Wall-crossing Functors and D-modules

We will be concerned here with infinite dimensional representations of a complex semisimple Lie algebra g . In more detail, let Ug be the universal enveloping algebra of g , and let Z(g) be the center of Ug . We consider the category of Ug -modules annihilated by a great enough (unspecified) power of a maximal ideal in Z(g) . It is natural to compare the categories corresponding to two different maximal ideals. This question was first studied by Jantzen a long time ago. In [Ja], Jantzen introduced certain functors between the two categories, called translation functors. He showed that if both maximal ideals satisfy certain regularity and integrality conditions, then the translation functor establishes an equivalence of the two categories. If one of the two ideals is regular while the other is not, the corresponding translation functor is no longer an equivalence. The composition of the translation functor that sends the category at a regular maximal ideal to the category at a non-regular maximal ideal with the translation functor acting in the opposite direction is called a wall-crossing functor. The terminology stems from the identification of (integral) maximal ideals of Z(g) with Weyl group orbits in the weight lattice of the maximal torus. Nonregular ideals correspond to the orbits contained in the union of walls of the Weyl chambers. In this paper we will be mainly interested in the ”most singular” case, where the non-regular maximal ideal corresponds to the fixed point of the Weyl group, that is the point contained in all the walls. Our study was partly motivated by trying to understand two important results, the ”Endomorphism-theorem” and the ”Structure-theorem”, proved by W. Soergel [S1] in the course of the proof of the Koszul duality conjecture, see [BG], [BGS]. Soergel’s argument was very clever,