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 $f_u = \sum_v \tilde{m} (u, v) \psi_v$ 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 $m (u, v) = \prod_{\alpha} \frac{1 - q^{- 1} \mathbf{z}^{\alpha}}{1 -\mathbf{z}^{\alpha}}$, 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 system of $G$) such that $u \leqslant v r_{\alpha} < v$ 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.

Certain polynomials in $n^2$ variables that serve as generating functions for symmetric group characters are sometimes called ($S_n$) character immanants. We point out a close connection between the identities of Littlewood--Merris--Watkins and Goulden--Jackson, which relate $S_n$ character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra $H_n(q)$, we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood--Merris--Watkins identities and selected Goulden--Jackson identities that relate $H_n(q)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis--L\^e--Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.