One-line formula for automorphic differential operators on Siegel modular forms

Abstract

We consider the Siegel upper half space $$H_{2m}$$ of degree 2m and a subset $$H_m\times H_m$$ of $$H_{2m}$$ consisting of two $$m\times m$$ diagonal block matrices. We consider two actions of $$Sp(m,{\mathbb R})\times Sp(m,{\mathbb R}) \subset Sp(2m,{\mathbb R})$$ , one is the action on holomorphic functions on $$H_{2m}$$ defined by the automorphy factor of weight k on $$H_{2m}$$ and the other is the action on vector valued holomorphic functions on $$H_m\times H_m$$ defined on each component by automorphy factors obtained by $$det^k \otimes \rho $$ , where $$\rho $$ is a polynomial representation of $$GL(n,{\mathbb C})$$ . We consider vector valued linear holomorphic differential operators with constant coefficients on holomorphic functions on $$H_{2m}$$ which give an equivariant map with respect to the above two actions under the restriction to $$H_m\times H_m$$ . In a previous paper, we have already shown that all such operators can be obtained either by a projection of the universal automorphic differential operator or alternatively by a vector of monomial basis corresponding to the partition $$2m=m+m$$ . Here in this paper, based on a completely different idea, we give much simpler looking one-line formula for such operators. This is obtained independently from our previous results. The proofs also provide more algorithmic approach to our operators.