Representation Growth of Compact Special Linear Groups of degree two

Abstract

We study the finite-dimensional continuous complex representations of $\mathrm{SL}_2$ over the ring of integers of non-Archimedean local fields of even residual characteristic. We prove that for characteristic two, the abscissa of convergence of the representation zeta function is $1$, resolving the last remaining open case of this problem. We additionally prove that, contrary to the expectation, the group algebras of $\mathbb C[\mathbb{SL}_2(\mathbb Z/(2^{2 r}))]$ and $\mathbb C[\mathbb{SL}_2(\mathbb F_2[t]/(t^{2r}))]$ are not isomorphic for any $r > 1$. This is the first known class of reductive groups over finite rings wherein the representation theory in the equal and mixed characteristic settings is genuinely different. From our methods, we explicitly obtain the primitive representation zeta polynomials of $\mathrm{SL}_2\left (\mathbb F_2[t]/(t^{2r}) \right) $ and $\mathrm{SL}_2\left (\mathbb Z/(2^{2r}) \right) $ for $1 \leq r \leq 3$.