Invariant differential operators on Siegel-Jacobi space and Maass-Jacobi forms

zbMATH Open1306.32017arXiv1107.0509MaRDI QIDQ2875835

Publication date: 12 August 2014

Abstract: For two positive integers

m

and

n

, we let

{m a t h b b H}_{n}

be the Siegel upper half plane of degree

n

and let

{m a t h b b C}^{(m, n)}

be the set of all

m i m e s n

complex matrices. In this article, we study differential operators on the Siegel-Jacobi space

{m a t h b b H}_{n} i m e s {m a t h b b C}^{(m, n)}

that are invariant under the natural action of the Jacobi group

S p (n, m a t h b b R l t i m e s H_{m a t h b b R}^{(n, m)}

on

{m a t h b b H}_{n} i m e s {m a t h b b C}^{(m, n)}

, where

H_{m a t h b b R}^{(n, m)}

denotes the Heisenberg group. We give some explicit invariant differential operators. We present important problems which are natural. We give some partial solutions for these natural problems.

Full work available at URL: https://arxiv.org/abs/1107.0509

zbMATH Keywords

invariant differential operators Siegel-Jacobi space

Mathematics Subject Classification ID

Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) (32M15)