Covariant bi-differential operators on matrix space

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 $m=1$)

Orbits of triples in the Shilov boundary of a bounded symmetric domain

Let ${\cal D}$ be a bounded symmetric domain of tube type, $S$ its Shilov boundary, and $G$ the neutral component of its group of biholomorphic transforms. We classify the orbits of $G$ in the set $S\times S\times S$

Singular conformally invariant trilinear forms and covariant differential operators on the sphere

Let $G=SO_0(1,n)$ be the conformal group acting on the $(n-1)$ dimensional sphere $S$, and let $(\pi_\lambda)_{\lambda\in \mathbb C}$ be the spherical principal series. For generic values of $\boldsymbol \lambda =(\lambda_1,\lambda_2,\lambda_3)$ in $\mathbb C^3$, there exits a (essentially unique) trilinear form on $\mathcal C^\infty(S)\times \mathcal C^\infty(S)\times \mathcal C^\infty(S)$ which is invariant under $\pi_{\lambda_1}\otimes \pi_{\lambda_2}\otimes \pi_{\lambda_3}$. Using differential operators on the sphere $S$ which are covariant under the conformal group $SO_0(1,n)$, we construct new invariant trilinear forms corresponding to singular values of $\boldsymbol \lambda$. The family of generic invariant trilinear forms depend meromorphically on the parameter $\boldsymbol \lambda$ and the new forms are shown to be residues of this family

Conformally Covariant Bi-Differential Operators on a Simple Real Jordan Algebra

For a simple real Jordan algebra $V,$ a family of bi-differential operators from $\mathcal{C}^\infty(V\times V)$ to $\mathcal{C}^\infty(V)$ is constructed. These operators are covariant under the rational action of the conformal group of $V.$ They generalize the classical {\em Rankin-Cohen} brackets (case $V=\mathbb{R}$)