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Let $(\pi_{\mathbf{z}},V_{\mathbf{z}})$ be an unramified principal seriesrepresentation of a reductive group over a nonarchimedean local field,parametrized by an element $\mathbf{z}$ of the maximal torus in the Langlandsdual group. If $v$ is an element of the Weyl group $W$, then the standardintertwining integral $\mathcal{A}_v$ maps $V_{\mathbf{z}}$ to$V_{v\mathbf{z}}$. Letting $\psi^{\mathbf{z}}_w$ with $w\in W$ be a suitablebasis of the Iwahori fixed vectors in $V_{\mathbf{z}}$, and$\widehat\psi^{\mathbf{z}}_w$ a basis of the contragredient representation, wedefine $\sigma(u,v,w)$ (for $u,v,w\in W$) to be $\langle\mathcal{A}_v\psi_u^{\mathbf{z}},\widehat\psi^{v\mathbf{z}}_w\rangle$. This isan 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 theresidue field. Indeed if $v>v_\hbox{min}$, then $\sigma(u,v,w)$ is a rationalfunction 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 independentinterest, relating the Bruhat order $\leqslant$ and the weak Bruhat order$\leqslant_R$. For example we will prove (for finite Coxeter groups) thefollowing "mixed meet" property. If $u, w$ are elements of $W$, then thereexists a unique element $m \in W$ that is maximal with respect to the conditionthat $m \leqslant_R u$ and $m \leqslant w$. Thus if $z \leqslant_R u$ and $z\leqslant w$, then $x \leqslant m$. The value $v_\hbox{min}$ is $m^{-1}u$.

Matrix coefficients of intertwining operators and the Bruhat order