Depth-zero supercuspidal \(L\)-packets and their stability
type: Item
instance of: scholarly article
title: Depth-zero supercuspidal \(L\)-packets and their stability
zbMATH Open document ID: 1193.11111
author: Stephen DeBacker
author: Mark Reeder
published in: Annals of Mathematics. Second Series
publication date: 2009-07-14T00:00:00Z
full work available at URL: p03.xhtml
review text: This is a paper on verifying the local Langlands correspondence for pure inner forms of unramified \(p\)-adic groups and tame Langlands parameters in ``general position''. Given each tame Langlands parameter, the authors construct, in a natural way, an \(L\)-packet of depth-zero supercuspidal representation of the appropriate \(p\)-adic group, and verify some expected properties of this \(L\)-packet. In particular, they check the stability of their \(L\)-packets, i.e., they prove that, with some conditions on the base field, the appropriate sum of characters of the representations in the \(L\)-packet is stable, and no proper subset of their \(L\)-packets can form a stable combination. In the spirit of local class field theory, the authors construct both the ``geometric'' and ``\(p\)-adic'' sides of local Langlands correspondence, and make an explicit connection between the two sides. Let \(k\) be a \(p\)-adic field of characteristic zero, and \(G\) be a reductive group over \(k\). Let \(K\) be the maximal unramified extension of \(k\). Assume \(G\) is quasi-split over \(k\), and split over \(K\). Let \(F\) be a Frobenius element in \(\text{Gal}(K/k)\). Then it induces an action on a root system, and hence induces an automorphism \(\sigma\) on the Langlands dual group \(G^\vee\). The authors start with the geometric side, from a Langlands parameter which is a continuous homomorphism from tame Galois group to the semiproduct of \(G^\vee\) and \(\langle\sigma\rangle\), they construct an \(L\)-packet, which is a finite set of irreducible representations of \(p\)-adic groups twisted by different inner forms which are induced from data on the Galois side. On the \(p\)-adic side, take a pair \((S,\theta)\), where \(S=S(K)\) is the set of \(K\)-points in an unramified \(k\)-anisotropic maximal torus \(S\) in \(G\), and \(\theta\) is a depth-zero character of \(S^F\), where \(F\) is the Frobenius. From such data, one can construct a class function \(R(G,S,\theta)\) on the se…
reviewed by: Jiuzu Hong
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 5578757
zbMATH Keywords: local Langlands correspondence
zbMATH Keywords: tame Langlands parameter
zbMATH Keywords: L-packet
zbMATH Keywords: supercuspidal representation
zbMATH Keywords: Bruhat-Tits building
zbMATH Keywords: Deligne-Lusztig representation
MaRDI profile type: Publication
OpenAlex ID: W2082824937
DOI: 10.4007/ANNALS.2009.169.795
description: scientific article