Toric periods for a p-adic quaternion algebra

IF 1 3区数学 Q1 MATHEMATICS

Mathematische Zeitschrift Pub Date : 2024-07-14 DOI:10.1007/s00209-024-03551-3

U. K. Anandavardhanan, Basudev Pattanayak

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Abstract

Let G be a compact group with two given subgroups H and K. Let $\pi $ be an irreducible 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 a given G-invariant inner product $\langle ~,~ \rangle _\pi $ on $\pi $. We are interested in calculating the correlation coefficient

$$\begin{aligned} c(\pi \text {;}\,H,K) = |\langle v_H,v_K \rangle _\pi |^2. \end{aligned}$$

In this paper, we compute the correlation coefficient of an irreducible representation of the multiplicative group of the p-adic quaternion algebra with respect to any two tori. In particular, if $\pi $ is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori H and K, then its root number $\varepsilon (\pi )$ is $\pm 1$ and $c(\pi \text {;}\, H, K)$ is non-vanishing precisely when $\varepsilon (\pi ) = 1$.

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p-adic 四元数代数的 Toric 周期

让 G 是一个紧凑群，有两个给定的子群 H 和 K。让 $\pi $ 是 G 的不可还原表示，使得它的 H 不变向量空间和 K 不变向量空间都是一维的。让 $v_H$ (resp. $v_K$) 表示给定 G 不变内积 $\langle ~,~ \rangle _\pi $ 在 $\pi $ 上的单位法的 H 不变（或 K 不变）向量。我们感兴趣的是计算相关系数 $$\begin{aligned} c(\pi \text {;}\,H,K) = |\langle v_H,v_K \rangle _\pi |^2。\end{aligned}$$ 在本文中，我们计算 p-adic 四元数代数的乘法群的不可还原表示与任意两个环的相关系数。特别地，如果$\pi $是这样一个奇数最小导体的不可还原表示，它对于两个环 H 和 K 具有非难变向量，那么它的根（(\varepsilon (\pi )\）是（\pm 1\ ），并且（c(\pi \text {;}\, H, K)\）恰好在（\(\varepsilon (\pi)= 1\ ）时是非递减的。

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