Dimension formulas for certain spaces of Drinfeld cusp forms

TL;DR

This paper derives explicit dimension formulas for spaces of Drinfeld cusp forms associated with harmonic cocycles and irreducible representations, extending previous results and providing asymptotic estimates.

Contribution

It introduces new dimension formulas for Drinfeld cusp form spaces using Brauer character theory, generalizing prior work with different methods.

Findings

Derived explicit dimension formulas for Drinfeld cusp form spaces.

Proved a simple asymptotic formula for these dimensions.

Extended previous results to broader classes of representations.

Abstract

In this short note, we derive dimension formulas for spaces of Drinfeld cusp forms corresponding to harmonic cocycles invariant under the group $\mathrm{SL}_2(\mathbb{F}_q[t])$ and with values in absolutely irreducible $\mathrm{SL}_2(\mathbb{F}_q(t))$-representations via the theory of Brauer characters. This generalizes results in [BGP21] obtained by different methods. In addition, we prove a simple asymptotic formula for these dimensions.