groups and elliptic representations for unitary groups
Author:
David Goldberg
Journal:
Proc. Amer. Math. Soc. 123 (1995), 12671276
MSC:
Primary 22E35; Secondary 22D30, 22E50
MathSciNet review:
1224616
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Abstract: We determine the reducibility and number of components of any representation of a quasisplit unitary group which is parabolically induced from a discrete series representation. The Rgroups are computed explicitly, in terms of reducibility for maximal parabolics. This gives a description of the elliptic representations.
 [1]
James
Arthur, On elliptic tempered characters, Acta Math.
171 (1993), no. 1, 73–138. MR 1237898
(94i:22038), http://dx.doi.org/10.1007/BF02392767
 [2]
I.
N. Bernstein and A.
V. Zelevinsky, Induced representations of reductive 𝔭adic
groups. I, Ann. Sci. École Norm. Sup. (4) 10
(1977), no. 4, 441–472. MR 0579172
(58 #28310)
 [3]
David
Goldberg, 𝑅groups and elliptic representations for
𝑆𝐿_{𝑛}, Pacific J. Math. 165
(1994), no. 1, 77–92. MR 1285565
(95h:22020)
 [4]
, Reducibility of induced representations for and , Amer. J. Math. 484 (1994), 6595.
 [5]
David
Goldberg, Some results on reducibility for unitary groups and local
Asai 𝐿functions, J. Reine Angew. Math. 448
(1994), 65–95. MR 1266747
(95g:22031), http://dx.doi.org/10.1515/crll.1994.448.65
 [6]
HarishChandra,
Harmonic analysis on reductive 𝑝adic groups, Harmonic
analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI,
Williams Coll., Williamstown, Mass., 1972), Amer. Math. Soc., Providence,
R.I., 1973, pp. 167–192. MR 0340486
(49 #5238)
 [7]
Rebecca
A. Herb, Elliptic representations for
𝑆𝑝(2𝑛) and 𝑆𝑂(𝑛),
Pacific J. Math. 161 (1993), no. 2, 347–358. MR 1242203
(94i:22040)
 [8]
Hervé
Jacquet, Generic representations, Noncommutative harmonic
analysis (Actes Colloq., MarseilleLuminy, 1976), Springer, Berlin, 1977,
pp. 91–101. Lecture Notes in Math., Vol. 587. MR 0499005
(58 #16985)
 [9]
Charles
David Keys, On the decomposition of reducible principal series
representations of 𝑝adic Chevalley groups, Pacific J. Math.
101 (1982), no. 2, 351–388. MR 675406
(84d:22032)
 [10]
C.
David Keys, 𝐿indistinguishability and 𝑅groups for
quasisplit groups: unitary groups in even dimension, Ann. Sci.
École Norm. Sup. (4) 20 (1987), no. 1,
31–64. MR
892141 (88m:22042)
 [11]
A.
W. Knapp and E.
M. Stein, Irreducibility theorems for the principal series,
Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971),
Springer, Berlin, 1972, pp. 197–214. Lecture Notes in Math.,
Vol. 266. MR
0422512 (54 #10499)
 [12]
G. I. Ol'sanskii, Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the genreal linear group over a locally compact division algebra, Math. USSRSb 22 (1974), 217254.
 [13]
Freydoon
Shahidi, The notion of norm and the representation theory of
orthogonal groups, Invent. Math. 119 (1995),
no. 1, 1–36. MR 1309970
(96e:22034), http://dx.doi.org/10.1007/BF01245173
 [14]
Freydoon
Shahidi, On certain 𝐿functions, Amer. J. Math.
103 (1981), no. 2, 297–355. MR 610479
(82i:10030), http://dx.doi.org/10.2307/2374219
 [15]
Allan
J. Silberger, The KnappStein dimension theorem for
𝑝adic groups, Proc. Amer. Math.
Soc. 68 (1978), no. 2, 243–246. MR 0492091
(58 #11245), http://dx.doi.org/10.1090/S00029939197804920915
 [16]
Allan
J. Silberger, Introduction to harmonic analysis on reductive
𝑝adic groups, Mathematical Notes, vol. 23, Princeton
University Press, Princeton, N.J., 1979. Based on lectures by
HarishChandra at the Institute for Advanced Study, 1971–1973. MR 544991
(81m:22025)
 [1]
 J. Arthur, On elliptic temptered characters, Acta Math. 171 (1993), 73138. MR 1237898 (94i:22038)
 [2]
 I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive padic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), 441472. MR 0579172 (58:28310)
 [3]
 D. Goldberg, Rgroups and elliptic representations for , Pacific J. Math. 165 (1994), 7792. MR 1285565 (95h:22020)
 [4]
 , Reducibility of induced representations for and , Amer. J. Math. 484 (1994), 6595.
 [5]
 , Some results on reducibility for unitary groups and local Asai Lfunctions, J. Reine Angew. Math. 448 (1994), 6595. MR 1266747 (95g:22031)
 [6]
 HarishChandra, Harmonic analysis on reductive padic groups, Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, RI, 1973, pp. 167192. MR 0340486 (49:5238)
 [7]
 R. A. Herb, Elliptic representations for and , Pacific J. Math. 161 (1993), 347358. MR 1242203 (94i:22040)
 [8]
 H. Jacquet, Generic representations, Non Commutative Harmonic Analysis, Lecture Notes in Math., vol. 587, SpringerVerlag, New York, Heidelberg, and Berlin, 1977, pp. 91101. MR 0499005 (58:16985)
 [9]
 C. D. Keys, On the decomposition of reducible principal series representations of padic Chevalley groups, Pacific J. Math. 101 (1982), 351388. MR 675406 (84d:22032)
 [10]
 , Lindistinguishability and Rgroups for quasi split groups: unitary groups in even dimension, Ann. Sci. École Norm. Sup. (4) 20 (1987), 3164. MR 892141 (88m:22042)
 [11]
 A. W. Knapp and E. M. Stein, Irreducibility theorems for the principal series, Conference on Harmonic Analysis, Lecture Notes in Math., vol. 266, SpringerVerlag, New York, Heidelberg, and Berlin, 1972, pp. 197214. MR 0422512 (54:10499)
 [12]
 G. I. Ol'sanskii, Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the genreal linear group over a locally compact division algebra, Math. USSRSb 22 (1974), 217254.
 [13]
 F. Shahidi, The notion of norm and the representation theory of orthogonal groups, Invent. Math, (to appear). MR 1309970 (96e:22034)
 [14]
 , On certain Lfunctions, Amer. J. Math. 103 (1981), 297355. MR 610479 (82i:10030)
 [15]
 A. J. Silberger, The KnappStein dimension theorem for padic groups, Proc. Amer. Math. Soc. 68 (1978), 243246; Correction, Proc. Amer. Math. Soc. 76 (1979), 169170. MR 0492091 (58:11245)
 [16]
 , Introduction to harmonic anaysis on reductive padic groups, Math. Notes, vol. 23, Princeton Univ. Press, Princeton, NJ, 1979. MR 544991 (81m:22025)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512246166
PII:
S 00029939(1995)12246166
Article copyright:
© Copyright 1995 American Mathematical Society
