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On vector valued automorphic forms for the Weil representation

Oliver Stein

Abstract

We develop a theory of vector valued automorphic forms associated to the Weil representation $ω_f$ and corresponding to vector valued modular forms transforming with the ``finite'' Weil representation $ρ_L$. For each prime $p$ we determine the structure of a vector valued spherical Hecke algebra depending on $ω_f$, which acts on the space of automorphic forms.

On vector valued automorphic forms for the Weil representation

Abstract

We develop a theory of vector valued automorphic forms associated to the Weil representation and corresponding to vector valued modular forms transforming with the ``finite'' Weil representation . For each prime we determine the structure of a vector valued spherical Hecke algebra depending on , which acts on the space of automorphic forms.

Paper Structure

This paper contains 8 sections, 17 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the "finite" Weil representation, vector valued modular forms and some notation
  3. The Weil representation on $\operatorname{GL}_2(\mathbb{A})$
  4. The Hecke algebra $\mathcal{H}(\mathcal{Q}_p//\mathcal{K}_p, \omega_p)$
  5. The case of primes $p$ dividing $|D|$
  6. The case of primes $p$ not dividing $|D|$
  7. Vector valued automorphic forms and vector valued modular forms
  8. The action of $\mathcal{H}(\mathcal{Q}_p//\mathcal{K}_p,\omega_p)$ on $A_{\kappa}(\omega_f)$

Key Result

Lemma 2.1

Theorems & Definitions (43)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Definition 3.5
  • Remark 3.6
  • Definition 4.1
  • Lemma 4.2
  • ...and 33 more