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

Abstract

We calculate the tempered O ( p , q ) O(p,q) -invariant eigendistributions of the O ( p , q ) O(p,q) -invariant Hermite operator − 1 2 ( Δ x − Δ y ) + 1 2 ( | x | 2 − | y | 2 ) ( x ∈ R p , y ∈ R q ) . \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 | |x|=|y| and are given elsewhere in terms of confluent hypergeometric functions.