The rigid-component conjecture for parabolic induction

Let $\mathbf{d}$ be a dimension vector, let $\mathbf{Comp}(\mathbf{d})$ denote the set of irreducible components under consideration, and let $G_{\mathbf{d}}$ act on the corresponding representation variety. The Deligne–Langlands correspondence gives a bijection $C\mapsto\pi(C)$ between $\mathbf{Comp}(\mathbf{d})$ and the relevant irreducible representations.

Rigid-component conjecture. For $C\in\mathbf{Comp}(\mathbf{d})$, the induced representation $\pi(C)\times\pi(C)$ is irreducible if and only if $C$ contains an open $G_{\mathbf{d}}$-orbit.

Irreducible components satisfying this condition are called rigid. The conjecture relates irreducibility of parabolic induction to the geometry of Lusztig's characteristic variety and is attributed in the source to Geiss–Schröer and Lapid–Mínguez.