X) is defined and we have L = σ′F♯. We have a
!
cartesian diagram
X −−−f˜−→ X¯ −−−σ¯−′→ C
x x
X −−−f−→ X¯
τ τ¯
 
 f′ 
T ×y T −−−−→ Ty
where X −→ X,X¯ −→ X¯ are the obvious imbeddings, f,f′,τ¯ are as in 40.6 and f˜
is the obvious map.
Assume first that 40.6(i) holds. Let m′ : T × X −→ X be the free T-action
t : (h U∗,h U∗,h U∗) 7→ (h U∗,h t−1U∗,h U∗). This restricts to a free T-
1 1 2 3 1 2 1 3
action m : T × X −→ X. Define a free T action m : T × (T × T) −→ T × T by
0
t : (t,t′) 7→ (t−1t,Ad(w˙′)−1(t )t′. Then m,m are compatible with τ. By our
1 1 1 0
assumption we have m∗(L ⊠ L′) = L ⊠ L ⊠ L′ where L ∈ s(T), L 6∼= Q¯ . It
0 0 0 0 l
follows that m∗(F) ∼= L ⊠ F. From the properties of intersection cohomology
0
we then have m′∗(F♯) ∼= L ⊠F♯. Let r : T ×X −→ X be the second projection.
0
Since L ∈ s(T), L 6∼= Q¯ , we have r (L ⊠F♯) = 0. Hence r m′∗(F♯) = 0. Since
0 0 l ! 0 !
m′,f′,r,f′ form a cartesian diagram we must have f′∗f′(F♯) = 0. Since f′ is
a principal T-bundle we deduce that f′(F♯) = 0. We have L = σ¯′f′(F♯) hence
! ! !
L = 0. In this case (a) is clear.
Assume next that 40.6(ii) holds. Then L ⊠ L′ = f′∗L′ and F = τ∗f′∗L′ =
f∗τ¯∗L′. Since f′ is a principal T-bundle and X = f′−1(X¯) it follows that F♯ =
f′∗IC(X¯,τ¯∗L′). Note that f′Q¯ ≎ {He(f′Q¯ )[−e],r ≤ e ≤ 2r},
! l ! l
r
He(f′Q¯ ) ≎ {Q¯ (r−e),...,Q¯ (r−e),( copies)}.
! l l l (cid:18)2r−e(cid:19)
Hence setting L¯ = σ¯′(IC(X¯,τ¯∗L′)) we have
!
L = σ′f′∗IC(X¯,τ¯∗L′) = σ¯′f′f′∗IC(X¯,τ¯∗L′) = σ¯′(IC(X¯,τ¯∗L′)⊗f′Q¯ ),
! ! ! ! ! l
r
L ≎ {L¯(r−e)[−e],...,L¯(r−e)[−e],( copies),r ≤ e ≤ 2r}.
(cid:18)2r−e(cid:19)
Since A ⊣ L, this shows that A ⊣ L¯. We regard L′ as a pure local system of weight
0. Then L¯ = σ¯′(IC(X¯,τ¯∗L′)) is again pure of weight 0, since σ¯′ is proper (see
!
[BBD]). Hence the coefficient with which A appears in the expansion of gr(L¯) is a