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On local zeta-integrals for GSp(4) and GSp(4) x GL(2)

David Loeffler

Abstract

We prove that Novodvorsky's definition of local L-factors for generic representations of GSp(4) x GL(2) is compatible with the local Langlands correspondence when the GL(2) representation is non-supercuspidal. We also give an interpretation in terms of Langlands parameters of the "exceptional" poles of the GSp(4) x GL(2) L-factor, and of the "subregular" poles of the GSp(4) L-factor studied in recent work of Roesner and Weissauer.

On local zeta-integrals for GSp(4) and GSp(4) x GL(2)

Abstract

We prove that Novodvorsky's definition of local L-factors for generic representations of GSp(4) x GL(2) is compatible with the local Langlands correspondence when the GL(2) representation is non-supercuspidal. We also give an interpretation in terms of Langlands parameters of the "exceptional" poles of the GSp(4) x GL(2) L-factor, and of the "subregular" poles of the GSp(4) L-factor studied in recent work of Roesner and Weissauer.

Paper Structure

This paper contains 34 sections, 30 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Compatibility of $L$-factors
  3. Exceptional poles for $\mathop{\mathrm{GSp}}\nolimits(4) \times \mathop{\mathrm{GL}}\nolimits(2)$
  4. Subregular poles for $\mathop{\mathrm{GSp}}\nolimits(4)$
  5. Distinction of representations
  6. Multiplicity one for reducible representations
  7. Acknowledgements
  8. General notation
  9. Principal series representations of GL(2)
  10. Definitions
  11. Godement--Siegel sections
  12. Whittaker functions
  13. Bessel models
  14. The Bessel model
  15. Piatetski-Shapiro's integral
  16. ...and 19 more sections

Key Result

Theorem A

conj:compat holds under the additional assumption that the $\mathop{\mathrm{GL}}\nolimits(2, F)$-representation $\sigma$ be non-super-cuspidal.

Theorems & Definitions (68)

  • Conjecture \greekconjecture
  • Theorem A
  • Conjecture \greekconjecture
  • Theorem B
  • Theorem C
  • Remark 1
  • Conjecture \greekconjecture
  • Theorem D
  • Remark 2
  • Conjecture \greekconjecture
  • ...and 58 more