Abstract
Let g be a semi-simple simply-connected Lie algebra and let U ℓ be the corresponding quantum group with divided powers, where ℓ is an even order root of unity. Let in addition u ℓ⊂ U ℓ be the corresponding “small” quantum group. In this paper we establish the following relation between the categories of representations of U ℓ and u ℓ. We show that the category of u ℓ-modules is naturally equivalent to the category of U ℓ-modules, which have a Hecke eigen-property with respect to representations lifted by means of the quantum Frobenius map U ℓ→U( g ̌ ) , where g ̌ is the Langlands dual Lie algebra. This description allows to express the regular linkage class in the category u ℓ-mod in terms of perverse sheaves on the affine flag variety with a Hecke eigen-property. Moreover, it can serve as a basis to the program to understand the connection between the category u ℓ-mod and the category of representations of the corresponding affine algebra at the critical level.
Highlights
The main result of this paper is Theorem 2.4, which states that there is a natural equivalence between C(AG, OG) and ul -mod
We introduce the category Ul -mod as follows: its objects are finite-dimensional representations M of Ul, for which the action of the Kt’s comes from an algebraic action of the torus T on M, and such that for λ ∈ X, the action of on the subspace of M of weight λ ∈ X is given by di the scalar αi, λ + m t
To prove Theorem 2.8, it suffices to check that the other adjunction map Res(Ind(M )) → M is an isomorphism for any M ∈ a -comod
Summary
Since we already know the geometric interpretation for modules over the big quantum group, it is a natural idea to first express ul -mod entirely in terms of Ul -mod This is exactly what we do in this paper. Ul -mod0 obtains an interpretation in terms of the geometric Langlands correspondence: it can be thought of as a categorical counterpart of the space of Iwahori-invariant vectors in a spherical representation In another direction, Theorem 2.4 has as a consequence the theorem that ul -mod is equivalent to the category of G[[t]]-integrable representations of the chiral Hecke algebra, introduced by Beilinson and Drinfeld.
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