A Remark on Bernstein–Zelevinsky Classification for General Linear Groups

Abstract

In this paper, we reveal a submodule relation among the standard modules of p-adic general linear groups. This observation has several applications. We use it to disprove a conjecture by Lapid and Mínguez, which stated that the unique maximal submodule of a standard module is the sum of certain smaller submodules in the Zelevinsky partial order. We also apply a fundamental lemma by Borel–Wallach to investigate the partial order structure of standard modules under the Zelevinsky partial order.

Keywords

Parabolic induction / standard module / Zelevinsky classification / Borel–Wallach partial order / general linear group / 22E50

References

[1] Bernstein IN, Zelevinsky AV. Induced representations of reductive p-adic groups, I. Ann. Sci. École Norm. Sup. (4), 1977, 10(4): 441-472

[2] Borel A, Wallach N. Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2000Second ÉditionProvidence, RI, American Mathematical Society 67

[3] Casselman W. Introduction to the Theory of Admissible Representations of p-adic Reductive Groups, 1995

[4] Lapid E, Mínguez A. On a determinantal formula of Tadić. Amer. J. Math., 2014, 136(1): 111-142

[5] Zelevinsky AV. Induced representations of reductive p-adic groups, II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. (4), 1980, 13(2): 165-210

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