Representation theory via geometry

Given a group G, a fundamental question is how to construct and understand its irreducible representations. A rich class of finite simple groups for which we can ask this question is the finite groups of Lie type, close cousins of algebraic groups over finite fields. In this talk, we will focus on the group SL_2(F_q) – through the process of parabolic induction, we obtain roughly half of its irreducible representations, so how can we excavate the others? Drinfeld realized that the answer to this lay in the geometry of an “an extremely symmetric” algebraic curve over F_q, a subtle and beautiful idea that inspired Deligne and Lusztig to prove far-reaching results about representations of all finite groups of Lie type by analyzing algebraic varieties that now bear their names. We retrace the roots of Deligne-Lusztig theory by studying SL_2(F_q), and if time permits, discuss more general developments.

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