Let $O$ be the ring of integers of a non-Archimedean local field such that the residue field has characteristic $p$. Let $P$ be the maximal ideal of $O$. For Char$(O)=0$, let $e$ be the ramification index of $O$, i.e., $2O = P^e$. Let $GL_n(O)$ be the group of $n \times n$ invertible matrices with entries from $O$ and $SL_n(O)$ be the subgroup of $GL_n(O)$ consisting of all determinant one matrices.

In this talk, our focus is on the construction of the continuous complex irreducible representations of the group $SL_2(O)$ and to describe the representation growth. Also, we will discuss some results about group algebras of $SL_2(O/P^r)$ for large $r$ and branching laws obtained by restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$.

**Construction:** For $r\geq 1$ the construction of irreducible representations
of $GL_2(O/P^r)$ and for $SL_2(O/P^r)$ with $p>2$ are known by the work of
Jaikin-Zapirain and Stasinski-Stevens. However, those methods do not work
for $p=2$. In this case we give a construction of all irreducible
representations of groups $SL_2(O/P^r)$, for $r \geq 1$ with Char$(O)=2$
and for $r \geq 4e+2$ with Char$(O)=0$.

**Representation Growth:** For a rigid group $G$, it is well known that
the abscissa of convergence $\alpha(G)$ of the representation zeta function
of $G$ gives precise information about its representation growth.
Jaikin-Zapirain and Avni-Klopsch-Onn-Voll proved that $\alpha( SL_2(O) )=1,$
for either $p > 2$ or Char$(O)=0$. We complete these results by proving that
$\alpha(SL_2(O))=1$ also for $p=2$ and Char$(O) > 0$.

**Group Algebras:** The groups $GL_2(O/P^r)$ and $GL_2(F_q[t]/(t^{r}))$ need
not be isomorphic, but the group algebras ‘$\mathbb{C}[GL_2(O/P^r)]$’ and
$\mathbb{C}[GL_2(F_q[t]/(t^{r}))]$ are known to be isomorphic. In parallel,
for $p >2$ and $r\geq 1,$ the group algebras $\mathbb{C}[SL_2(O/P^r)]$ and
$\mathbb{C}[SL_2(F_q[t]/(t^{r}))]$ are also isomorphic. We show that for $p=2$
and Char$(O)=0$, the group algebras $\mathbb{C}[SL_2(O/P^{2m})]$ and
$\mathbb{C}[SL_2(F_q[t]/(t^{2m}))]$ are NOT isomorphic for $m > e$. As a
corollary we obtain that the group algebras
$\mathbb{C}[SL_2(\mathbb{Z}/2^{2m}\mathbb{Z})]$ and $\mathbb{C}[SL_2(F_2[t]/(t^{2m}))]$
are NOT isomorphic for $m>1$.

**Branching Laws:** We give a description of the branching laws obtained by
restricting irreducible representations of $GL_2(O/P^r)$ to $SL_2(O/P^r)$ for
$p=2$. In this case, we again show that many results for $p=2$ are quite different
from the case $p > 2$.

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Last updated: 09 Aug 2024