a number of endoscopic groups G', namely the ones associated to those automorphisms of LGO which differ by an inner automorphism. One would hope to compare a (twisted) trace formula for G with some combination of trace formulas for the relevant groups G'. There now exists a (twisted) trace formula for general groups. The last few years have also seen progress on other questions, motivated by a comparison of trace formulas. The purpose 1 det(s-)M tr(M(s,O)pp,t(O,f)) M sEW(aM)reg in which pp,, is a representation induced from the discrete spectrum of M, and M(s, 0) is an intertwining operator. (See §2.9 for a fuller description of the notation, and, in particular, the role of the real number t.) Theorem B of Chapter 2 implies an identity between the discrete parts of the trace formulas of G and G'. We shall describe this more precisely. Chapter 1 The same construction applies to the Lie algebra (which is also naturally an associative matrix algebra): we define Gx,, and Gu in the same manner, and Gu is an E/F-form of G,,,. Hilbert's Theorem 90 (cf. [35, Exercise 2, p. 160]) then gives H1(, EG, (E)) = 0. But then an easy cocycle computation gives (ii). | We will say that z E G(E) is a-semi-simple if the class NAf is semisimple. In that case, of course, Gu is a semi-simple algebra, isomorphic to r a product fi M, (Fi) where Fi/F are field extensions; Gu is isomorphic to i=l n GL(ni, Fi) seen as an F-group, and Gx,a is an inner form of this group which defines a product of central simple algebras. Assume now that F is a global field. We will need to extend the definition of the local norms to the places of F which are not inert in E. This is easy and we do not give details. Assume for example that v is a place of F which splits in E. Then E® F, = Fv * * D F, (e factors), E acting by cyclic permutations; we set N(gl,...g9) = (91,...g9)(g2,... gi) * (9 (,91 .. g9-1) = (9192 ' gt,92 ' 1, 9 ...,g gl''' gt-1) It is conjugate in G(Ev) to an element of the form (h,h,... h) E G(F,). The general case is an obvious composite of the split case and the inert case. LEMMA 1.2: Assume F is a global field. Then, if u E G(F), u = Nx has a solution in G(E) if and only if it has a solution in G(E,) for any place v ofF. Proof. Only the "if" part need be proved. We will first treat the case of a semi-simple u. We may write u as a diagonal matrix U' U1 U2 U2 Uk Uk Local Results 5 where ui generates a field extension Fi/F of degree mi, embedded in GL(mi,F). The centralizer gu is then (if ul y u2 **- .-uk) a product of matrix algebras Mki(Fi) C Mk,,,(F), where ui appears ki times. It is easy to see that the problem actually takes place in H Mimi,(F); thus we i may assume that u has only one eigenvalue, say ul E F;. We set k = kl, m = m1.
We shall outline a classification [A] of the automorphic representations of special orthogonal and symplectic groups in terms of those of general linear groups. This necessarily includes a classification of local L-packets of representations. It also requires a classification of the extended packets that are the local constituents of nontempered automorphic representations. Our description will be brief. In particular, we will restrict it to quasisplit * orthogonal and symplectic groups G, even though at least some of the results can be extended (not without effort) to inner twists of G. The methods rest ultimately on two comparisons of trace formulas. One is the spectral identity that is the end product of the stabilization of the trace formula for G. This was established some years ago [A1], under the assumption of the fundamental lemma. It now holds without condition, thanks to the work of Waldspurger [W1][W2], the recent breakthrough by Ngo [N], and the extensions of Chaudouard and Laumon [CL1] [CL2]. The other is the spectral identity given by the stabilization of the twisted trace formula for GL(N). This formula is still conditional. The relevant twisted fundamental lemmas are now known [W3], [W4], at least up to the twisted variants of [CL1] and [CL2]. The problem is to develop twisted generalizations of the techniques of [A1] and related papers. Until this is done, the results we describe here have also to be regarded as conditional. We take F to be a local or global field of characteristic 0, and G to be a quasisplit, special orthogonal or symplectic group over F. Then G has a complex dual group G, and a corresponding L-group L G = G Γ E/F. We are taking Γ E/F = Gal(E/F) to be the Galois group of a suitable finite extension E/F. If G is split, for example, the absolute Galois group Γ = Γ F = Γ F /F acts trivially on G, and we can take E = F. There are three general possibilities for G, which correspond to the three infinite families of simple groups B n , C n and D n. They are as follows * It is understood that G is "classical", in the sense that it is not an outer twist of the split group SO(8) by a triality automorphishm of order 3.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.