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Arthur's groups $\mathcal S$ in local Langlands correspondence for certain covering groups of algebraic tori

Yuki Nakata

TL;DR

The paper addresses parametrizing packets in the local Langlands correspondence for Brylinski-Deligne covering groups of tori by identifying a finite packet group $\mathcal S_{\widetilde{T}}$. Using lattice data $Y$, its sublattices $Y^{\#}$ and $Y^{\Gamma\#}$, and the bilinear form $B_{\widetilde{T}}$, it reduces the problem to a Galois-cohomological analysis of the center image $Z^{\dagger}$ and derives an explicit description of the packet group. The main result proves that, under the coprimality condition $\gcd(n,e)=1$, the packet group is $\mathcal S_{\widetilde{T}}=\iota((Y^{\Gamma\#}\otimes\overline{\mathcal O}^{\times})^{\Gamma})/\iota((Y^{\#}\otimes\overline{\mathcal O}^{\times})^{\Gamma})$, with the residue-field version involving $\overline{\mathbf f}^{\times}$; this generalizes Weissman’s unramified torus description to ramified tori. The approach connects center data to Langlands parameters via $T^{\#}$ and Tate duality, advancing the Langlands program for covering groups by providing a concrete packet parametrization in the ramified setting.

Abstract

We determine the finite group $\mathcal S$ parametrizing a packet in the local Langlands correspondence for a Brylinski-Deligne covering group of an algebraic torus, under some assumption on ramification. Especially, this work generalizes Weissman's result on covering groups of tori that split over an unramified extension of the base field.

Arthur's groups $\mathcal S$ in local Langlands correspondence for certain covering groups of algebraic tori

TL;DR

The paper addresses parametrizing packets in the local Langlands correspondence for Brylinski-Deligne covering groups of tori by identifying a finite packet group . Using lattice data , its sublattices and , and the bilinear form , it reduces the problem to a Galois-cohomological analysis of the center image and derives an explicit description of the packet group. The main result proves that, under the coprimality condition , the packet group is , with the residue-field version involving ; this generalizes Weissman’s unramified torus description to ramified tori. The approach connects center data to Langlands parameters via and Tate duality, advancing the Langlands program for covering groups by providing a concrete packet parametrization in the ramified setting.

Abstract

We determine the finite group parametrizing a packet in the local Langlands correspondence for a Brylinski-Deligne covering group of an algebraic torus, under some assumption on ramification. Especially, this work generalizes Weissman's result on covering groups of tori that split over an unramified extension of the base field.

Paper Structure

This paper contains 5 sections, 17 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Brylinski-Deligne covering groups of a torus
  3. Local Langlands correspondence for a covering group
  4. Description of the image $\mathrm{Z}^\dagger$ of the center
  5. Main theorem

Key Result

Theorem 1.1

Let $F$ be a non-archimedean local field of characteristic zero, $\mathbf f$ the residue field of the valuation ring in $F$, and $\mu_n\subset \mathbf f^\times$ the cyclic subgroup of order $n$. Suppose that an algebraic torus $T$ defined over $F$ splits over an unramified extension $L/F$, and that of subgroups of $(Y\otimes_{\mathbb Z}\overline{\mathbf f}^\times)^\Gamma$.

Theorems & Definitions (39)

  • Theorem 1.1: Weissman Weiss_loc_tori
  • Definition 2.1: Brylinski-Deligne Bry_Del
  • Theorem 2.2: Classification Theorem of covers Bry_Del
  • Definition 2.3
  • Example 3.1: central character
  • Definition 3.2
  • Definition 3.3
  • Proposition 3.5
  • Definition 3.6
  • Proposition 3.7: role of $\mathcal{S}_{\widetilde{T}}$
  • ...and 29 more