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On families of degenerate representations of GL_n(F)

Johannes Girsch, David Helm

TL;DR

The paper develops a refined structural picture for representations of GL_n(F) by stratifying the category via the highest derivative partition and linking each stratum to a Serre quotient that is controlled by a commutative endomorphism ring E_{\lambda}. Central innovations include constructing projective generators from degenerate Whittaker models, proving that the endomorphism rings decompose as products of smooth commutative algebras (in characteristic zero), and describing a universal theory of families of type-\lambda that yields a geometric interpretation of tangent spaces as Ext^1-classes. The combination of Zelevinsky classification, degenerate Whittaker models, and a detailed Ext-calculus leads to a precise, Bernstein–Deligne–style description of the Bernstein-like center for each partition, and a potential for modular and mixed-characteristic extensions. The single-segment case and the appendix on the truncate theorem provide essential technical backbone, enabling the main commutativity and product-decomposition results that give a concrete, computable model for the Serre quotients as module categories over smooth commutative algebras.

Abstract

We consider the stratification of the category of smooth representations of $\mbox{GL}_n(F)$ (for $F$ a $p$-adic field) induced by degenerate Whittaker models. We show that, remarkably, the successive quotient categories in this stratification turn out to be module categories over commutative rings. In fact these rings turn out to be infinite products of smooth algebras over the ground field. We further obtain explict descriptions of these rings in terms of the Zelevinsky classification; these descriptions closely resemble the Bernstein-Deligne description of the Bernstein center.

On families of degenerate representations of GL_n(F)

TL;DR

The paper develops a refined structural picture for representations of GL_n(F) by stratifying the category via the highest derivative partition and linking each stratum to a Serre quotient that is controlled by a commutative endomorphism ring E_{\lambda}. Central innovations include constructing projective generators from degenerate Whittaker models, proving that the endomorphism rings decompose as products of smooth commutative algebras (in characteristic zero), and describing a universal theory of families of type-\lambda that yields a geometric interpretation of tangent spaces as Ext^1-classes. The combination of Zelevinsky classification, degenerate Whittaker models, and a detailed Ext-calculus leads to a precise, Bernstein–Deligne–style description of the Bernstein-like center for each partition, and a potential for modular and mixed-characteristic extensions. The single-segment case and the appendix on the truncate theorem provide essential technical backbone, enabling the main commutativity and product-decomposition results that give a concrete, computable model for the Serre quotients as module categories over smooth commutative algebras.

Abstract

We consider the stratification of the category of smooth representations of (for a -adic field) induced by degenerate Whittaker models. We show that, remarkably, the successive quotient categories in this stratification turn out to be module categories over commutative rings. In fact these rings turn out to be infinite products of smooth algebras over the ground field. We further obtain explict descriptions of these rings in terms of the Zelevinsky classification; these descriptions closely resemble the Bernstein-Deligne description of the Bernstein center.

Paper Structure

This paper contains 15 sections, 51 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. The highest derivative partition
  3. The Zelevinsky classification
  4. Degenerate Whittaker models
  5. The subcategory $\operatorname{Rep}_{\mathcal{O}}(G_n)^{\preceq \lambda}$ and its localization
  6. The endomorphism ring $E_{\lambda,\overline{\mathcal{K}}}$
  7. Families of type $\lambda$
  8. Tangent vectors
  9. The single-segment case
  10. Self-Ext algebras and dimensions of tangent spaces
  11. Proof of Theorem \ref{['thm:main']}
  12. Appendix: Proof of Theorem \ref{['thm:truncate']}
  13. Prerequisites
  14. Technical Result
  15. Proof of Theorem \ref{['thm:truncate']}

Key Result

Lemma 2.1

Let $V$ be an irreducible representation of $G_n$, and $k_1,\dots,k_r$ its highest derivative sequence. Then $k_1 \geq \dots\geq k_r$; that is, the $k_i$ are a partition of $n$.

Theorems & Definitions (91)

  • Lemma 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • ...and 81 more