Colloquium abstracts

Arindam Jana
TIFR, Mumbai
February 3, 2022

Orthogonality of invariant vectors:  Let $(pi,V)$ be an irreducible complex representation of a finite group $G$ and let $langle~, ~ angle_pi$ 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 $c(pi; H, K)=|langle v_H, v_K angle_pi|^2.$ In this talk we analyze the Correlation constant $c(pi; H, K),$ when $G={ m GL}_2(mathbb{F}_q),$ 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.