  Let $(\pi_{\mathbf{z}},V_{\mathbf{z}})$ be an unramified principal series
representation of a reductive group over a nonarchimedean local field,
parametrized by an element $\mathbf{z}$ of the maximal torus in the Langlands
dual group. If $v$ is an element of the Weyl group $W$, then the standard
intertwining integral $\mathcal{A}_v$ maps $V_{\mathbf{z}}$ to
$V_{v\mathbf{z}}$. Letting $\psi^{\mathbf{z}}_w$ with $w\in W$ be a suitable
basis of the Iwahori fixed vectors in $V_{\mathbf{z}}$, and
$\widehat\psi^{\mathbf{z}}_w$ a basis of the contragredient representation, we
define $\sigma(u,v,w)$ (for $u,v,w\in W$) to be $\langle
\mathcal{A}_v\psi_u^{\mathbf{z}},\widehat\psi^{v\mathbf{z}}_w\rangle$. This is
an interesting function and we initiate its study. We show that given $u$ and
$w$, there is a minimal $v$ such that $\sigma(u,v,w)\neq 0$. Denoting this $v$
as $v_\hbox{min}=v_\hbox{min}(u,w)$, we will prove that
$\sigma(u,v_\hbox{min},w)$ is a polynomial of the cardinality $q$ of the
residue field. Indeed if $v>v_\hbox{min}$, then $\sigma(u,v,w)$ is a rational
function of $\mathbf{z}$ and $q$, whose denominator we describe. But if
$v=v_\hbox{min}$, the dependence on $\mathbf{z}$ disappears. We will express
$\sigma(u,v_\hbox{min},w)$ as the Poincar\'e polynomial of a Bruhat interval.
The proof leads to fairly intricate considerations of the Bruhat order.
  Thus our results require us to prove some facts that may be of independent
interest, relating the Bruhat order $\leqslant$ and the weak Bruhat order
$\leqslant_R$. For example we will prove (for finite Coxeter groups) the
following "mixed meet" property. If $u, w$ are elements of $W$, then there
exists a unique element $m \in W$ that is maximal with respect to the condition
that $m \leqslant_R u$ and $m \leqslant w$. Thus if $z \leqslant_R u$ and $z
\leqslant w$, then $x \leqslant m$. The value $v_\hbox{min}$ is $m^{-1}u$.

Representation Theory;Combinatorics