Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Representations of metaplectic groups II: Hecke algebra correspondences

Authors: Wee Teck Gan and Gordan Savin
Journal: Represent. Theory 16 (2012), 513-539
MSC (2010): Primary 22E50; Secondary 11F27
Published electronically: October 11, 2012
MathSciNet review: 2982417
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The metaplectic group is defined by its oscillator or Weil representation. Using the types of the Weil representations we define two Hecke algebras that govern two Bernstein's components containing the even and the odd Weil representation, respectively.

References [Enhancements On Off] (What's this?)

  • [Au] A. M. Aubert, Description de la correspondance de Howe en termes de classification de Kazhdan-Lusztig. Invent. Math. 103 (1991) 379-415. MR 1085113 (92h:22036)
  • [Be] J. N. Bernstein (rédigé par P. Deligne), Le ``centre'' de Bernstein. Représentations des groupes réductif sur un corps local (Hermann, Paris, 1984) 1-32. MR 771671 (86e:22028)
  • [Bo] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup. Invent. Math. 35, (1976) 233-259. MR 0444849 (56:3196)
  • [BK] C. J. Bushnell and P. C. Kutzko, Smooth representations of reductive $ p$-adic groups: structure theory via types. Proc. London Math. Soc. (3) 77 (1998) 582-634. MR 1643417 (2000c:22014)
  • [GI] W. T. Gan and A. Ichino, Formal degrees and local theta correspondences. Preprint.
  • [GS] W. T. Gan and G. Savin, Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence. To appear in Comp. Math.
  • [IM] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $ p$-adic Chevalley groups. Pub. Math. IHES, 25 (1965) 5-48. MR 0185016 (32:2486)
  • [KL] D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras. Invent. Math. 87 (1987) 153-215. MR 862716 (88d:11121)
  • [Ku1] S. Kudla, On the local theta correspondence. Invent. Math. 83 (1986) 229-255. MR 818351 (87e:22037)
  • [Ku2] S. Kudla, Notes on the local theta correspondence. Lecture Notes from the European School of Group Theory, available at˜skudla/ssk.research.html
  • [Lu] G. Lusztig, Unipotent representations of simple $ p$-adic groups. Inter. Math. Res. Not. 11 (1995) 517-589. MR 1369407 (98b:22034)
  • [Mc] I. G. Macdonald, The Poincaré Series of a Coxeter Group. Math. Ann. 199 (1972) 161-174. MR 0322069 (48:433)
  • [Ma] H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés. Ann. Sci. École Norm. Sup. Sér. 4, 2 no. 1 (1969) 1-62 . MR 0240214 (39:1566)
  • [Mo] C. C. Moore, Group extensions of $ p$-adic and adelic linear groups. Pub. Math. IHES, 35 (1968) 5-70. MR 0244258 (39:5575)
  • [Ro] J. Rogawski, On modules over the Hecke algebra of a $ p$-adic group. Invent. Math. 79 (1985) 443-465. MR 782228 (86j:22028)
  • [Sa] G. Savin, On unramified representations of covering groups. J. Reine and Angew. Math. 566 (2004) 111-134. MR 2039325 (2005a:22014)
  • [St] R. Steinberg, Lectures on Chevalley groups. Yale University (1968). MR 0466335 (57:6215)
  • [Ti] J. Tits, Reductive groups over local fields. Automorphic forms, representations and $ L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis Ore., 1977), Part 1, pp 29-69, Proc. Sympos. Pure Math., XXXIII, AMS, Providence, R.I., 1979. MR 546588 (80h:20064)
  • [We] A. Weil, Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964) 143-211. MR 0165033 (29:2324)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 22E50, 11F27

Retrieve articles in all journals with MSC (2010): 22E50, 11F27

Additional Information

Wee Teck Gan
Affiliation: Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093 — and — Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076

Gordan Savin
Affiliation: Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112

Received by editor(s): April 28, 2011
Received by editor(s) in revised form: May 9, 2012, and June 6, 2012
Published electronically: October 11, 2012
Additional Notes: The first author was partially supported by NSF grant DMS0801071
The second author was partially supported by DMS 0852429
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society