Orthogonality of invariant vectors

Abstract: Let $(\pi,V)$ be an irreducible complex representation of a
finite group $G$ and let $\langle~, ~\rangle_\pi$ be the standard
$G$-invariant inner product on $\pi.$ Let $H$ and $K$ be subgroups of $G$
such that the space of $H$-invariant vectors as well as the space of
$K$-invariant vectors of $\pi$ are one dimensional. Fix an $H$-invariant
unit vector $v_H$ and a $K$-invariant unit vector $v_K.$ Benedict Gross
defines the Correlation constant $c(\pi; H, K)$ of $H$ and $K$ with
respect to $\pi.$ It turn out that $c(\pi; H, K)=|\langle v_H,
v_K\rangle_\pi|^2.$

In this talk we analyze the Correlation constant $c(\pi; H, K),$ when
$G={\rm GL}_2(\mathbb{F}_q),$ where $\mathbb{F}_q$ is the finite field
with $q=p^f$ elements for some odd prime $p,$ $H$ (resp. $K$) is the split
(resp. non split) torus of $G.$ This is joint with U. K. Anandavardhanan.