Demazure Descent and Representations of Reductive Groups

We introduce the notion of Demazure descent data on a triangulated category and define the descent category for such data. We illustrate the definition by our basic example. Let be a reductive algebraic group with a Borel subgroup . Demazure functors form Demazure descent data on and the descent category is equivalent to . 1. Motivation The present paper is the first one in a series devoted to various cases of categorical descent. Philosophically, our interest in the subject grew out of attempts to understand the main construction from the recent paper by Ben-Zvi and Nadler [1] in plain terms that would not involve higher category theory. 1.1. Beilinson-Bernstein Localization and Derived Descent Let be a reductive algebraic group with the Lie algebra . Denote the Flag variety of by . A major part of Geometric Representation Theory originated in the seminal work of Be？linson and Bernstein [2] devoted to investigation of the globalization functor . This functor turns out to be fully faithful and provides geometric and topological tools to investigate a wide class of -modules, in particular the ones from the famous category . Various generalizations of this result lead to the investigation of the categories of twisted D-modules on the Flag variety and on the base affine space for and of their derived categories. Ben-Zvi and Nadler define a certain comonad acting on a higher categorical version for the derived category of D-modules on the base affine space. In fact, the functor is built into the higher categorical treatment of Beilinson-Bernstein localization-globalization construction. Using the heavy machinery of Barr-Beck-Lurie descent, the authors argue that the derived category of -modules is equivalent to the category of D-modules equivariant with respect to this comonad. Thus the global sections functor becomes equivariant with respect to the action. The comonad is called the Hecke comonad. It provides a categorification for the classical action of the Weyl group on various homological and K-theoretic invariants of the Flag variety. Notice that the descent construction fails to work on the level of the usual triangulated categories. Ideally one would like to replace it by a categorical action of the Weyl group or rather of the Braid group on categories of D-modules related to the Flag variety. One would need to define a notion of “invariants” with respect to such action. 1.2. Descent in Equivariant -Theory Another source of inspiration for the present paper, which is in a way closer to our work, is a recent article of Harada et al. [3]. Given a