Singularities of Steinberg deformation rings

Abstract Let l and p be distinct primes, let F be a local field with residue field of characteristic p , and let X $\mathfrak {X}$ German upper X be the irreducible component of the moduli space of Langlands parameters for G L 3 $GL_3$ upper G upper L 3 over Z l $\mathbb {Z}_l$ double struck upper Z Subscript l corresponding to parameters of Steinberg type. We show that X $\mathfrak {X}$ German upper X is Cohen–Macaulay and compute explicit equations for it. We also compute the Weil divisor class group of the special fibre of X $\mathfrak {X}$ German upper X , motivated by work of Manning for G L 2 $GL_2$ upper G upper L 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.