Construction of even-level representations of SL2(o) with residue field of characteristic two

Abstract

We construct the finite-dimensional continuous complex representations of ${\mathrm{SL}}_2$ over compact discrete valuation rings of even residual characteristic, assuming the level is large enough compared to the ramification index, in the mixed characteristic case. 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 $C\left[{\mathrm{SL}}_2\right(Z/2^{r}Z\left)\right]$ and $C\left[{\mathrm{SL}}_2\right(F_2\left[t\right]/\left(t^r\right)\left)\right]$ are not isomorphic for any $r\ge 4$.