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.

W. Casselman defined a basis ${{f}_{u}}$ of Iwahori fixed vectors of a spherical representation $(\pi ,\,V)$ of a split semisimple $p$-adic group $G$ over a nonarchimedean local field $F$ by the condition that it be dual to the intertwining operators, indexed by elements $u$ of the Weyl group $W$. On the other hand, there is a natural basis ${{\psi }_{u}}$, and one seeks to find the transition matrices between the two bases. Thus, let ${{f}_{u}}\,=\,{{\sum }_{v}}\overset{\tilde{\ }}{\mathop{m}}\,(u,\,v){{\psi }_{v}}$ and ${{\psi }_{u}}\,=\,{{\sum }_{v}}m(u,\,v){{f}_{v}}$. Using the Iwahori–Hecke algebra we prove that if a combinatorial condition is satisfied, then $m(u,\,v)\,=\,{{\Pi }_{\alpha }}\,\frac{1-{{q}^{-1}}\,{{z}^{\alpha }}}{1-{{z}^{\alpha }}}$ , where $\mathbf{z}$ are the Langlands parameters for the representation and $\alpha $ runs through the set $S(u,\,v)$ of positive coroots $\alpha \,\in \,\hat{\Phi }$ (the dual root systemof $G$) such that $u\,\le \,v{{r}_{\alpha }}\,<\,v$ with ${{r}_{\alpha }}$ the reflection corresponding to $\alpha $. The condition is conjecturally always satisfied if $G$ is simply-laced and the Kazhdan–Lusztig polynomial ${{P}_{{{w}_{0}}v,\,{{w}_{0}}u}}\,=\,1$ with ${{w}_{0}}$ the long Weyl group element. There is a similar formula for $\tilde{m}$ conjecturally satisfied if ${{P}_{u,\,v}}\,=\,1$. This leads to various combinatorial conjectures.