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.