Gurbir Dhillon, Stanford University
"Double cosets, character formulas, and the Virasoro algebra"
We will explain a localization theorem for the Virasoro algebra. This will be a statement of the form: "sheaves with flat connection on a double coset space sit fully faithfully inside the category of representations of the Virasoro algebra." Our principal goal, however, will be to try to review some motivation and context for these types of theorems. As such, the talk will be mostly expository, and will assume only minimal familiarity with representation theory. We will start with finite groups, where double cosets arise in character formulas relating induction and restriction (Mackey theory). We will then move on to algebraic groups, and review an analog of Mackey theory. We will discuss an important special case of this, the Weyl character formula, which determines the characters of semisimple algebraic groups (e.g. SL_n). By inspecting the summands in the Weyl character formula, we will be led to an analogous discussion for representations of semisimple Lie algebras. In particular, we will discuss the seminal localization theorem of Beilinson and Bernstein. Finally, we will state and discuss the localization theorem for Virasoro.