Twisted Verma modules

Using principal series Harish-Chandra modules, local cohomology with support in Schubert cells and twisting functors we construct certain modules parametrized by the Weyl group and a highest weight in the subcategory O of the category of representations of a complex semisimple Lie algebra. These are in a sense modules between a Verma module and its dual. We prove that the three different approaches lead to the same modules. Moreover, we demonstrate that they possess natural Jantzen type filtrations with corresponding sum formulae. Let g be a finite dimensional complex semisimple Lie algebra with a Cartan subalgebra h ⊂ g and Weyl group W . In this paper we consider twisted Verma modules. These are in a sense representations between a Verma module and its dual. Fix a highest weight λ ∈ h. The twisted Verma modules M(λ) corresponding to λ are parametrized by the Weyl group W . They have the same formal character as the Verma module M(λ) (but in general not the same module structure). In the affine Kac-Moody setting these modules (turning out to be Wakimoto modules) have been studied by Feigin and Frenkel [6]. We give three rather different ways of constructing twisted Verma modules. First we obtain them as images of principal series Harish-Chandra modules (under the Bernstein-Gelfand-Joseph-Enright equivalence). In this setting Irving [7] applied wall crossing functors to describe principal series modules in a regular block inductively (Irving uses the term shuffled Verma module for a principal series module in a regular block). His inductive procedure inspired this work. Let G be a complex semisimple algebraic group with Lie algebra g, B a Borel subgroup in G, X = G/B the flag manifold of G, where B denotes the Borel subgroup opposite to B. Let w0 denote the longest word and e the identity element in the Weyl group W of G. We let C(w) = BwB/B ⊆ X denote the Schubert cell corresponding to w ∈ W . Notice that codimC(w) = l(w). It is known that the Verma module M(λ) with integral highest weight λ can be realized as the top local cohomology group H l(w0) C(w0) (X,L(w0 · λ)) of the line bundle L(w0 · λ) with support in the point Supported in part by the TMR programme “Algebraic Lie Representations” (ECM Network Contract No. ERB FMRX-CT 97/0100) 2 H. H. Andersen and N. Lauritzen C(w0). The dual Verma module can be realized as the bottom local cohomology group HC(e)(X,L(λ)) of the line bundle L(λ) with support in the big cell C(e). Our second construction of twisted Verma modules (which was the starting point of this work) are the intermediate local cohomology groups of the line bundle L(w ·λ) with support in an arbitrary Bruhat cell C(w) — these are the modules in the global Grothendieck-Cousin complex [12]. The intermediate local cohomology groups H l(w) C(w)(X,L(λ)) are isomorphic to dual Verma modules for dominant weights λ. In this case the global Grothendieck-Cousin complex is the dual BGG-resolution. Let us be more precise about the link from local cohomology to principal series modules. Fix a regular antidominant integral weight λ. The principal series modules M(x, y) in the block Oλ (under the Bernstein-Gelfand-Joseph-Enright equivalence) are parametrized by (x, y) ∈ W ×W . Let C(w) = Bwx0 ⊆ X. Then our result says that M(x, y) ∼= H l(x) C(x)(X,L(y · λ)) as g-modules. Our isomorphism is constructed using wall translation functors and gives a very explicit algorithm for obtaining the g-structure of the intermediate local cohomology groups (starting from a Verma module). Notice that the local cohomology approach only makes sense for integral weights. Following Arkhipov we may for each w ∈ W define twisting functors Tw of O (by tensoring with the “semiregular” U(g)-bimodule Sw — see Section 6.1). When applied to a Verma module, Tw produces a twisted Verma module. Again it follows quite easily that the modules obtained in this way satisfy Irving’s inductive procedure. This setup is probably the most powerful for studying twisted Verma modules and turns out to be the key for showing that the three approaches are isomorphic: the derived functor LTw is a self-equivalence of the bounded derived category D (O). This implies that twisted Verma modules only have constant g-endomorphisms (and therefore that they are indecomposable g-modules). This property allows us to deduce the required isomorphisms between the three approaches. The twisting functors also give the required deformation theory for constructing Jantzen filtrations and proving sum formulas for twisted Verma modules (which turn out to be twisted versions of the original Jantzen sum formula). At the end of the paper we have used the sum formula to compute the structure of all twisted Verma modules in the B2-case. Acknowledgment. We are grateful to S. Arkhipov for pointing out the paper [6] of Feigin and Frenkel and for explaining twisting functors to us during his stay in Aarhus, January 2001. We also thank M. Kashiwara and C. Stroppel for discussions that influenced this work. 1. Twisted Verma modules 3 1 Notation and preliminaries Fix a complex semisimple Lie algebra g with a Cartan subalgebra h. Let R ⊆ h be the root system associated with (g, h) and ZR the lattice of roots in h. Fix a basis S of simple roots and let R be the positive roots with respect to S. Let n = ∑