The Lafforgue variety and irreducibility of induced representations

TL;DR

This paper introduces the Lafforgue variety, a scheme parametrizing simple modules of certain non-commutative algebras, and uses it to characterize irreducibility of induced representations via a generalized discriminant.

Contribution

It constructs the Lafforgue variety for non-commutative algebras and generalizes the Hilbert scheme, providing new tools for analyzing representation irreducibility.

Findings

Constructed the Lafforgue variety for algebras with finitely generated center.

Derived a criterion for irreducibility of induced representations using a generalized discriminant.

Explicitly computed the discriminant for Iwahori-Hecke algebras of split reductive p-adic groups.

Abstract

We construct the Lafforgue variety, an affine scheme equipped with an open dense subscheme parametrizing the simple modules of a non-commutative unital algebra $R$ over any field $k$, provided that the center $Z(R)$ is finitely generated and $R$ is finitely generated as a $Z(R)$-module. Our main technical tool is a generalization of the Hilbert scheme for non-commutative algebras, which may be of independent interest. 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.