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
\[c(\pi;H,K) = |\langle v_H,v_K \rangle_\pi|^2.\]
In this talk, 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 the root number \(\varepsilon(\pi)\) of \(\pi\) is \(\pm 1\) and \(c(\pi; H, K)\) is non-vanishing precisely when \(\varepsilon(\pi) = 1\). This is joint work with U. K. Anandavardhanan.