Dimension formulas for certain spaces of Drinfeld cusp forms

In this short note, we derive dimension formulas for spaces of Drinfeld cusp forms corresponding to harmonic cocycles invariant under the group ${\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_q[t])$ and with values in absolutely irreducible ${\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_q(t))$-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.