Hecke eigenvalues and relations for Siegel Eisenstein series of arbitrary degree, level, and character

TL;DR

This paper explicitly evaluates Hecke operators on Siegel Eisenstein series across various degrees, levels, and characters, providing diagonalization results and explicit eigenvalues, thus advancing understanding of their algebraic structure.

Contribution

It introduces explicit calculations of Hecke eigenvalues for Siegel Eisenstein series of arbitrary degree, level, and character, including multiplicity-one results for square-free levels.

Findings

Diagonalization of Hecke operators for square-free levels

Explicit eigenvalues for all Hecke operators at primes not dividing the level

Multiplicity-one result for the space of Eisenstein series

Abstract

We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalise the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalise the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.