Class numbers and invariant characters of $\mathfrak{sl}_2(\mathbb{F}_p)$

Abstract

Let $p$ be a prime and let $S_2(\Gamma(p))$ be the space of weight $2$ cusp forms for the principal congruence subgroup $\Gamma(p)$. Then $\mathrm{SL}_2(\mathbb{F}_p)$ acts on $S_2(\Gamma(p))$ in a natural way. Around 1928, Hecke proved that if $p>3$ and $p\equiv 3\mod 4$, then the class number of $\mathbb{Q}(\sqrt{-p})$ is equal to the difference between the multiplicities of two particular irreducible representations of $\mathrm{SL}_2(\mathbb{F}_p)$ in $S_2(\Gamma(p))$. In this paper we prove a Lie algebra analogue of this result. As an application we extend Hecke's result to $\mathrm{SL}_2(\mathbb{Z}/p^r)$ (acting on $S_2(\Gamma(p^r))$) for any $r\geq 2$.