Program Results

Introduction to the event

I have been mainly working on representation theory of p-adic reductive groups in relation to number theory, especially, from the viewpoint of Langlands program. This year I focused on joint research with Charlotte Chan (University of Michigan) on the positive-depth Deligne-Lusztig theory.

One of the ultimate goals of representation theory of p-adic reductive groups is to construct and classify all irreducible representations of p-adic reductive groups. Around 2000, Jiu-Kang Yu established a methodology of explicitly constructing a lot of irreducible representations of p-adic reductive groups. Since then, in the context of the Langlands program, Yu's theory has played significant roles.

On the other hand, for finite classical groups, it is known that all irreducible representations can be constructed by means of geometry; it is called the Deligne-Lusztig theory. Finite reductive groups can be thought of as a toy model of p-adic reductive groups. Hence it is natural to seek a version of the Deligne-Lusztig theory in the context of representation theory of p-adic reductive groups. Recently, Chan-Ivanov developed a p-adic version of DeligneLusztig theory, which we call "positive-depth Deligne-Lusztig theory". They introduced a geometric object which enables us to produce representations of p-adic reductive groups.

The natural direction suggested by the work of Chan-Ivanov is to investigate what kinds of, and how many, representations of p-adic groups can be obtained by their construction. This is exactly what we achieved this year. More precisely, we established a precise comparison result between positive-depth Deligne-Lusztig representations and the representations obtained by Yu's construction in an algebraic manner. As a consequence, we gave a geometric interpretation to various algebraic results on representations of p-adic reductive groups, for example, explicit character formulas.

Concerning this result, we released a preprint, which is available on "arXiv" (https://arxiv.org/abs/2506.04449). Also, I reported this result at 5 international conferences/workshops in total.