Casselman's basis of Iwahori vectors and Kazhdan-Lusztig polynomials

2017/10/09 by Daniel Bump, Bump, Daniel, Maki Nakasuji +1
Mathematics · #05E15 #20F55 #22E50 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT #msc:05E15 #msc:20F55 #msc:22E50

paper · pdf · doi:10.48550/arxiv.1710.03185

arxiv created 2017/10/09 · openalex publication_date 2017/10/09 · arxiv updated 2017/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A problem in representation theory of p-adic groups is the computation of the Casselman basis of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix (m_u,v) of the Casselman basis to another natural basis in terms of certain polynomials which are deformations of the Kazhdan-Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality q to q^-1. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix (m_u,v) to its inverse.

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