The unitary -spherical dual for split -adic groups of type

Author:
Dan Ciubotaru

Journal:
Represent. Theory **9** (2005), 94-137

MSC (2000):
Primary 22E50

DOI:
https://doi.org/10.1090/S1088-4165-05-00206-2

Published electronically:
February 1, 2005

MathSciNet review:
2123126

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is known that the determination of the Iwahori-spherical unitary dual for -adic groups can be reduced to the classification of unitary representations with real infinitesimal character for the associated Hecke algebras. In this setting, I determine the Iwahori-spherical unitary dual for split groups of type .

**[A]**D. Alvis*Induce/Restrict matrices for exceptional Weyl groups*, preprint.**[B1]**Dan Barbasch,*The spherical dual for 𝑝-adic groups*, Geometry and representation theory of real and 𝑝-adic groups (Córdoba, 1995) Progr. Math., vol. 158, Birkhäuser Boston, Boston, MA, 1998, pp. 1–19. MR**1486132**, https://doi.org/10.1007/s10107-015-0911-4**[B2]**-*Unitary spherical spectrum for split classical groups*(to appear).**[B3]**Dan Barbasch,*Unipotent representations for real reductive groups*, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 769–777. MR**1159263****[BM1]**Dan Barbasch and Allen Moy,*A unitarity criterion for 𝑝-adic groups*, Invent. Math.**98**(1989), no. 1, 19–37. MR**1010153**, https://doi.org/10.1007/BF01388842**[BM2]**Dan Barbasch and Allen Moy,*Reduction to real infinitesimal character in affine Hecke algebras*, J. Amer. Math. Soc.**6**(1993), no. 3, 611–635. MR**1186959**, https://doi.org/10.1090/S0894-0347-1993-1186959-0**[BM3]**Dan Barbasch and Allen Moy,*Unitary spherical spectrum for 𝑝-adic classical groups*, Acta Appl. Math.**44**(1996), no. 1-2, 3–37. Representations of Lie groups, Lie algebras and their quantum analogues. MR**1407038**, https://doi.org/10.1007/BF00116514**[BW]**A. Borel, N. Wallach*Continuous cohomology, discrete subgroups and representations of reductive groups*, volume 94 of Annals of Mathematics Studies, Princeton University Press, Princeton, 1980. MR**0554917 (83c:22018)****[Ca]**R. Carter*Finite groups of Lie type*, Wiley-Interscience, New York, 1985. MR**0794307 (87d:20060)****[K]**Takeshi Kondo,*The characters of the Weyl group of type 𝐹₄*, J. Fac. Sci. Univ. Tokyo Sect. I**11**(1965), 145–153 (1965). MR**0185018****[KZ]**A. Knapp, G. Zuckerman*Classification theorems for representations of semisimple Lie groups*, in*Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976)*, Lecture Notes in Math., Vol. 587, Springer, 1977, pp. 138-159. MR**476923 (57 #16474)****[Kn]**A. Knapp*Representation theory of semisimple groups: an overview based on examples*, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1986. MR**0855239 (87j:22022)****[KL]**D. Kazhdan, G. Lusztig*Proof of the Deligne-Langlands conjecture for Hecke algebras*, Invent. Math. 87, 1987, pp. 153-215. MR**0862716 (88d:11121)****[L1]**G. Lusztig*Affine Hecke algebras and their graded version*, Jour. AMS 2, 1989, pp. 599-635. MR**0991016 (90e:16049)****[L2]**-*Intersection cohomology complexes on a reductive group*, Invent. Math. 75, 1984, pp. 205-272. MR**0732546 (86d:20050)****[L3]**-*Cuspidal local systems and graded algebras I*, Publ. Math. IHES 67, 1988, pp. 145-202. MR**0972345 (90e:22029)****[L4]**George Lusztig,*Cuspidal local systems and graded Hecke algebras. II*, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 217–275. With errata for Part I [Inst. Hautes Études Sci. Publ. Math. No. 67 (1988), 145–202; MR0972345 (90e:22029)]. MR**1357201**, https://doi.org/10.1090/S1088-4165-02-00172-3**[L5]**-*Characters of reductive groups over finite fields*, Annals of Math. Studies, vol. 107, Princeton University Press. MR**0742472 (86j:20038)****[M]**Goran Muić,*The unitary dual of 𝑝-adic 𝐺₂*, Duke Math. J.**90**(1997), no. 3, 465–493. MR**1480543**, https://doi.org/10.1215/S0012-7094-97-09012-8**[T]**M. Tadic*Classification of unitary representations in irreducible representations of the general linear group (nonarchimedean case)*, Ann. Scient. Ec. Norm. Sup. 19 (1986), pp. 335-382. MR**0870688 (88b:22021)****[V1]**David A. Vogan Jr.,*The unitary dual of 𝐺₂*, Invent. Math.**116**(1994), no. 1-3, 677–791. MR**1253210**, https://doi.org/10.1007/BF01231578**[V2]**-*Computing the unitary dual*, notes at atlas.math.umd.edu/papers

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2000):
22E50

Retrieve articles in all journals with MSC (2000): 22E50

Additional Information

**Dan Ciubotaru**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853

Address at time of publication:
Massachusetts Institute of Technology, Department of Mathematics, Room 2-180, Cambridge, Massachusetts 02139

Email:
ciubo@math.mit.edu

DOI:
https://doi.org/10.1090/S1088-4165-05-00206-2

Received by editor(s):
August 21, 2003

Received by editor(s) in revised form:
September 21, 2004

Published electronically:
February 1, 2005

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.