Rigid inner forms of real and \(p\)-adic groups. (English) Zbl 1393.22009

Kaletha, Tasho

Ann. Math. (2) 184, No. 2, 559-632 (2016).

For a reductive group \(G\) over a local field \(F\), the local Langlands conjecture is a description of the irreducible, admissible representations of \(G(F)\) in terms of the Weil-Deligne group \(W_F'\) and the complex Langlands dual group \(\hat{G}\). Given tempered Langlands parameters \(\phi : W_F' \rightarrow \hat{G}\), the expectation is that the corresponding \(L\)-packets \(\pi_{\phi}^G\) satisfy certain properties like their internal parametrization and endoscopic character identities. Shahidi’s tempered \(L\)-packet conjecture states that for a fixed Whittaker datum \(\vartheta\) of \(G\), the \(L\)-packet \(\pi_{\phi}^G\) should contain a unique generic \(\vartheta\)-representation. There are further conjectural expectations. These are known for connected, quasi-split groups. It is known that the situation is much more complicated for general reductive groups that are not quasi-split.
For this purpose, the author introduces a new type of cohomology set for (affine) algebraic groups by replacing the Galois group with a certain Galois gerbe that is canonically associated to any local field \(F\) of characteristic \(0\). When \(G\) is connected, and reductive over \(F\), and \(Z\) is a finite central subgroup containing the center of the derived subgroup of \(G\), the author defines the cohomology set \(H^1(u \rightarrow W, Z \rightarrow G)\) which is a finite set with some remarkable properties. Here, \(u\) is a certain pro-algebraic group and \(W\) is a certain extension of the absolute Galois group by \(u(\bar{F})\). In particular, \(H^1(\mathrm{Gal},G)\) injects into it and this cohomology set surjects onto \(H^1(\mathrm{Gal},G/Z(G))\). Therefore, every inner twist can be equipped with an element of this cohomology set. The author is able to give a very precise conjectural description of the internal structure of tempered \(L\)-packets and character identities satisfied by them. In particular, he provides a solution to problem 9.3 mentioned by D. A. Vogan jun. [Contemp. Math. 145, 305–379 (1993; Zbl 0802.22005)]. Further, he is able to provide explicit formulae for Arthur’s conjectural spectral transfer factors and mediating functions.
Already, these local results have been applied to global problems. Studying the cohomology of certain Galois gerbes over number fields, the author has provided in later work, a bridge between refined local endoscopy as introduced here, and classical global endoscopy. As particular applications, he has expressed the canonical adelic transfer factor that governs the stabilization of the Arthur-Selberg trace formula as a product of normalized local transfer factors, and gives an explicit constriction of the pairing between an adelic \(L\)-packet and the corresponding \(S\)-group based on the conjectural pairings in the local setting that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group.
The reader is referred to the author’s nine and a half pages long admirable introduction which gives a very clear description of all subtleties.

Reviewer: Balasubramanian Sury (Bangalore)

MSC:

22E50

Representations of Lie and linear algebraic groups over local fields

Keywords:

local Langlands; endoscopic transfer; tempered \(L\)-packet; \(p\)-adic groups

Citations:

Zbl 0802.22005

References:

