Epipelagic representations and invariant theory

Reeder, Mark; Yu, Jiu-Kang

doi:10.1090/S0894-0347-2013-00780-8

Epipelagic representations and invariant theory

Authors: Mark Reeder and Jiu-Kang Yu
Journal: J. Amer. Math. Soc. 27 (2014), 437-477
MSC (2010): Primary 22E50, 11S15, 11S37
DOI: https://doi.org/10.1090/S0894-0347-2013-00780-8
Published electronically: August 5, 2013
MathSciNet review: 3164986
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Abstract: We introduce a new approach to the representation theory of reductive $p$-adic groups $G$, based on the geometric invariant theory (GIT) of Moy-Prasad quotients. Stable functionals on these quotients are used to give a new construction of supercuspidal representations of $G$ having small positive depth, called epipelagic. With some restrictions on $p$, we classify the stable and semistable functionals on Moy-Prasad quotients. The latter classification determines the nondegenerate $K$-types for $G$ as well as the depths of irreducible representations of $G$. The main step is an equivalence between Moy-Prasad theory and the theory of graded Lie algebras, whose GIT was analyzed by Vinberg and Levy. Our classification shows that stable functionals arise from $\mathbb {Z}$-regular elliptic automorphisms of the absolute root system of $G$. These automorphisms also appear in the Langlands parameters of epipelagic representations, in accordance with the conjectural local Langlands correspondence.

References

Jeffrey D. Adler, Refined anisotropic $K$-types and supercuspidal representations, Pacific J. Math. 185 (1998), no. 1, 1–32. MR 1653184, DOI https://doi.org/10.2140/pjm.1998.185.1
Jeffrey D. Adler and Stephen DeBacker, Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive $p$-adic group, Michigan Math. J. 50 (2002), no. 2, 263–286. MR 1914065, DOI https://doi.org/10.1307/mmj/1028575734
Armand Borel, Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tohoku Math. J. (2) 13 (1961), 216–240 (French). MR 147579, DOI https://doi.org/10.2748/tmj/1178244298
A. Borel and J.-P. Serre, Sur certains sous-groupes des groupes de Lie compacts, Comment. Math. Helv. 27 (1953), 128–139 (French). MR 54612, DOI https://doi.org/10.1007/BF02564557
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629
F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923
F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
Colin J. Bushnell and Guy Henniart, The local Langlands conjecture for $\rm GL(2)$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Springer-Verlag, Berlin, 2006. MR 2234120

Colin J. Bushnell and Guy Henniart Langlands parameters for epipelagic representations of $GL_n$, preprint 2013.

Henri Carayol, Représentations supercuspidales de ${\rm GL}_{n}$, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 1, A17–A20 (French, with English summary). MR 522009

Stephen DeBacker, Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. (2) 156 (2002), no. 1, 295–332. MR 1935848, DOI https://doi.org/10.2307/3597191

Wee Teck Gan and Gordan Savin, Endoscopic lifts from ${\rm PGL}_3$ to $G_2$, Compos. Math. 140 (2004), no. 3, 793–808. MR 2041781, DOI https://doi.org/10.1112/S0010437X03000678

Benedict H. Gross, Irreducible cuspidal representations with prescribed local behavior, Amer. J. Math. 133 (2011), no. 5, 1231–1258. MR 2843098, DOI https://doi.org/10.1353/ajm.2011.0035

Benedict H. Gross and Mark Reeder, Arithmetic invariants of discrete Langlands parameters, Duke Math. J. 154 (2010), no. 3, 431–508. MR 2730575, DOI https://doi.org/10.1215/00127094-2010-043

Benedict H. Gross and Gordan Savin, The dual pair ${\rm PGL}_3\times G_2$, Canad. Math. Bull. 40 (1997), no. 3, 376–384. MR 1464847, DOI https://doi.org/10.4153/CMB-1997-045-0

Jochen Heinloth, Bao-Châu Ngô, and Zhiwei Yun, Kloosterman sheaves for reductive groups, Ann. of Math. (2) 177 (2013), no. 1, 241–310. MR 2999041, DOI https://doi.org/10.4007/annals.2013.177.1.5

