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Erschienen in:
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2019 | OriginalPaper | Buchkapitel

6. Langlands Parametrization

verfasst von : Gaëtan Chenevier, Jean Lannes

Erschienen in: Automorphic Forms and Even Unimodular Lattices

Verlag: Springer International Publishing

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Abstract

This chapter starts with a general discussion of reductive groups and their root data. Our aim is to review the Satake isomorphism for the Hecke ring of a reductive group scheme over the p-adic integers \(\mathbb {Z}_p\), as well as the Harish-Chandra isomorphism for the center of the universal enveloping algebra of a complex reductive Lie algebra. In both cases, we explain Langlands’ point of view on these isomorphisms in terms of the so-called Langlands dual group. This allows to define Langlands parameters and state the important Arthur-Langlands conjecture. As another application, we obtain a deeper understanding of the Hecke rings of classical \(\mathbb {Z}\)-groups.

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Vorheriges Kapitel Theta Series and Even Unimodular Lattices
Nächstes Kapitel A Few Cases of the Arthur–Langlands Conjecture
Fußnoten
1
Recall that an A-group is a group scheme over A which is affine and of finite type.
 
2
Let us emphasize that this assumption is not part of the axioms in the references listed above; it will help us avoid certain difficulties.
 
3
For α ∈ Φ(G, T), let T α ⊂ T be the neutral component of the kernel of \(\alpha \colon T \rightarrow \mathbb {G}_m\), and let Z α be the derived subgroup of the centralizer of T α in G. It is a k-group isomorphic to SL2 or PGL2. Recall that the coroot α ∨∈X∗(T) is the unique cocharacter with image in Z α such that 〈α, α ∨〉 = 2.
 
4
Strictly speaking, we should replace Hp(G) by the opposite ring Hp(G)opp in this isomorphism. Since the existence of the latter implies the commutativity of Hp(G), we will leave out this decoration.
 
5
Let V be a \(G(\mathbb {Q}_p)\)-module and \(\pi \colon V^{G(\mathbb {Z}_p)} \rightarrow V_{N(\mathbb {Q}_p)}\) the canonical projection. The ring H(G) acts on \(V^{G(\mathbb {Z}_p)}\) (Sect. 4.​2.​2). By the construction of s 2, we have π ∘ T = s 2(T) ∘ π for every T ∈Hp(G). The assertion follows by considering \(V=\mathbb {Z}\) and recalling the shift by ρ in the definition of the Satake homomorphism.
 
6
We use λ(p) to denote the image of p by the morphism \(\mathbb {Q}_p^\times \rightarrow T(\mathbb {Q}_p)\) induced by λ.
 
7
This is because this outer conjugate admits a vector \(\mathcal {C}^\infty \) that is annihilated by \(\mathfrak {p}^+\) and generates W under the action of K (lowest weight). Its \((\mathfrak {g},K)\)-module can be studied in a manner completely analogous to that of \(\pi ^{\prime }_W\): it can be isomorphic to that of \(\pi ^{\prime }_W\) only if it is finite-dimensional, that is, if \(\pi ^{\prime }_W\) (and therefore W) is trivial. This does not occur because the trivial representation of \({\mathrm{Sp}}_{2g}(\mathbb {R})\) does not occur in \(\mathcal {A}_{\mathrm{cusp}}({\mathrm{PGSp}}_{2g})\).
 
8
Gross’ argument is the following. It is a general fact that the natural action of \({\mathrm{Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})\) on \(\varPsi (G_{\overline {\mathbb {Q}}})\) factors into a faithful action of the Galois group of a number field K that is Galois over \(\mathbb {Q}\). The reductivity of G over \(\mathbb {Z}_p\) implies that K is unramified at p, and therefore \(K=\mathbb {Q}\) by a famous result of Minkowski. This, in turn, implies that G is split over \(\mathbb {Z}_p\) and the rest of the assertions above.
 
9
In general, a condition that is conjecturally automatic is added on π ∞; we omit it here.
 
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Metadaten
Titel
Langlands Parametrization
verfasst von
Gaëtan Chenevier
Jean Lannes
Copyright-Jahr
2019
Buch
Automorphic Forms and Even Unimodular Lattices

Print ISBN: 978-3-319-95890-3

Electronic ISBN: 978-3-319-95891-0

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-319-95891-0_6

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