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Shalika newforms for GL(n)

Takeo Okazaki

TL;DR

This work develops a comprehensive Shalika-model framework for GL(n) over p-adic fields with even n, focusing on generic, especially supercuspidal, representations. It introduces and analyzes pre-Shalika models on mirabolic subgroups, constructs essential pre-Shalika forms, and links Shalika data to Whittaker models via explicit maps, enabling precise control of values and invariants. Through a Hecke-theoretic treatment, it derives recursive expressions for Whittaker newforms, which in turn illuminate the structure of Shalika newforms. Finally, it builds Godement–Jacquet zeta integrals in the Shalika setting and proves that their Dirichlet series reproduce the L-functions L(s, \pi) and the dual L( s, \pi^\vee) up to the root number, establishing a robust bridge between Shalika models, Whittaker models, and standard L-functions for GL(n).

Abstract

Let (pi,V) be a generic irreducible representation of a general linear group over a p-adic field. Jacquet, Piatetski-Shapiro, and Shalika gave an open compact subgroup K, so that the subspace V^K consisting of v in V fixed by K is one-dimensional. If pi has a Shalika model Lambda, then we call vectors in Lambda(V) the Shalika forms of pi, and those in Lambda(V^{K}) the Shalika newforms. In this article, we give a method to determine all values of the Shalika newforms on the mirabolic subgroup in the case where pi is supercuspidal. Using this result, we give another Shalika form with nice properties, which is not fixed by K in the case where the character defining the Shalika model is ramified.

Shalika newforms for GL(n)

TL;DR

This work develops a comprehensive Shalika-model framework for GL(n) over p-adic fields with even n, focusing on generic, especially supercuspidal, representations. It introduces and analyzes pre-Shalika models on mirabolic subgroups, constructs essential pre-Shalika forms, and links Shalika data to Whittaker models via explicit maps, enabling precise control of values and invariants. Through a Hecke-theoretic treatment, it derives recursive expressions for Whittaker newforms, which in turn illuminate the structure of Shalika newforms. Finally, it builds Godement–Jacquet zeta integrals in the Shalika setting and proves that their Dirichlet series reproduce the L-functions L(s, \pi) and the dual L( s, \pi^\vee) up to the root number, establishing a robust bridge between Shalika models, Whittaker models, and standard L-functions for GL(n).

Abstract

Let (pi,V) be a generic irreducible representation of a general linear group over a p-adic field. Jacquet, Piatetski-Shapiro, and Shalika gave an open compact subgroup K, so that the subspace V^K consisting of v in V fixed by K is one-dimensional. If pi has a Shalika model Lambda, then we call vectors in Lambda(V) the Shalika forms of pi, and those in Lambda(V^{K}) the Shalika newforms. In this article, we give a method to determine all values of the Shalika newforms on the mirabolic subgroup in the case where pi is supercuspidal. Using this result, we give another Shalika form with nice properties, which is not fixed by K in the case where the character defining the Shalika model is ramified.
Paper Structure (8 sections, 41 theorems, 299 equations)

This paper contains 8 sections, 41 theorems, 299 equations.

Table of Contents

  1. Main results
  2. Preparation
  3. Pre-Shalika model
  4. Uniqueness of pre-Shalika models
  5. Essential pre-Shalika form
  6. Whittaker model
  7. Hecke operators
  8. Zeta integrals

Key Result

Theorem 1.1

Let $\pi$ be a generic, irreducible, supercuspidal representation of $G_{n}$ realized in ${\mathbb{S}}_\pi(\chi)$. Let $e$ be the conductor of $\chi$. Let $P_n(\mathfrak{o}) \subset P_n$ denote the subgroup consisting matrices $p \in P_n$ with entries in $\mathfrak{o}$. Assume that $\psi(\mathfrak{o

Theorems & Definitions (69)

  • Theorem 1.1
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 59 more