  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.
