Hermite distributions associated to the group $O(p,q)$

Abstract:

We calculate the tempered $O(p,q)$-invariant eigendistributions of the $O(p,q)$-invariant Hermite operator \begin{equation*}-{\textstyle {\frac {1}{2}}}(\Delta _{x}- \Delta _{y}) +{\textstyle {\frac {1}{2}}}(|x|^{2}-|y|^{2})\qquad (x\in \mathbb {R}^{p}, y\in \mathbb {R}^{q}).\end{equation*} They are singular on the cone $|x|=|y|$ and are given elsewhere in terms of confluent hypergeometric functions.