PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC 15

For any variety X over k, we write k[X] for the ring of regular functions on

X. If H is an algebraic group acting on X, we denote by

CohH

(X) the abelian

category of H-equivariant coherent sheaves on X. For any F ∈

DbCohH

(X), its

derived global sections RΓ(F) may be regarded as an object of DbRep(H).

4.2. Graded objects. Let Rep(H) be the category of graded rational H-

representations. This is, of course, equivalent to Rep(H × Gm). Repf (H) is deﬁned

similarly. For homogeneous components and Tate twists of objects of Rep(H), we

retain the conventions introduced in Section 2.2 for Vectk.

Consider k m , the graded H-representation consisting of the trivial H-module

concentrated in degree m. We can regard this as an (H × Gm)-equivariant sheaf

on pt = Spec k. If X is a variety equipped with an action of H × Gm, we put

OX m =

a∗k

m , where a : X → pt is the constant map. More generally, for any

F ∈

DbCohH×Gm

(X), we put

Fm = F

L

⊗ OX m .

We write U for any of the various functors that forget gradings. In particular,

for coherent sheaves, we have U :

DbCohH×Gm

(X) →

DbCohH

(X).

4.3. Reductive groups. Throughout the rest of the paper, G will be a simply

connected semisimple algebraic group over k, and the characteristic of k will be

assumed to be good for G. Fix a Borel subgroup B ⊂ G and a maximal torus

T ⊂ B. Let U ⊂ B be the unipotent radical, and let u be the Lie algebra of U.

Recall that there is a T -equivariant isomorphism of varieties

(4.1) e : u → U.

Let Λ be the weight lattice of T . We will think of B as the “negative” Borel:

we deﬁne

Λ+

⊂ Λ to be the set of dominant weights determined by declaring the

weights of T on u to be the negative roots. Let W be the Weyl group, and let

w0 ∈ W be the longest element. For any λ ∈ Λ, let dom(λ) denote the unique

dominant weight in the W -orbit of λ.

Let ≤ denote the usual partial order on Λ. That is, for μ, λ ∈ Λ, we say that

μ ≤ λ if λ − μ is a nonnegative integer linear combination of positive roots. We

also deﬁne a preorder on Λ as follows: μ λ if dom(μ) ≤ dom(λ). Obviously, ≤

and coincide on

Λ+.

For any λ ∈ Λ, let kλ denote the 1-dimensional T -representation of weight

λ. We may also regard this as a B-representation on which U acts trivially. For

λ ∈

Λ+,

let L(λ), M(λ), and N(λ) denote the simple module, Weyl module, and

dual Weyl module, respectively, of highest weight λ. These representations may

sometimes be regarded as graded by placing them in degree 0.

4.4. Nilpotent cone and Springer resolution. Let N be the variety of

nilpotent elements in the Lie algebra of G. We will also work with the flag variety

B = G/B and the Springer resolution

˜

N = G

×B

u. All these varieties are acted

on by G. Let Gm act on N by (z, x) → z2x, where z ∈ Gm and x ∈ N . This

action commutes with the action of G. The same formula deﬁnes an action on u

commuting with that of B, and so an action on

˜

N commuting with that of G.

Finally, let Gm act trivially on B. The obvious projection maps, which we denote

N

π

←−

˜

N

p

−→ B,

15