[1]	H. Atobe, On the uniqueness of generic representations in an \(L\)-packet, 2015. · Zbl 1365.11050
[2]	J. Adams and D. A. Vogan Jr., ”\(L\)-groups, projective representations, and the Langlands classification,” Amer. J. Math., vol. 114, iss. 1, pp. 45-138, 1992. · Zbl 0760.22013
[3]	J. Adams, D. Barbasch, and D. A. Vogan Jr., The Langlands Classification and Irreducible Characters for Real Reductive Groups, Boston: Birkhäuser, 1992, vol. 104. · Zbl 0756.22004
[4]	J. Adams and J. F. Johnson, ”Endoscopic groups and packets of non-tempered representations,” Compositio Math., vol. 64, iss. 3, pp. 271-309, 1987. · Zbl 0647.22008
[5]	J. Arthur, ”On the transfer of distributions: weighted orbital integrals,” Duke Math. J., vol. 99, iss. 2, pp. 209-283, 1999. · Zbl 0938.22019
[6]	J. Arthur, ”A note on \(L\)-packets,” Pure Appl. Math. Q., vol. 2, iss. 1, Special Issue: In honor of John H. Coates. Part 1, pp. 199-217, 2006. · Zbl 1158.22017
[7]	J. Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, Providence, RI: Amer. Math. Soc., 2013, vol. 61. · Zbl 1310.22014
[8]	A. Borel, ”Automorphic \(L\)-functions,” in Automorphic Forms, Representations and \(L\)-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. XXXIII, pp. 27-61. · Zbl 0412.10017
[9]	N. Bourbaki, Lie Groups and Lie Algebras. Chapters 4-6, New York: Springer-Verlag, 2002. · Zbl 1145.17001
[10]	M. Borovoi, ”Abelian Galois cohomology of reductive groups,” Mem. Amer. Math. Soc., vol. 132, iss. 626, p. viii, 1998. · Zbl 0918.20037
[11]	D. Blasius and J. D. Rogawski, ”Zeta functions of Shimura varieties,” in Motives, Providence, RI: Amer. Math. Soc., 1994, vol. 55, pp. 525-571. · Zbl 0827.11033
[12]	S. DeBacker and M. Reeder, ”Depth-zero supercuspidal \(L\)-packets and their stability,” Ann. of Math., vol. 169, iss. 3, pp. 795-901, 2009. · Zbl 1193.11111
[13]	K. Hiraga and H. Saito, ”On \(L\)-packets for inner forms of \(SL_n\),” Mem. Amer. Math. Soc., vol. 215, iss. 1013, p. vi, 2012. · Zbl 1242.22023
[14]	T. Kaletha, ”Endoscopic character identities for depth-zero supercuspidal \(L\)-packets,” Duke Math. J., vol. 158, iss. 2, pp. 161-224, 2011. · Zbl 1220.22013
[15]	T. Kaletha, ”Supercuspidal \(L\)-packets via isocrystals,” Amer. J. Math., vol. 136, iss. 1, pp. 203-239, 2014. · Zbl 1295.22022
[16]	T. Kaletha, ”Epipelagic \(L\)-packets and rectifying characters,” Invent. Math., vol. 202, iss. 1, pp. 1-89, 2015. · Zbl 1428.11201
[17]	T. Kaletha, Global rigid inner forms and multiplicities of discrete automorphic representations, 2015. · Zbl 1415.11079
[18]	T. Kaletha, Rigid inner forms vs isocrystals, 2015. · Zbl 1430.11071
[19]	M. Kneser, ”Galois-Kohomologie halbeinfacher algebraischer Gruppen über \(\mathfrakp\)-adischen Körpern. I,” Math. Z., vol. 88, pp. 40-47, 1965. · Zbl 0143.04702
[20]	T. Konno, ”Twisted endoscopy and the generic packet conjecture,” Israel J. Math., vol. 129, pp. 253-289, 2002. · Zbl 1004.22008
[21]	B. Kostant, ”On Whittaker vectors and representation theory,” Invent. Math., vol. 48, iss. 2, pp. 101-184, 1978. · Zbl 0405.22013
[22]	R. E. Kottwitz, ”Rational conjugacy classes in reductive groups,” Duke Math. J., vol. 49, iss. 4, pp. 785-806, 1982. · Zbl 0506.20017
[23]	R. E. Kottwitz, ”Sign changes in harmonic analysis on reductive groups,” Trans. Amer. Math. Soc., vol. 278, iss. 1, pp. 289-297, 1983. · Zbl 0538.22010
[24]	R. E. Kottwitz, ”Stable trace formula: Cuspidal tempered terms,” Duke Math. J., vol. 51, iss. 3, pp. 611-650, 1984. · Zbl 0576.22020
[25]	R. E. Kottwitz, ”Isocrystals with additional structure,” Compositio Math., vol. 56, iss. 2, pp. 201-220, 1985. · Zbl 0597.20038
[26]	R. E. Kottwitz, ”Stable trace formula: Elliptic singular terms,” Math. Ann., vol. 275, iss. 3, pp. 365-399, 1986. · Zbl 0577.10028
[27]	R. E. Kottwitz, ”Isocrystals with additional structure. II,” Compositio Math., vol. 109, iss. 3, pp. 255-339, 1997. · Zbl 0966.20022
[28]	R. E. Kottwitz and D. Shelstad, Foundations of twisted endoscopy, Paris: Math. Soc. France, 1999, vol. 255. · Zbl 0958.22013
