Covariant bi-differential operators on matrix space

TL;DR

This paper constructs a family of covariant bi-differential operators on matrix spaces, generalizing classical operators like transvectants and Rankin-Cohen brackets, under the action of the group SL(2m, R).

Contribution

It introduces a new family of covariant bi-differential operators on matrix spaces that generalize classical operators for the case m=1.

Findings

Operators are covariant under SL(2m, R) action.

Generalization of transvectants and Rankin-Cohen brackets.

Framework applicable to matrix spaces of arbitrary size.

Abstract

A family of bi-differential operators from $C^\infty\big(\Mat(m,\mathbb R)\times\Mat(m,\mathbb R)\big)$ into $C^\infty\big(\Mat(m,\mathbb R)\big)$ which are covariant for the projective action of the group $SL(2m,\mathbb R)$ on $\Mat(m,\mathbb R)$ is constructed, generalizing both the \emph{transvectants} and the \emph{Rankin-Cohen brackets} (case $m=1$).