The Lafforgue Variety and Topics in Geometric Representation Theory

This thesis consists of two separate projects. In the first project, we construct the Lafforgue variety, an affine scheme parametrizing the simple modules of a non-commutative algebra $R$ over any field $k$, provided that the center $Z(R)$ is finitely generated and $R$ is finite as a $Z(R)$-module. Applying our construction in the case of Hecke algebras of Bernstein components, we derive a characterization for the irreducibility of induced representations in terms of the vanishing of a generalized discriminant on the Bernstein variety. We explicitly compute the discriminant in the case of an Iwahori-Hecke algebra of a split reductive $p$-adic group. We additionally give potential applications to the Local Langlands conjecture via comparison of Hecke algebras on the group and Galois sides, as in the ABPS conjectures. In particular, we construct a Bernstein variety for the Galois side of the Local Langlands correspondence and conjecture that the Lafforgue varieties of the two sides are isomorphic. In the second project, we prove that character sheaves have nilpotent singular support in any characteristic, partially extending the work of \cite{mirvo88} and \cite{gin89} to positive characteristic. We do this by introducing a category of tame perverse sheaves and studying its functorial properties.