Construction of representations of compact special linear groups of degree two

TL;DR

This paper constructs finite-dimensional complex representations of SL_2 over certain compact rings and shows that their group algebras depend on the ring's characteristic, highlighting non-isomorphism in specific cases.

Contribution

It introduces new constructions of representations for SL_2 over compact discrete valuation rings and demonstrates characteristic-dependent differences in their group algebras.

Findings

Representations constructed for SL_2 over compact discrete valuation rings.

Group algebras depend on the characteristic of the ring.

Non-isomorphism of group algebras for specific rings.

Abstract

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 $r\geq 4.$