Irreducible vector-valued modular forms of dimension less than six

Abstract

An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, $T$-unitarizable representations of the full modular group, of dimension less than six. For representations of dimension less than four, it is shown that the associated space of vector-valued modular forms is a cyclic module over a certain skew polynomial ring of differential operators. For dimensions four and five, a complete list of possible Hilbert–Poincaré series is given, using the fact that the space of vector-valued modular forms is a free module over the ring of classical modular forms for the full modular group. A mild restriction is then placed on the class of representation considered in these dimensions, and this again yields an explicit determination of the associated Hilbert–Poincaré series.