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Supercuspidal representations: An exhaustion theorem

Author(s): Ju-Lee Kim
Journal: J. Amer. Math. Soc. 20 (2007), 273-320.
MSC (2000): Primary 22E50; Secondary 22E35, 20G25.
Posted: October 24, 2006
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Abstract: Let $ G$ be a reductive $ p$-adic group. We prove that all supercuspidal representations of $ G$ arise through Yu's construction subject to certain hypotheses on $ k$ (depending on $ G$). As a corollary, under the same hypotheses, we see that any supercuspidal representation is compactly induced from a representation of an open subgroup which is compact modulo the center.

References:

1.
J. Adler, Refined anisotropic $ K$-types and supercuspidal representations, Pacific J. Math. 185 (1998), 1-32. MR 1653184 (2000f:22019)

2.
J. Adler and S. DeBacker, Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive $ p$-adic group, Michigan J. Math. 50 (2002), no. 2, 263-286. MR 1914065 (2003g:22016)

3.
-, Murnaghan-Kirillov theory for supercuspidal representations of tame general linear groups, J. Reine Angew. Math. 575 (2004), 1-35. MR 2097545 (2005j:22008)

4.
J. Adler and A. Roche, An intertwining result for $ p$-adic groups, Canad. J. Math. 52 (2000), no. 3, 449-467. MR 1758228 (2001m:22032)

5.
C. Bushnell and P. Kutzko, The admissible dual of $ GL(N)$ via compact open subgroups, 129, Annals of Math Studeis, Princeton Univ. Press, 1993. MR 1204652 (94h:22007)

6.
-, The admissible dual of $ SL(N)$ I, 26, Ann. Sci. ENS, 261-280. MR 1209709 (94a:22033)

7.
-, The admissible dual of $ SL(N)$ II, Proc. London Math. Soc. 68 (1994), 317-379. MR 1253507 (94k:22035)

8.
-, Smooth representations of reductive $ p$-adic groups: structure theory via types, Proc. London Math. Soc. 77 (1998), no. 3, 582-634. MR 1643417 (2000c:22014)

9.
F. Bruhat and J. Tits, Groupes réductifs sur un corps local I, Publ. Math. I.H.E.S. 41 (1972), 5-251. MR 0327923 (48:6265)

10.
W. Casselman, Introduction to the theory of admissible representation of $ p$-adic reductive groups, preprint.

11.
S. DeBacker, Homogeneity results for invariant distributions of a reductive $ p$-adic group, Ann. Sci. École Norm. Sup. (4) 35 (2002), 391-422. MR 1914003 (2003i:22019)

12.
-, Parameterizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. 156 (2002), 295-331. MR 1935848 (2003i:20086)

13.
J. Dixmier, Les $ C^*$-algèbres et leurs représentations, Éditions Jacques Gabay, Paris, 1996. MR 1452364 (98a:46066)

14.
J. Hakim and F. Murnaghan, Distinguished tame supercuspidal representations, work in progress.

15.
Harish-Chandra, Admissible invariant distributions on reductive $ p$-adic groups, Preface and notes by Stephen DeBacker and Paul J. Sally, Jr., University Lecture Series, 16, American Mathematical Society, Providence, RI, 1999. MR 1702257 (2001b:22015)

16.
-, Eisenstein series over finite fields, Collected papers, Springer Verlag, New York, 1984. MR 0457579 (56:15784)

17.
-, The Plancherel formula for reductive $ p$-adic groups, Collected Papers, 4, Springer-Verlag, Berlin, 1976 MR 0726026 (85e:01061d)

18.
R. Howe, The Fourier transform and germs of characters (case of Gl$ _n$ over a $ p$-adic field), Math. Ann. 208 (1974), 305-322.MR 0342645 (49:7391)

19.
-, Some qualitative results on the representation theory of $ Gl_n$ over a $ p$-adic field, Pacific J. of Math. 73 (1977), 479-538.MR 0492088 (58:11242)

20.
-, Tamely ramified supercuspidal representations of $ Gl_n$, Pacific J. of Math. 73 (1977), 437-460. MR 0492087 (58:11241)

21.
R. Howe with collaboration of A. Moy, Harish Chandra Homomorphisms for $ p$-adic groups, vol. 59, CBMS regional conference series, AMS, Providence, RI, 1985.MR 0821216 (87h:22023)

22.
R. Howe and A. Moy, Hecke algebra isomorphisms for $ GL(n)$ over a p-adic field, J. Algebra 131 (1990), 388-424. MR 1058553 (91f:22031)

23.
R. Howlett and G. Lehrer, On Harish-Chandra induction and restriction for modules of Levi subgroups, J. Algebra 165 (1994), 172-183.MR 1272585 (95d:20025)

24.
J. L. Kim, Hecke algebras of classical groups over $ p$-adic fields and supercuspidal representations, Amer. J. Math. 121 (1999), 967-1029.MR 1713299 (2001g:11075)

25.
-, Dual blobs and Plancherel formulas, Bull. Soc. Math. France 132 (2004), no. 1, 55-80. MR 2075916 (2005f:22025)

26.
J. L. Kim and F. Murnaghan, Character expansions and unrefined minimal $ {\mathsf K}$-types, Amer. J. Math. 125 (2003), 1199-1234.MR 2018660 (2004k:22024)

27.
-, $ {\mathsf K}$-types and $ \Gamma$-asymptotic expansions, J. Reine Angew. Math. 592 (2006), 189-236. MR 2222734

28.
L. Morris, Level zero $ G$-types, Compositio Math. 118 (1999), 135-157.MR 1713308 (2000g:22029)

29.
A. Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 108 (1986), 863-930. MR 0853218 (88b:11081)

30.
A. Moy and G. Prasad, Unrefined minimal $ {K}$-types for $ p$-adic groups, Inv. Math. 116 (1994), 393-408. MR 1253198 (95f:22023)

31.
-, Jacquet functors and unrefined minimal $ {K}$-types, Comment. Math. Helvetici 71 (1996), 98-121. MR 1371680 (97c:22021)

32.
G. Prasad, Galois fixed points in the Bruhat-Tits buildings of a reductive group, Bulletin Soc. Math. France 129 (2001), no. 2, 169-174.MR 1871292 (2002j:20088)

33.
G. Rousseau, Immeubles des groupes réductifs sur les corps locaux, Thèse, Paris XI (1977).MR 0491992 (58:11158)

34.
S. Stevens, Semisimple characters for $ p$-adic classical groups, Duke Math. J. 127 (2005), no. 1, 123-173. MR 2126498 (2006a:22017)

35.
M. Tadic, Geometry of dual spaces of reductive groups, J. d'Analyse Math. 51 (1988), 139-181. MR 0963153 (90c:22057)

36.
J.-L. Waldspurger, Homogénéité de certaines distributions sur les groupes $ p$-adiques, Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 25-72.MR 1361755 (98f:22023)

37.
-, La formule de Plancherel pour les groupes $ p$-adiques d'après Harish-Chandra, J. Inst. Math. Jussieu 2 (2003), no. 2, 235-333.MR 1989693 (2004d:22009)

38.
J. K. Yu, Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), no. 3, 579-622. MR 1824988 (2002f:22033)

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Additional Information:

Ju-Lee Kim
Affiliation: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 2-275, Cambridge, Massachusetts 02139
Email: julee@math.uic.edu, julee@math.mit.edu

DOI: 10.1090/S0894-0347-06-00544-3
PII: S 0894-0347(06)00544-3
Keywords: Supercuspidal representations, $\bK$-types, $p$-adic groups
Received by editor(s): February 20, 2004
Posted: October 24, 2006
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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