[2309.10401] On submodules of standard modules Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Representation Theory arXiv:2309.10401 (math) [Submitted on 19 Sep 2023 ( v1 ), last revised 24 Apr 2026 (this version, v3)] Title: On submodules of standard modules Authors: Maarten Solleveld View a PDF of the paper titled On submodules of standard modules, by Maarten Solleveld View PDF HTML (experimental) 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. Comments: In version 3, the claims about tempered representations occurring only as subrepresentations of standard modules were removed, because counterexamples had surfaced. The main changes are in Section 2, where now triviality of the enhancement of an L-parameters is used several times. Later sections have been modified accordingly Subjects: Representation Theory (math.RT) MSC classes: 20C08, 14F08, 22E57 Cite as: arXiv:2309.10401 [math.RT] (or arXiv:2309.10401v3 [math.RT] for this version) https://doi.org/10.48550/arXiv.2309.10401 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Maarten Solleveld [ view email ] [v1] Tue, 19 Sep 2023 08:04:11 UTC (38 KB) [v2] Thu, 16 May 2024 08:17:03 UTC (41 KB) [v3] Fri, 24 Apr 2026 08:01:58 UTC (44 KB) Full-text links: Access Paper: View a PDF of the paper titled On submodules of standard modules, by Maarten Solleveld View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2023-09 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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