Toric periods for a p-adic quaternion algebra

Let G be a compact group with two given subgroups H and K. Let \(\pi \) $\pi$ 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 \(v_H\) $v_H$ (resp. \(v_K\) $v_K$ ) denote an H-invariant (resp. K-invariant) vector of unit norm in a given G-invariant inner product \(\langle ~,~ \rangle _\pi \) ${⟨,⟩}_{\pi }$ on \(\pi \) $\pi$ . We are interested in calculating the correlation coefficient \(\begin{aligned} c(\pi \text {;}\,H,K) = |\langle v_H,v_K \rangle _\pi |^2. \end{aligned}\) $$\begin{array}{c}\hfill c(\pi \text{;}H,K)=|{⟨v_H,v_K⟩}_{\pi }{|}^2.\end{array}$$ 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 \(\pi \) $\pi$ is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori H and K, then its root number \(\varepsilon (\pi )\) $\epsilon (\pi )$ is \(\pm 1\) $\pm 1$ and \(c(\pi \text {;}\, H, K)\) $c(\pi \text{;}H,K)$ is non-vanishing precisely when \(\varepsilon (\pi ) = 1\) $\epsilon (\pi )=1$ .