Let $\mathfrak{O}$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Then the abscissa of convergence of representation zeta function of Special Linear group $\mathrm{SL}_2(\mathfrak{O})$ is $1.$ The case $p\neq 2$ is already known in the literature. For $p=2$ we need more tools to prove the result. In this talk I will discuss the difference between those cases and give an outline of the proof for $p=2.$

Let $\mathfrak{p}$ be the maximal ideal of $\mathfrak{O}$ and
$|\mathfrak{O}/\mathfrak{p}|=q.$ It is already shown in literature that
for $r \geq 1,$ the group algebras $\mathbb
C[\mathrm{GL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb
C[\mathrm{GL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. Also for
$2\nmid q,$ the group algebras $\mathbb
C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{r})]$ and $\mathbb
C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{r}))]$ are isomorphic. In this talk I
will also show that if $2\mid q$ and $\mathrm{char}(\mathfrak{O})=0$
then, the group algebras,
$\mathbb C[\mathrm{SL}_2(\mathfrak{O}/\mathfrak{p}^{2\ell})]$ and
$\mathbb C[\mathrm{SL}_2(\mathbb F_q[t]/(t^{2\ell}))]$ are *not*
isomorphic for $\ell > \mathrm{e}$, where $\mathrm{e}$ is the
ramification index of $\mathfrak{O}.$

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Last updated: 17 Aug 2019