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Supercuspidal representations: construction, classification, and characters

Jessica Fintzen

TL;DR

This survey provides an explicit, comprehensive framework for constructing, classifying, and computing the characters of supercuspidal representations of p-adic groups. It unifies Moy–Prasad/Bruhat–Tits theory with Yu’s construction, depth analysis, and generic character theory, and it frames regular supercuspidals via a streamlined data pair $(S,\theta)$ that mirrors discrete series parameters for real groups. The work also integrates a quadratic twist to refine parameterizations and develops Harish-Chandra character formulas for both topologically semisimple elements and general regular semisimple cases, connecting to Langlands program and endoscopy. Collectively, these results yield a near-complete program for understanding supercuspidals, their parametrizations, and their characters, with clear routes to local Langlands correspondences for large classes of representations.

Abstract

The building blocks for irreducible smooth representations of p-adic groups are the supercuspidal representations. In these notes that are an expansion of a lecture series given during the IHES summer school 2022 we will explore an explicit exhaustive construction of these supercuspidal representations and their character formulas and observe a striking parallel between a large class of these representations and discrete series representations of real algebraic Lie groups. A key ingredient for the construction of supercuspidal representations is the Bruhat-Tits theory and Moy-Prasad filtration, which we will recall in this survey.

Supercuspidal representations: construction, classification, and characters

TL;DR

This survey provides an explicit, comprehensive framework for constructing, classifying, and computing the characters of supercuspidal representations of p-adic groups. It unifies Moy–Prasad/Bruhat–Tits theory with Yu’s construction, depth analysis, and generic character theory, and it frames regular supercuspidals via a streamlined data pair that mirrors discrete series parameters for real groups. The work also integrates a quadratic twist to refine parameterizations and develops Harish-Chandra character formulas for both topologically semisimple elements and general regular semisimple cases, connecting to Langlands program and endoscopy. Collectively, these results yield a near-complete program for understanding supercuspidals, their parametrizations, and their characters, with clear routes to local Langlands correspondences for large classes of representations.

Abstract

The building blocks for irreducible smooth representations of p-adic groups are the supercuspidal representations. In these notes that are an expansion of a lecture series given during the IHES summer school 2022 we will explore an explicit exhaustive construction of these supercuspidal representations and their character formulas and observe a striking parallel between a large class of these representations and discrete series representations of real algebraic Lie groups. A key ingredient for the construction of supercuspidal representations is the Bruhat-Tits theory and Moy-Prasad filtration, which we will recall in this survey.
Paper Structure (28 sections, 13 theorems, 51 equations, 2 figures, 1 table)

This paper contains 28 sections, 13 theorems, 51 equations, 2 figures, 1 table.

Table of Contents

  1. Motivation and connection between representations of real and $p$-adic reductive groups
  2. Representations of real algebraic Lie groups
  3. Connection between representations of real and $p$-adic groups
  4. Moy--Prasad filtration and Bruhat--Tits theory
  5. The split case
  6. Properties of the Moy--Prasad filtration
  7. The Bruhat--Tits building
  8. The non-split (tame) case
  9. The enlarged Bruhat--Tits building
  10. The depth of a representation
  11. Construction of supercuspidal representations
  12. A non-exhaustive overview of some historic developments
  13. Generalities about the construction of supercuspidal representations
  14. Depth-zero supercuspidal representations
  15. An example of a positive-depth supercuspidal representations
  16. ...and 13 more sections

Key Result

Theorem 1.1.2

The equivalence classes of essentially square-integrable representations of $G(\mathbb{R})$ are parameterized by triples $(S, \theta, \Phi^+)$ up to conjugacy, where Moreover, the representations in the equivalence class attached to the conjugacy class of $(S, \theta, \Phi^+)$ are determined by their character $\Theta_{(S, \theta, \Phi^+)}$ satisfying for every regular element $\gamma$ of $S(\ma

Figures (2)

  • Figure 1: Excerpt of an apartment for $\mathop{\mathrm{SL}}\nolimits_3$ with hyperplanes, where $\alpha_{i, j}$ is the root corresponding to $\text{diag}(t_1, t_2, t_3) \mapsto t_it_j^{-1}$
  • Figure 2: Excerpt of the Bruhat--Tits building $\mathscr{B}(\mathop{\mathrm{SL}}\nolimits_2, \mathbb{Q}_3)$

Theorems & Definitions (37)

  • Remark 1.1.1
  • Theorem 1.1.2
  • Definition 2.1.2
  • Definition 2.1.4
  • Definition 2.1.5
  • Definition 2.1.6: see KP-BTbook
  • Definition 2.2.3
  • Definition 2.4.1
  • Lemma 3.2.1
  • Lemma 3.2.3
  • ...and 27 more