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

  • Daniel Bump (a1) and Maki Nakasuji (a2)

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.

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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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Canadian Journal of Mathematics
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