Hecke operators on half-integral weight Siegel Eisenstein series

Abstract

We construct a basis for the space of half-integral weight Siegel Eisenstein series of level [Formula: see text] where [Formula: see text] is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of [Formula: see text], commenting on why this restriction is necessary for our methods. We directly apply to these forms all Hecke operators attached to odd primes, and we realize the images explicitly as linear combinations of Siegel Eisenstein series. Using this information, we diagonalize the subspace of Eisenstein series generated from elements of [Formula: see text], obtaining a multiplicity-one result.