Iwahori-invariants of principle series, stable envelopes and motivic Chern classes

Abstract:  Let G be a split reductive p-adic group. In the Iwahori-invariants of an unramified principle series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove a conjecture of Bump--Nakasuji--Naruse about certain transition matrix between these two bases. The ingredients of the proof include Maulik--Okounkov's K theoretic stable envelopes and Brasselet--Schurmann--Yokura's motivic Chern classes for the Langlands dual groups. These objects are related by the affine Hecke algebra. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.