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Local newforms and spherical characters for unitary groups

Gefei Dang

Abstract

We prove a smooth transfer statement analogous to Jacquet-Rallis's fundamental lemma, use it to compute a local spherical character appearing in the Ichino-Ikeda conjecture, and prove a statement on the existence of local newforms for unitary groups as a corollary.

Local newforms and spherical characters for unitary groups

Abstract

We prove a smooth transfer statement analogous to Jacquet-Rallis's fundamental lemma, use it to compute a local spherical character appearing in the Ichino-Ikeda conjecture, and prove a statement on the existence of local newforms for unitary groups as a corollary.
Paper Structure (16 sections, 20 theorems, 135 equations)

This paper contains 16 sections, 20 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Notations and measures
  3. Notations
  4. Measures
  5. An explicit transfer statement
  6. Orbital integrals and smooth transfers
  7. An explicit transfer statement
  8. Proof of \ref{['inhom_fl']}
  9. Calculation of local characters
  10. Local Langlands correspondence and base change
  11. Essential Whittaker function
  12. Proof of \ref{['thm:mainthm']}
  13. Newforms
  14. Proof of \ref{['exist_newform']}
  15. Uniqueness of newforms for two-dimensional unitary groups
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $V$ be a hermitian space over $E$. In every tempered Vogan $L$-packet of $U(V)(F)$ there exists a unique representation that contains newforms.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Remark
  • Corollary 1.3
  • Remark
  • Theorem 3.1
  • Remark
  • Theorem 3.2: Jacquet--Rallis's Fundamental Lemma
  • Theorem 3.3: homogeneous version
  • ...and 28 more