Construction of representations of compact special linear groups of degree two

We construct the finite-dimensional continuous complex representations of ${\mathrm{S}\mathrm{L}}_2$ over compact discrete valuation rings of even residual characteristic. We also prove that the complex group algebras of ${\mathrm{S}\mathrm{L}}_2$ over finite quotient rings of such compact discrete valuation rings depend on the characteristic of the ring. In particular, we prove that the group algebras $\mathbb{C}[{\mathrm{S}\mathrm{L}}_2(\mathbb{Z}\mathrm{/}{2}^r\mathbb{Z})]$ and $\mathbb{C}[{\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_2[t]\mathrm{/}(t^r))]$ are not isomorphic for any $r\ge 4.$