Localization for derived categories of (𝔤,𝔎)-modules

Abstract

0.1. This paper arose from an attempt to solve the following problem. Let (g, K) be a Harish-Chandra pair, i.e. g is a complex reductive Lie algebra, and K is an algebraic group with an action K -Aut(g) and an embedding k = Lie K c g, satisfying some standard conditions (see 1.1 below). Let Z be the center of the enveloping algebra U(g) . Fix a regular character 0: Z -C. Let ((g, K) be the category of (g, K)-modules and t,(g, K) c l((g, K) the subcategory consisting of modules annihilated by Ker 0 . Then by the localization theorem this category X (g, K) can be described geometrically. Namely, fix a Borel subalgebra b c g and a dominant weight A corresponding to 0. Consider the algebra D, of twisted differential operators on the flag space X of g. Then X4/g, K) = J((DZ, K), the K-equivariant D,-modules on X. This result allows us to study many properties of Harish-Chandra modules geometrically. But it does not give a geometric interpretation of Ext-groups of modules in t, (g, K). Namely, let M, N E 4t (gi, K). From the point of view of representation theory the interesting objects are Ext.,, (, K) (M, N), the Ext-groups in the category of all (g, K)-modules. But these Ext's do not admit localization since arbitrary (g, K)-modules do not localize. Our main result is a geometric interpretation of these Ext-groups and, more precisely, of the corresponding derived category. Let us describe it. Let to(g, K) c l((g, K) be the subcategory of 0-finite modules. That is, each element m of M E Ito(g, K) is annihilated by some power of Ker 0. Recall the localization for the category to(g, K) (precise definitions will be given later). Let G be the algebraic group of automorphisms of g, H c G a maximal torus, [ = Lie H. The flag variety X has a natural H-monodromic structure X -X. Let At(Dk) denote the category of weakly H-equivariant Dk-modules. Elements of 1#(Dk) are called monodromic D-modules on X.