Geodesic
Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials
Canadian journal of mathematics, Tome 71 (2019) no. 6, pp. 1351-1366

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DOI

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.
DOI : 10.4153/CJM-2018-011-1
Mots-clés : Kazhdan-Lusztig Polynomial, Iwahori fixed vector, Bruhat order 
Bump, Daniel; Nakasuji, Maki. Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials. Canadian journal of mathematics, Tome 71 (2019) no. 6, pp. 1351-1366. doi: 10.4153/CJM-2018-011-1
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     author = {Bump, Daniel and Nakasuji, Maki},
     title = {Casselman{\textquoteright}s {Basis} of {Iwahori} {Vectors} and {Kazhdan{\textendash}Lusztig} {Polynomials}},
     journal = {Canadian journal of mathematics},
     pages = {1351--1366},
     year = {2019},
     volume = {71},
     number = {6},
     doi = {10.4153/CJM-2018-011-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2018-011-1/}
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