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Extended Weil representations: the non-dyadic local field cases

Chun-Hui Wang

Abstract

Let F be a non-archimedean local field of odd residual characteristic. Let W be a symplectic vector space over F. It is known that there are different Weil representations of a Meteplectic covering group Mp(W). By some twisted actions, we reorganize them as a representation of $PGMp^{\pm}(W)$, which is a covering group related to the projective similitude symplectic group.

Extended Weil representations: the non-dyadic local field cases

Abstract

Let F be a non-archimedean local field of odd residual characteristic. Let W be a symplectic vector space over F. It is known that there are different Weil representations of a Meteplectic covering group Mp(W). By some twisted actions, we reorganize them as a representation of , which is a covering group related to the projective similitude symplectic group.
Paper Structure (31 sections, 43 theorems, 239 equations)

This paper contains 31 sections, 43 theorems, 239 equations.

Table of Contents

  1. Introduction to the main result
  2. Preliminaries
  3. Notations and conventions
  4. $2$-cocycles
  5. Perrin-Rao's $2$-cocycle
  6. Barthel's $2$-cocycle I
  7. Barthel's $2$-cocycle II
  8. Example
  9. $F^{\times}$
  10. The group $\operatorname{PGMp}^{\pm}(W)$: Case $|k_F|\equiv 3(\bmod4)$
  11. $\widetilde{F^{\times}}$
  12. $\nu(-,-)$
  13. Example
  14. $\operatorname{PGMp}^{\pm}(W)$
  15. Example
  16. ...and 16 more sections

Key Result

Lemma 2.4

For two automorphisms $\alpha_1, \alpha_2$, $\nu(\alpha_1\alpha_2, g)=\nu(\alpha_1,g)\nu(\alpha_2, g^{\alpha_1})$.

Theorems & Definitions (95)

  • Example 2.1
  • proof
  • Example 2.2
  • proof
  • Example 2.3: Barthel
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • proof
  • ...and 85 more