Hecke eigenvalues of Klingen--Eisenstein series of squarefree level

TL;DR

This paper derives explicit formulas for the action of Hecke operators on Klingen--Eisenstein series of squarefree level, using intertwining relations and Fourier expansion techniques, enhancing understanding of modular forms at arbitrary levels.

Contribution

It provides new explicit intertwining relations between Hecke operators and Siegel lowering operators for modular forms of arbitrary level and character, specifically extending to squarefree levels.

Findings

Formulas for Hecke operators on Fourier expansions

Intertwining relations for each cusp of the modular variety

Explicit action formulas for Klingen--Eisenstein series at squarefree levels

Abstract

We compute the intertwining relation between the Hecke operators and the Siegel lowering operators on Siegel modular forms of arbitrary level $N$ and character $\chi$ by using formulas for the action of the Hecke operators on Fourier expansions. Using an explicit description of the Satake compactification of $\Gamma_0^{(n)}(N) \backslash \mathfrak{H}_n$ when $N$ is squarefree we extend this to give intertwining relations for each cusp. As an application we give formulas for the action of Hecke operators on the space of Klingen--Eisenstein series of squarefree level $N$, for primes $p \nmid N$.