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
(X) the abelian
category of H-equivariant coherent sheaves on X. For any F
(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 =
m , where a : X pt is the constant map. More generally, for any
(X), we put
Fm = F
OX m .
We write U for any of the various functors that forget gradings. In particular,
for coherent sheaves, we have U :
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 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
For any λ Λ, let denote the 1-dimensional T -representation of weight
λ. We may also regard this as a B-representation on which U acts trivially. For
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
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
−→ B,
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