Simultaneous nonvanishing of the correlation constant

For $q=p^m$, where $p$ is an odd prime number, we study the correlation coefficient $c(\pi ;H,K)$ of an irreducible (complex) representation $\pi$ of $G={\mathrm{G}\mathrm{L}}_2({\mathbb{F}}_q)$ with respect to a split torus $H$ and a non-split torus $K$. We consider a family of non-split tori $K_{\alpha ,u}$ indexed by $u\in {\mathbb{F}}_q$ and $\alpha \in {\mathbb{F}}_q^{\times }\setminus {\mathbb{F}}_q^{\times 2}$. We show that under any identification of $\mathbb{C}$ with ${\bar{\mathbb{Q}}}_p$, and writing $\pi ={\pi }_r$ where $0\le r\le (q-1)\mathrm{/}2$ depending on this identification, we have $$c({\pi }_r;H,K_{\alpha ,u})\equiv [P_r(u\mathrm{/}\sqrt{\alpha }){]}^2\mathrm{m}\mathrm{o}\mathrm{d}p,$$ where $P_r(X)\in \mathbb{Z}[\frac{1}{2}][X]$ is the $r$-th Legendre polynomial. As a corollary, when $m\ge 2$, we prove that there exists $u\in {\mathbb{F}}_q^{\times }$ such that $c(\pi ;H,K_{\alpha ,u})\mathrm{\ne }0$ for all irreducible representations $\pi$ of $G$ admitting fixed vectors for both $H$ and $K$.