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

Daniel Bump; Maki Nakasuji

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arXiv · 1710.03185

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

Daniel Bump·Maki Nakasuji

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Abstract

A problem in representation theory of $p$-adic groups is the computation of the \textit{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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Daniel Bump, Maki Nakasuji. 2017-10-09. Casselman's basis of Iwahori vectors and Kazhdan-Lusztig polynomials. https://arxiv.org/abs/1710.03185

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