We construct the finite-dimensional continuous complex representations of $\mathrm{ SL}_2$ over compact discrete valuation rings of even residual characteristic. We also prove that the complex group algebras of $\mathrm{ SL}_2$ 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 $\mathbb{ C}[\mathrm{ SL}_2 (\mathbb{ Z}/2^r \mathbb{ Z})]$ and $\mathbb{ C}[\mathrm{ SL}_2 (\mathbb{ F}_2 [t]/(t^r ))]$ are not isomorphic for any $r\geq 4.$
Categories: math.RT