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Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials

Part of: Lie groups Special aspects of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  07 January 2019

Daniel Bump  and
Maki Nakasuji
Daniel Bump
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305-2125 Email: bump@math.stanford.edu
Maki Nakasuji
Affiliation:
Department of Information and Communication Science, Faculty of Science, Sophia University, 7-1 Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan Email: nakasuji@sophia.ac.jp
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Abstract

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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 that 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.

Keywords

Kazhdan-Lusztig PolynomialIwahori fixed vectorBruhat order

MSC classification

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields
Secondary: 20C08: Hecke algebras and their representations 20F55: Reflection and Coxeter groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1351 - 1366
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This work was supported by NSF grant DMS-1601026 and JSPS Grant-in-Aid for Young Scientists (B) 15K17519. We thank the referee for careful reading.

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