The conjugation representation of $\operatorname{GL}_{2}$ and $\operatorname{SL}_{2}$ over finite local rings

TL;DR

This paper extends the understanding of the conjugation representation of al_{2} and al_{2} over finite local rings, confirming a conjecture for all primes and broadening previous results to rings with odd characteristic residue fields.

Contribution

It generalizes Tiep's results on conjugation representations from prime power rings to all finite local principal ideal rings with odd characteristic residue fields.

Findings

Confirmed the Hain--Tiep question for all primes.

Established conjugation representation properties over broader rings.

Extended previous results to rings with odd characteristic residue fields.

Abstract

The conjugation representation of a finite group $G$ is the complex permutation module defined by the action of $G$ on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura integrals, Tiep proved that for $G=\operatorname{SL}_{2}(\mathbb{Z}/p^{r})$, where $r\geq1$ and $p\geq5$ is a prime, any irreducible representation of $G$ that is trivial on the centre of $G$ is contained in the conjugation representation. Moreover, Tiep asked whether this can be generalised to $p=2$ or $3$. We answer the Hain--Tiep question in the affirmative and also prove analogous statements for $\operatorname{SL}_{2}$ and $\operatorname{GL}_{2}$ over any finite local principal ideal ring with residue field of odd characteristic.