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- W4300485791 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 $-1$." @default.
- W4300485791 created "2022-10-03" @default.
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- W4300485791 date "2021-06-03" @default.
- W4300485791 modified "2023-09-26" @default.
- W4300485791 title "Orthogonality of invariant vectors" @default.
- W4300485791 doi "https://doi.org/10.48550/arxiv.2106.01929" @default.
- W4300485791 hasPublicationYear "2021" @default.
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