Singularities of Steinberg deformation rings

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.