is of orthogonal type if it preserves a suitable orthogonal (i.e. symmetric bilinear) form on the ambient 2n+ 1-dimensional vector space, and this is equivalent to the condition that ϕv factors throughξ (up to conjugation by an element ofGL2n+1(C)):
LFv
ϕv //
%%K
KK KK KK KK
K GL2n+1(C)
SO2n+1(C)
ξoooooo77 oo oo o
Note that all irreducible self-dual representations into GL2n+1(C) must be of orthogonal type, since the existence of a symplectic form on the space would imply that the ambient space is even- dimensional.
To each localL-parameter
ϕv:LFv →SO2n+1(C)
or localL-parameter forGL2n+1of symplectic type, Arthur attaches anL-packet Πϕv, which consists of finitely many irreducible representations ofSp2n(Fv); we have the analogous results onL-packets and discrete parameters. For a discrete automorphic representationτ of Sp2n(AF), Arthur shows that there exists a self-dual isobaric automorphic representationπofGL2n+1(AF) which is a func- torial transfer ofτ along the standard embeddingξ, with the analogous correspondence condition for representations that are generic in the sense of Arthur.
tation of G(Fv)). Then there exists a cuspidal automorphic representation ofGLN(AF) satisfying those local conditions such that:
(i) π is unramified away fromv∈S; and
(ii) π≃π∨, that is,π is self-dual
Proof. Here, as in the proof of the globalization theorem above, our case divides into the even and odd cases; ifN = 2nis even letH =SO2n+1 and ifN = 2n+ 1 is odd, letH =Sp2n.
Apply the generalized version of Proposition8.7 with ourS and where eachUv∧ is prescribable forv∈S. Thus, there exists a cuspidal automorphic representationτ ofH(AF) such that
1. τv is unramified away fromS;
2. τv∈Uv∧ for allv∈S;
3. either τ∞ is a discrete series whose infinitesimal character is sufficiently regular, or
3’. τ∞is any discrete series, but we lose control of the prescribed representation at two auxiliary primes.
The functorial transferπofτthen has the desired properties. For example, to see thatπis cuspidal, by condition (3) or (3’), thenπ is generic in the sense of Arthur. To see that the condition on the central character holds, note that the central character is trivial at almost all finite places. Indeed, the central character corresponds to the determinant of the Langlands parameter forπat each place via local class feld, but the determinant is trivial since the parameter factors throughH.b
Bibliography
[AB67] M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86(1967), 374–407.
[AdRDSW15] Sara Arias-de Reyna, Luis V. Dieulefait, Sug Woo Shin, and Gabor Wiese, Compat- ible systems of symplectic Galois representations and the inverse Galois problem III.
Automorphic construction of compatible systems with suitable local properties, Math.
Ann.361(2015), no. 3-4, 909–925.
[Art88a] James Arthur, The invariant trace formula. I. Local theory, J. Amer. Math. Soc.1 (1988), no. 2, 323–383.
[Art88b] ,The invariant trace formula. II. Global theory, J. Amer. Math. Soc.1(1988), no. 3, 501–554.
[Art13] ,The endoscopic classification of representations, American Mathematical So- ciety Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013, Orthogonal and symplectic groups.
[BLS96] A. Borel, J.-P. Labesse, and J. Schwermer, On the cuspidal cohomology of S- arithmetic subgroups of reductive groups over number fields, Compositio Math.102 (1996), no. 1, 1–40.
[Bor79] A. Borel, Automorphic L-functions, Automorphic forms, representations and L- functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61.
[Bou87] Abderrazak Bouaziz,Sur les caract`eres des groupes de Lie r´eductifs non connexes, J.
Funct. Anal.70(1987), no. 1, 1–79.
[BW80] Armand Borel and Nolan R. Wallach,Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980.
[BZ76] I. N. Bernˇste˘ın and A. V. Zelevinski˘ı,Representations of the group GL(n, F), where F is a local non-Archimedean field, Uspehi Mat. Nauk31(1976), no. 3(189), 5–70.
[Car12] Ana Caraiani,Local-global compatibility and the action of monodromy on nearby cy- cles, Duke Math. J.161(2012), no. 12, 2311–2413.
[CC09] G. Chenevier and L. Clozel, Corps de nombres peu ramifi´es et formes automorphes autoduales, J. Amer. Math. Soc.22(2009), no. 2, 467–519.
[CD90] Laurent Clozel and Patrick Delorme, Le th´eor`eme de Paley-Wiener invariant pour les groupes de Lie r´eductifs. II, Ann. Sci. ´Ecole Norm. Sup. (4) 23 (1990), no. 2, 193–228.
[Clo90] Laurent Clozel,Motifs et formes automorphes: applications du principe de fonctori- alit´e, Automorphic forms, Shimura varieties, andL-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 77–159.
[Clo13] ,Purity reigns supreme, Int. Math. Res. Not. IMRN (2013), no. 2, 328–346.
[CR10] Ga¨etan Chenevier and David Renard,On the vanishing of some non-semisimple or- bital integrals, Expo. Math.28(2010), no. 3, 276–289.
[HT01] Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001, With an appendix by Vladimir G. Berkovich.
[Kna94] A. W. Knapp,Local Langlands correspondence: the Archimedean case, Motives (Seat- tle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 393–410.
[Kna01] Anthony W. Knapp, Representation theory of semisimple groups, Princeton Land- marks in Mathematics, Princeton University Press, Princeton, NJ, 2001, An overview based on examples, Reprint of the 1986 original.
[Kos59] Bertram Kostant,A formula for the multiplicity of a weight, Trans. Amer. Math. Soc.
93(1959), 53–73.
[Kot88] Robert E. Kottwitz,Tamagawa numbers, Ann. of Math. (2)127(1988), no. 3, 629–
646.
[KR00] Robert E. Kottwitz and Jonathan D. Rogawski, The distributions in the invariant trace formula are supported on characters, Canad. J. Math. 52 (2000), no. 4, 804–
814.
[KS99] Robert E. Kottwitz and Diana Shelstad,Foundations of twisted endoscopy, Ast´erisque (1999), no. 255, vi+190.
[Lab91] J.-P. Labesse, Pseudo-coefficients tr`es cuspidaux et K-th´eorie, Math. Ann. 291 (1991), no. 4, 607–616.
[Lab99] Jean-Pierre Labesse, Cohomologie, stabilisation et changement de base, Ast´erisque (1999), no. 257, vi+161, Appendix A by Laurent Clozel and Labesse, and Appendix B by Lawrence Breen.
[Mez04] Paul Mezo,Twisted trace Paley-Wiener theorems for special and general linear groups, Compos. Math. 140(2004), no. 1, 205–227.
[PR12] Dipendra Prasad and Dinakar Ramakrishnan, Self-dual representations of division algebras and Weil groups: a contrast, Amer. J. Math.134(2012), no. 3, 749–767.