Singularities of Steinberg deformation rings

TL;DR

This paper investigates the geometric and algebraic properties of a specific component of the moduli space of Langlands parameters for $GL_3$, showing it is Cohen-Macaulay, providing explicit equations, and analyzing its divisor class group.

Contribution

It provides the first explicit equations and Cohen-Macaulay property for the Steinberg deformation ring component for $GL_3$, extending previous work from $GL_2$.

Findings

$rak{X}$ is Cohen-Macaulay.

Explicit equations for $rak{X}$ are computed.

The Weil divisor class group of the special fibre is determined.

Abstract

Let $l$ and $p$ be distinct primes, let $F$ be a local field with residue field of characteristic $p$, and let $\mathfrak{X}$ be the irreducible component of the moduli space of Langlands parameters for $GL_3$ over $\mathbb{Z}_l$ corresponding to parameters of Steinberg type. We show that $\mathfrak{X}$ is Cohen-Macaulay and compute explicit equations for it. We also compute the Weil divisor class group of the special fibre of $\mathfrak{X}$, motivated by work of Manning for $GL_2$. Our methods involve the calculation of the cohomology of certain vector bundles on the flag variety, and build on work of Snowden, Vilonen-Xue, and Ngo.