Orthogonality of invariant vectors

TL;DR

This paper investigates the correlation between invariant vectors in irreducible representations of GL2 over finite fields, providing explicit formulas and conditions for their orthogonality or correlation, with implications for automorphic forms and base change phenomena.

Contribution

It offers new explicit formulas and criteria for the correlation of invariant vectors in representations of GL2 over finite fields, including mod p analysis and behavior under base change.

Findings

Explicit formula for the correlation constant modulo p.

Sufficient conditions for orthogonality and non-orthogonality of vectors.

Analysis of correlation behavior under Shintani base change.

Abstract

Let $G$ be a finite group with given subgroups $H$ and $K$. Let $\pi$ be an irreducible complex representation of $G$ such that its space of $H$-invariant vectors as well as the space of $K$-invariant vectors are both one dimensional. Let $v_H$ (resp. $v_K$) denote an $H$-invariant (resp. $K$-invariant) vector of unit norm in the standard $G$-invariant inner product $\langle ~,~ \rangle_\pi$ on $\pi$. Our interest is in computing the square of the absolute value of $\langle v_H,v_K \rangle_\pi$. This is the correlation constant $c(\pi;H,K)$ defined by Gross. In this paper, we give a sufficient condition for $\langle v_H, v_K \rangle_\pi$ to be zero and a sufficient condition for it to be non-zero (i.e., $H$ and $K$ are correlated with respect to $\pi$), when $G={\rm GL}_2(\mathbb F_q)$, where $\mathbb F_q$ is the finite field of $q=p^f$ elements of odd characteristic $p$, $H$ is its split torus and $K$ is a non-split torus. The key idea in our proof is to analyse the mod $p$ reduction of $\pi$. We give an explicit formula for $|\langle v_H,v_K \rangle_\pi|^2$ modulo $p$. Finally, we study the behaviour of $\langle v_H,v_K \rangle_\pi$ under the Shintani base change and give a sufficient condition for $\langle v_H,v_K \rangle_\pi$ to vanish for an irreducible representation $\pi={\rm BC}(\tau)$ of ${\rm PGL}_2(\mathbb E)$, in terms of the epsilon factor of the base changing representation $\tau$ of ${\rm PGL}_2(\mathbb F)$, where $\mathbb E/\mathbb F$ is a finite extension of finite fields. This is reminiscent of the vanishing of $L(1/2, {\rm BC}(\tau))$, in the theory of automorphic forms, when the global root number of $\tau$ is $-1$.