On the dimensions of certain spaces of vector-valued cusp forms

TL;DR

This paper determines the dimension of vector-valued cusp form spaces associated with irreducible representations of SL_2(F_q), linking it to the multiplicity in classical cusp forms for certain subgroups.

Contribution

It provides an explicit formula for the dimension of vector-valued cusp forms for odd prime q, connecting representation theory with classical modular forms.

Findings

Dimension equals the multiplicity of the representation in classical cusp forms.

Explicit dimension formula for vector-valued cusp forms.

Connection established between representation theory and modular form spaces.

Abstract

Given an irreducible representation of $SL_2(F_q)$ for an odd prime $q\geq 5$, we find the dimension of the space of cusp forms with respect to the full modular group taking values in the representation space. The dimension equals the multiplicity of the representation in the space of classical cusp forms with respect to the principal congruence subgroup of level $q$.