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A problem in representation theory of $p$-adic groups is the computation ofthe \textit{Casselman basis} of Iwahori fixed vectors in the sphericalprincipal series representations, which are dual to the intertwining integrals.We shall express the transition matrix $(m_{u,v})$ of the Casselman basis toanother natural basis in terms of certain polynomials which are deformations ofthe Kazhdan-Lusztig R-polynomials. As an application we will obtain certain newfunctional equations for these transition matrices under the algebraicinvolution sending the residue cardinality $q$ to $q^{-1}$. We will also obtaina new proof of a surprising result of Nakasuji and Naruse that relates thematrix $(m_{u,v})$ to its inverse.

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