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Local theta correspondence and Galois periods

Chong Zhang

Abstract

We study the behavior of Galois periods under the local theta correspondence for even orthogonal and symplectic groups. Specifically, we compare their multiplicities and construct explicit transfer maps. Furthermore, we establish both an adjoint relation and a relative character relation for these periods.

Local theta correspondence and Galois periods

Abstract

We study the behavior of Galois periods under the local theta correspondence for even orthogonal and symplectic groups. Specifically, we compare their multiplicities and construct explicit transfer maps. Furthermore, we establish both an adjoint relation and a relative character relation for these periods.
Paper Structure (43 sections, 40 theorems, 286 equations)

This paper contains 43 sections, 40 theorems, 286 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Main results
  4. The method
  5. Few remarks
  6. Structure of this paper
  7. Notation and convention
  8. Acknowledgements
  9. Base change doubling
  10. Base change doubled spaces
  11. Invariants of the double cosets
  12. Open orbits and their stabilizers
  13. Negligible orbits and their stabilizers
  14. Explicit orbits and stabilizers
  15. The symplectic case
  16. ...and 28 more sections

Key Result

Theorem 1.1

If $m>n$, then $\dim {\mathrm{Hom}}_{{\mathrm{Sp}}(W_F)}(\pi,{\mathbb {C}})=\dim{\mathrm{Hom}}_{{\mathrm{O}}(V_F)}(\sigma,{\mathbb {C}})$.

Theorems & Definitions (79)

  • Theorem 1.1: Theorem \ref{['thm symp-orth']}
  • Theorem 1.2: Theorem \ref{['thm transf1']}
  • Theorem 1.3: Theorem \ref{['thm adjoint']}
  • Theorem 1.4: Theorem \ref{['thm rel-char']}
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • Example 2.5
  • ...and 69 more