Branching rules for the restriction of regular representations of ${\mathrm{GL}}_2(\mathfrak{o}/{\mathfrak{p}}^r)$ to ${\mathrm{SL}}_2(\mathfrak{o}/{\mathfrak{p}}^r).$

Abstract

Let $\mathfrak{o}$ be a compact discrete valuation ring with maximal ideal $\mathfrak{p}$ such that the finite residue field $\mathfrak{o}/\mathfrak{p}$ has characteristic $p.$ For $r\ge 2$ and $p=2,$ we obtain the branching rules for the restriction of a regular representation of ${\mathrm{GL}}_2(\mathfrak{o}/{\mathfrak{p}}^r)$ to ${\mathrm{SL}}_2(\mathfrak{o}/{\mathfrak{p}}^r).$ These results have different behaviour than that of the known case of $p\ne 2.$

7 pages