On the Space Defined by the Modular Transformation Law for Multiple Eisenstein Series

Abstract

Motivated by a modular transformation law for multiple Eisenstein series established by Yu, we introduce spaces of holomorphic functions satisfying this law together with an additional filtration condition. We show that these spaces contain the spaces spanned by admissible multiple Eisenstein series, are finitedimensional in each weight, and share several structural properties with them. We also obtain an explicit formula for the generating function of their dimensions. However, the inclusions are known to be strict for certain weights. Thus, the defining conditions do not suffice to characterize the spaces spanned by admissible multiple Eisenstein series.