A binary operation on irreducible components of Lusztig's nilpotent varieties {II}: applications and conjectures for representations of $GL_n$ over a non-archimedean local field

TL;DR

This paper explores a binary operation on irreducible components of Lusztig's nilpotent varieties, proposing conjectures linking it to representation theory of GL_n over non-archimedean fields, with partial verifications.

Contribution

It introduces a new binary operation on Lusztig's nilpotent varieties and conjectures its compatibility with parabolic induction in GL_n representations, supported by partial evidence.

Findings

Conjecture on compatibility with socle of parabolic induction for rigid components.

Verification of the conjecture in specific cases.

Foundation for further research on geometric and representation-theoretic connections.

Abstract

In the first part of the paper we defined and studied a binary operation on the set of irreducible components of Lusztig's nilpotent varieties of a quiver. For type $A$ we conjecture, following Geiss and Schr\"oer, that this operation is compatible with taking the socle of parabolic induction of representations of general linear groups over a local non-archimedean field, at least when one of the irreducible components is rigid. We verify this conjecture in special cases.