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Newforms in Cuspidal Representations

Johannes Girsch, Robert Kurinczuk

TL;DR

This work extends newform theory to $ ell$-modular representations of $ ext{GL}_n(F)$, establishing conductor invariance under modulo-$ ell$ reduction for depth-zero and ramified supercuspidal cases. It develops depth-zero and minimax results through Mackey theory, Bushnell–Kutzko types, Bessel functions, and Whittaker models, yielding explicit newform vectors and matrix coefficients. Key contributions include existence and uniqueness of depth-zero and minimax newforms in modular settings, explicit formulae for depth-zero Whittaker and matrix coefficients, and a precise description of how conductors behave under congruences and epsilon-factor relations. The findings enhance modular local Langlands-type understanding and provide practical test vectors for $ ell$-modular representations in automorphic and p-adic contexts.

Abstract

We consider newform vectors in cuspidal representations of $p$-adic general linear groups. We extend the theory from the complex setting to include~$\ell$-modular representations with~$\ell\neq p$, and prove that the conductor is compatible with congruences modulo~$\ell$ for (ramified) supercuspidal~$\ell$-modular representations and for depth zero cuspidals. In the complex and modular setting, we prove explicit formulae for depth zero and minimax cuspidal representations of integral depth, in Bushnell-Kutzko and Whittaker models.

Newforms in Cuspidal Representations

TL;DR

This work extends newform theory to -modular representations of , establishing conductor invariance under modulo- reduction for depth-zero and ramified supercuspidal cases. It develops depth-zero and minimax results through Mackey theory, Bushnell–Kutzko types, Bessel functions, and Whittaker models, yielding explicit newform vectors and matrix coefficients. Key contributions include existence and uniqueness of depth-zero and minimax newforms in modular settings, explicit formulae for depth-zero Whittaker and matrix coefficients, and a precise description of how conductors behave under congruences and epsilon-factor relations. The findings enhance modular local Langlands-type understanding and provide practical test vectors for -modular representations in automorphic and p-adic contexts.

Abstract

We consider newform vectors in cuspidal representations of -adic general linear groups. We extend the theory from the complex setting to include~-modular representations with~, and prove that the conductor is compatible with congruences modulo~ for (ramified) supercuspidal~-modular representations and for depth zero cuspidals. In the complex and modular setting, we prove explicit formulae for depth zero and minimax cuspidal representations of integral depth, in Bushnell-Kutzko and Whittaker models.

Paper Structure

This paper contains 23 sections, 24 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Smooth representations
  4. Bessel functions
  5. Newforms for cuspidal representations
  6. Conductors and newforms
  7. Conductors modulo $\ell$
  8. Newforms vectors in models
  9. Whittaker models
  10. Matrix coefficients in the cuspidal case
  11. $\varepsilon$-factor conductor
  12. Newform vectors in depth zero cuspidal representations
  13. Mackey theory
  14. Coset representatives
  15. Proof of Theorem \ref{['depthzerocoset']}
  16. ...and 8 more sections

Key Result

Theorem 1

Let $n\geqslant 2$, and $\pi$ be a depth zero cuspidal $\mathrm{R}$-representation containing the cuspidal $\mathrm{R}$-type $(\mathrm{K}_n,\tau)$.

Theorems & Definitions (47)

  • Theorem : Theorem \ref{['depthzerocoset']}, Propositions \ref{['matrixcoeffsdepthzero']} and \ref{['depthzerowhittakernewform']}
  • Definition 3.1
  • Theorem 3.2: JPSSJacquetMatringe
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Theorem 3.6: Matringe, Miyauchi
  • Lemma 3.7
  • proof
  • ...and 37 more