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On dual groups of symmetric varieties and distinguished representations of $p$-adic groups

Shuichiro Takeda

Abstract

Let $X=H\backslash G$ be a spherical variety over a $p$-adic field. Assume $G$ is split. Let $\widehat{G}$ be the Langlands dual group of $G$. There is a complex group $\widehat{G}_X$ whose root datum is the little Weyl group of $X$. It was proposed by Sakellaridis-Venkatesh and fully proven by Knop and Schalke that there is a homomorphism $\widehat{\varphi}_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$. Conjecturally, this detects the $H$-distinguished representations of $G$. In this strictly utilitarian note, assuming $X$ is a symmetric variety, we give a more conceptual way of constructing the homomorphism $\widehat{\varphi}_X:\widehat{G}_X\times\operatorname{SL}_2(\mathbb{C})\to \widehat{G}$, and make a few conjectures on how $\widehat{\varphi}_X$ is related to $H$-distinguished representations of $G$ by using various known examples and conjectures, especially in the framework of the theory of Kato-Takano and Lagier on relative cuspidality and relative square integrability. We will also show that the local Langlands parameter of the trivial representation of $G$ factors through $\widehat{\varphi}_X$ for any symmetric variety $X=H\backslash G$.

On dual groups of symmetric varieties and distinguished representations of $p$-adic groups

Abstract

Let be a spherical variety over a -adic field. Assume is split. Let be the Langlands dual group of . There is a complex group whose root datum is the little Weyl group of . It was proposed by Sakellaridis-Venkatesh and fully proven by Knop and Schalke that there is a homomorphism . Conjecturally, this detects the -distinguished representations of . In this strictly utilitarian note, assuming is a symmetric variety, we give a more conceptual way of constructing the homomorphism , and make a few conjectures on how is related to -distinguished representations of by using various known examples and conjectures, especially in the framework of the theory of Kato-Takano and Lagier on relative cuspidality and relative square integrability. We will also show that the local Langlands parameter of the trivial representation of factors through for any symmetric variety .
Paper Structure (45 sections, 24 theorems, 237 equations, 1 table)

This paper contains 45 sections, 24 theorems, 237 equations, 1 table.

Table of Contents

  1. Introduction
  2. Folding of a root datum
  3. Additively closed root sub-datum
  4. Strongly orthogonal roots
  5. Folding a root datum
  6. Root datum with involution
  7. Involution on root datum
  8. Two tori
  9. Admissible involution
  10. $\theta$-basis
  11. $\theta$-invariant root sub-datum
  12. A lemma
  13. Principal $\operatorname{SL}_2$ of $G^+$
  14. $\theta$-split root datum
  15. Dynkin diagram of $\theta$ and some examples
  16. ...and 30 more sections

Key Result

Proposition 2.1

Assume $\Phi$ is a root datum of a split reductive group $G$. Then a root sub-datum $\Psi$ of $\Phi$ generates a subgroup in $G$ whose root datum is $\Psi$ if and only if $\Psi$ is additively closed, in which case the subgroup is generated by $T$ and the root subgroups $U_\alpha$ for all $\alpha\in

Theorems & Definitions (57)

  • Conjecture 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • ...and 47 more