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Conductors and local newforms for the metaplectic group of rank 1

Hiroshi Ishimoto

Abstract

In an earlier paper of W. Casselman, the theory of local newforms and conductors was initiated. Later, Roberts and Schmidt studied local newforms for the metaplectic group of rank 1. In this paper we define and calculate conductors of irreducible genuine representations of the metaplectic group of rank 1 over non-archimedean local field of characteristic zero and of odd residual characteristic. Moreover, we shall give an explicit formulae for dimensions of spaces of local newforms, and show a compatibility with the local theta correspondence.

Conductors and local newforms for the metaplectic group of rank 1

Abstract

In an earlier paper of W. Casselman, the theory of local newforms and conductors was initiated. Later, Roberts and Schmidt studied local newforms for the metaplectic group of rank 1. In this paper we define and calculate conductors of irreducible genuine representations of the metaplectic group of rank 1 over non-archimedean local field of characteristic zero and of odd residual characteristic. Moreover, we shall give an explicit formulae for dimensions of spaces of local newforms, and show a compatibility with the local theta correspondence.

Paper Structure

This paper contains 15 sections, 42 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Variants of Gauss sum
  3. The metaplectic group
  4. The metaplectic group
  5. Representations of the metaplectic group
  6. Definition of conductors
  7. The result of Roberts--Schmidt
  8. Non-supercuspidal representations
  9. Principal series representations
  10. Even Weil representations and Steinberg representations
  11. Supercuspidal representations
  12. Classifications of supercuspidal representations
  13. Some results of Lansky--Raghuram
  14. Main results for supercuspidal representations
  15. Behaviour under the local theta correspondence

Key Result

Theorem 1.1

Assume that the conductor of $\eta$ is less than or equal to $c^\varepsilon_{\min}(\pi)/2$ if $\pi$ is supercuspidal. We have explicit formulae for the dimensions of the spaces $\pi^{K^\varepsilon_m}_\eta$, $\pi^{K^\varepsilon_m, \mathrm{new}}_\eta$. In particular, $c^\varepsilon_{\min}(\pi)$ and $c

Theorems & Definitions (76)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark
  • Lemma 3.1
  • ...and 66 more