[29]	R. E. Kottwitz and D. Shelstad, On splitting invariants and sign conventions in endoscopic transfer, 2012.
[30]	R. P. Langlands, ”On the classification of irreducible representations of real algebraic groups,” in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Providence, RI: Amer. Math. Soc., 1989, vol. 31, pp. 101-170. · Zbl 0741.22009
[31]	R. P. Langlands, Les Débuts d’une Formule des Traces Stable, Paris: Université de Paris VII, U.E.R. de Mathématiques, 1983, vol. 13. · Zbl 0532.22017
[32]	R. P. Langlands and M. Rapoport, ”Shimuravarietäten und Gerben,” J. Reine Angew. Math., vol. 378, pp. 113-220, 1987. · Zbl 0615.14014
[33]	R. P. Langlands and D. Shelstad, ”On the definition of transfer factors,” Math. Ann., vol. 278, iss. 1-4, pp. 219-271, 1987. · Zbl 0644.22005
[34]	R. P. Langlands and D. Shelstad, ”Descent for transfer factors,” in The Grothendieck Festschrift, Vol. II, Boston: Birkhäuser, 1990, vol. 87, pp. 485-563. · Zbl 0743.22009
[35]	J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of Number Fields, Second ed., New York: Springer-Verlag, 2008, vol. 323. · Zbl 1136.11001
[36]	V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Boston: Academic Press, 1994, vol. 139. · Zbl 0841.20046
[37]	M. Reeder and J. Yu, ”Epipelagic representations and invariant theory,” J. Amer. Math. Soc., vol. 27, iss. 2, pp. 437-477, 2014. · Zbl 1284.22011
[38]	F. Shahidi, ”A proof of Langlands’ conjecture on Plancherel measures; complementary series for \(p\)-acid groups,” Ann. of Math., vol. 132, iss. 2, pp. 273-330, 1990. · Zbl 0780.22005
[39]	D. Shelstad, ”Orbital integrals and a family of groups attached to a real reductive group,” Ann. Sci. École Norm. Sup., vol. 12, iss. 1, pp. 1-31, 1979. · Zbl 0433.22006
[40]	D. Shelstad, ”Characters and inner forms of a quasi-split group over \({\mathbf R}\),” Compositio Math., vol. 39, iss. 1, pp. 11-45, 1979. · Zbl 0431.22011
[41]	D. Shelstad, ”Notes on \(L\)-indistinguishability (based on a lecture of R. P. Langlands),” in Automorphic Forms, Representations and \(L\)-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. XXXIII, pp. 193-203. · Zbl 0421.12015
[42]	D. Shelstad, ”Embeddings of \(L\)-groups,” Canad. J. Math., vol. 33, iss. 3, pp. 513-558, 1981. · Zbl 0457.22006
[43]	D. Shelstad, ”\(L\)-indistinguishability for real groups,” Math. Ann., vol. 259, iss. 3, pp. 385-430, 1982. · Zbl 0506.22014
[44]	D. Shelstad, ”Tempered endoscopy for real groups. I. Geometric transfer with canonical factors,” in Representation Theory of Real Reductive Lie Groups, Providence, RI: Amer. Math. Soc., 2008, vol. 472, pp. 215-246. · Zbl 1157.22008
[45]	D. Shelstad, ”Tempered endoscopy for real groups. II. Spectral transfer factors,” in Automorphic Forms and the Langlands Program, Somerville, MA: Int. Press, 2010, vol. 9, pp. 236-276. · Zbl 1198.22008
[46]	D. Shelstad, ”Tempered endoscopy for real groups. III. Inversion of transfer and \(L\)-packet structure,” Represent. Theory, vol. 12, pp. 369-402, 2008. · Zbl 1159.22007
[47]	J. Tate, ”Duality theorems in Galois cohomology over number fields,” in Proc. Internat. Congr. Mathematicians, Djursholm: Inst. Mittag-Leffler, 1963, pp. 288-295. · Zbl 0126.07002
[48]	J. Tate, ”The cohomology groups of tori in finite Galois extensions of number fields,” Nagoya Math. J., vol. 27, pp. 709-719, 1966. · Zbl 0146.06501
[49]	J. Tate, ”Number theoretic background,” in Automorphic Forms, Representations and \(L\)-Functions, Providence, R.I.: Amer. Math. Soc., 1979, vol. XXXIII, pp. 3-26. · Zbl 0422.12007
[50]	D. A. Vogan Jr., ”Gel\cprime fand-Kirillov dimension for Harish-Chandra modules,” Invent. Math., vol. 48, iss. 1, pp. 75-98, 1978. · Zbl 0389.17002
[51]	D. A. Vogan Jr., ”The algebraic structure of the representation of semisimple Lie groups. I,” Ann. of Math., vol. 109, iss. 1, pp. 1-60, 1979. · Zbl 0424.22010
[52]	D. A. Vogan Jr., ”The local Langlands conjecture,” in Representation Theory of Groups and Algebras, Providence, RI: Amer. Math. Soc., 1993, vol. 145, pp. 305-379. · Zbl 0802.22005