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.