Kaoru Hiraga, Atsushi Ichino, and Tamotsu Ikeda, Formal degrees and adjoint $\gamma $-factors, J. Amer. Math. Soc. 21 (2008), no. 1, 283–304. MR 2350057, DOI https://doi.org/10.1090/S0894-0347-07-00567-X

T. Kaletha, Epipelagic $L$-packets and rectifying characters, preprint 2012, arXiv:1209.1720

Chandrashekhar Khare, Michael Larsen, and Gordan Savin, Functoriality and the inverse Galois problem. II. Groups of type $B_n$ and $G_2$, Ann. Fac. Sci. Toulouse Math. (6) 19 (2010), no. 1, 37–70 (English, with English and French summaries). MR 2597780

Ju-Lee Kim, Supercuspidal representations: an exhaustion theorem, J. Amer. Math. Soc. 20 (2007), no. 2, 273–320. MR 2276772, DOI https://doi.org/10.1090/S0894-0347-06-00544-3

Andrew Knightly and Charles Li, Modular $L$-values of cubic level, Pacific J. Math. 260 (2012), no. 2, 527–563. MR 3001804, DOI https://doi.org/10.2140/pjm.2012.260.527

Alexei I. Kostrikin and Phạm Hũ’u Tiệp, Orthogonal decompositions and integral lattices, De Gruyter Expositions in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1994. MR 1308713

P. C. Kutzko, Mackey’s theorem for nonunitary representations, Proc. Amer. Math. Soc. 64 (1977), no. 1, 173–175. MR 442145, DOI https://doi.org/10.1090/S0002-9939-1977-0442145-3

Paul Levy, Vinberg’s $\theta $-groups in positive characteristic and Kostant-Weierstrass slices, Transform. Groups 14 (2009), no. 2, 417–461. MR 2504929, DOI https://doi.org/10.1007/s00031-009-9056-y

George Lusztig, Some examples of square integrable representations of semisimple $p$-adic groups, Trans. Amer. Math. Soc. 277 (1983), no. 2, 623–653. MR 694380, DOI https://doi.org/10.1090/S0002-9947-1983-0694380-4

Allen Moy and Gopal Prasad, Unrefined minimal $K$-types for $p$-adic groups, Invent. Math. 116 (1994), no. 1-3, 393–408. MR 1253198, DOI https://doi.org/10.1007/BF01231566

Allen Moy and Gopal Prasad, Jacquet functors and unrefined minimal $K$-types, Comment. Math. Helv. 71 (1996), no. 1, 98–121. MR 1371680, DOI https://doi.org/10.1007/BF02566411

David Mumford, Stability of projective varieties, Enseign. Math. (2) 23 (1977), no. 1-2, 39–110. MR 450272

Mark Reeder, On the Iwahori-spherical discrete series for $p$-adic Chevalley groups; formal degrees and $L$-packets, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 4, 463–491. MR 1290396

Mark Reeder, Torsion automorphisms of simple Lie algebras, Enseign. Math. (2) 56 (2010), no. 1-2, 3–47. MR 2674853, DOI https://doi.org/10.4171/LEM/56-1-1

Mark Reeder, Elliptic centralizers in Weyl groups and their coinvariant representations, Represent. Theory 15 (2011), 63–111. MR 2765477, DOI https://doi.org/10.1090/S1088-4165-2011-00377-0

Mark Reeder, Paul Levy, Jiu-Kang Yu, and Benedict H. Gross, Gradings of positive rank on simple Lie algebras, Transform. Groups 17 (2012), no. 4, 1123–1190. MR 3000483, DOI https://doi.org/10.1007/s00031-012-9196-3

J.-P. Serre, Coordonnées de Kac, Oberwolfach Reports, 3 (2006), pp. 1787-1790.

T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198. MR 354894, DOI https://doi.org/10.1007/BF01390173

Robert Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92. MR 354892, DOI https://doi.org/10.1016/0001-8708%2875%2990125-5

J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588

È. B. Vinberg, The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 488–526, 709 (Russian). MR 0430168

Jiu-Kang Yu, Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), no. 3, 579–622. MR 1824988, DOI https://doi.org/10.1090/S0894-0347-01-00363-0