On submodules of standard modules

TL;DR

This paper proves the generalized injectivity conjecture for standard modules of quasi-split reductive p-adic groups using geometric methods related to Langlands parameters and graded Hecke algebras.

Contribution

It introduces a geometric approach involving open Langlands parameters and reduces the problem to graded Hecke algebras, proving the conjecture in this setting and transferring it to p-adic groups.

Findings

Proved the generalized injectivity conjecture for standard modules.

Established that generic modules correspond to open L-parameters.

Connected geometric properties of graded Hecke algebras to representation theory of p-adic groups.

Abstract

Consider a standard representation $\pi_{st}$ of a quasi-split reductive p-adic group G. The generalized injectivity conjecture, posed by Casselman and Shahidi, asserts that any generic irreducible subquotient $\pi$ of $\pi_{st}$ is necessarily a subrepresentation of $\pi_{st}$. We will prove this conjecture, improving on the verification for many groups by Dijols. We study this in a geometric way, motivated by favourable properties of Langlands parameters which are open (which means that the nilpotent element from the L-parameter belongs to an appropriate open orbit). Since we do not want to assume a local Langlands correspondence, we involve similar parameters via reduction to Hecke algebras. It does not suffice to pass from G to an affine Hecke algebra, we further reduce to graded Hecke algebras and from there to algebras defined in terms of certain equivariant perverse sheaves. It is in the geometric setting of graded Hecke algebras from cuspidal local systems on nilpotent orbits that we can finally put the ``open" condition on L-parameters to good use. The closure relations between the involved nilpotent orbits provide useful insights in the internal structure of standard modules, which highlight the representations associated with open L-parameters and in particular those for which the enhancement of the L-parameter is trivial. We show that, in the parametrization of irreducible modules of geometric graded Hecke algebras, generic modules always have ``open L-parameters". This leads to a proof of a version of the generalized injectivity conjecture for graded Hecke algebras of geometric type, which is then transferred to reductive p-adic groups.