Deligne-Lusztig theory: from classical to p-adic

Prof. Dr. Alexander Ivanov, Ruhr-Universität Bochum

Finite groups of Lie type form a very rich and important class of finite groups, including, for example, groups like GL_n (F_p). Deligne-Lusztig theory, developped in 1970-80's, plays a central role in the representation theory of these groups. It provides a systematic algebraic-geometric tool to construct, classify and study the properties of all irreducible representations.

Similarly, but more complicated, one is also interested in the (smooth) representations of p-adic reductive groups (like, for example, GL_n (Q_p) where Q_p is the field of p-adic numbers) and in the closely related local Langlands correspondences. This is a highly active area of research, where recent breakthroughs were achieved (notably, in the seminal work of Fargues-Scholze). As for groups over finite fields, one can also develop a Deligne-Lusztig theory for p-adic groups.

In the talk I plan to give an introduction into the classical Deligne-Lusztig theory and its p-adic counterpart. If time permits, I also explain a conjectural application (interpolation between Fargues-Scholze's conjecture and the algebraic constructions of Yu, Kaletha and others).