Covariant bi-differential operators on matrix space

Clerc, Jean-Louis

Annales de l'Institut Fourier, Tome 67 (2017) no. 4, pp. 1427-1455.

On construit une famille d’opérateurs bi-différentiels de $C^{\infty }(\mathrm{Mat}(m,ℝ)\times \mathrm{Mat}(m,ℝ))$ dans $C^{\infty }\left(\mathrm{Mat}(m,ℝ)\right)$ qui sont covariants pour l’action projective du groupe $SL(2m,ℝ)$ sur $\mathrm{Mat}(m,ℝ)$. Dans le cas $m=1$, cette construction fournit une nouvelle approche des transvectants et des crochets de Rankin–Cohen.

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

Classification : 22E45, 58J70
Keywords: Covariant differential operators, Knapp–Stein intertwining operators, Zeta functional equation, transvectants, Rankin–Cohen brackets
Mot clés : Opérateurs différentiels covariants, opérateurs d’entrelacement de Knapp–Stein, équation fonctionnelle de Zeta, transvectants, crochets de Rankin–Cohen

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