Fourier inversion for unipotent invariant integrals
Author:
Dan Barbasch
Journal:
Trans. Amer. Math. Soc. 249 (1979), 5183
MSC:
Primary 22E30; Secondary 10D40
MathSciNet review:
526310
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Abstract: Consider G a semisimple Lie group and a discrete subgroup such that . An important problem for number theory and representation theory is to find the decomposition of into irreducible representations. Some progress in this direction has been made by J. Arthur and G. Warner by using the Selberg trace formula, which expresses the trace of a subrepresentation of in terms of certain invariant distributions. In particular, measures supported on orbits of unipotent elements of G occur. In order to obtain information about representations it is necessary to expand these distributions into Fourier components using characters of irreducible unitary representations of G. This problem is solved in this paper for real rank . In particular, a relationship between the semisimple orbits and the nilpotent ones is made explicit generalizing an earlier result of R. Rao.
 [1]
Shôrô
Araki, On root systems and an infinitesimal classification of
irreducible symmetric spaces, J. Math. Osaka City Univ.
13 (1962), 1–34. MR 0153782
(27 #3743)
 [2]
D. Barbasch, Fourier inversion formulas of orbital integrals, thesis, University of Illinois at UrbanaChampaign, 1976.
 [3]
Armand
Borel, Properties and linear representations of Chevalley
groups, Seminar on Algebraic Groups and Related Finite Groups (The
Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in
Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 1–55. MR 0258838
(41 #3484)
 [4a]
HarishChandra,
A formula for semisimple Lie groups, Amer. J. Math.
79 (1957), 733–760. MR 0096138
(20 #2633)
 [4b]
HarishChandra,
Invariant distributions on Lie algebras, Amer. J. Math.
86 (1964), 271–309. MR 0161940
(28 #5144)
 [4c]
HarishChandra,
Discrete series for semisimple Lie groups. I. Construction of invariant
eigendistributions, Acta Math. 113 (1965),
241–318. MR 0219665
(36 #2744)
 [4d]
HarishChandra,
Discrete series for semisimple Lie groups. II. Explicit determination
of the characters, Acta Math. 116 (1966),
1–111. MR
0219666 (36 #2745)
 [4e]
HarishChandra,
Two theorems on semisimple Lie groups, Ann. of Math. (2)
83 (1966), 74–128. MR 0194556
(33 #2766)
 [5]
Sigurđur
Helgason, Differential geometry and symmetric spaces, Pure and
Applied Mathematics, Vol. XII, Academic Press, New York, 1962. MR 0145455
(26 #2986)
 [6]
B. Herb, Fourier inversion of invariant integrals on semisimple real Lie groups (preprint).
 [7]
R.
A. Herb and P.
J. Sally Jr., Singular invariant eigendistributions
as characters, Bull. Amer. Math. Soc.
83 (1977), no. 2,
252–254. MR 0480875
(58 #1024), http://dx.doi.org/10.1090/S000299041977142870
 [8a]
Bertram
Kostant, The principal threedimensional subgroup and the Betti
numbers of a complex simple Lie group, Amer. J. Math.
81 (1959), 973–1032. MR 0114875
(22 #5693)
 [8b]
Bertram
Kostant and Stephen
Rallis, On orbits associated with symmetric
spaces, Bull. Amer. Math. Soc. 75 (1969), 879–883. MR 0257284
(41 #1935), http://dx.doi.org/10.1090/S000299041969123372
 [9]
G.
D. Mostow, Some new decomposition theorems for semisimple
groups, Mem. Amer. Math. Soc. 1955 (1955),
no. 14, 31–54. MR 0069829
(16,1087g)
 [10]
S. Osborne and G. Warner, Multiplicities of the integrable discrete series: The case of a nonuniform lattice in an Rrank one semisimple group (preprint).
 [11a]
R. Rao, Results on even nilpotents, unpublished.
 [11b]
, Orbital integrals in reductive Lie groups, Ann. of Math. 96 (1972), 505510.
 [12]
Paul
J. Sally Jr. and Garth
Warner, The Fourier transform on semisimple Lie groups of real rank
one, Acta Math. 131 (1973), 1–26. MR 0450461
(56 #8755)
 [13]
Hans
Samelson, Notes on Lie algebras, Van Nostrand Reinhold
Mathematical Studies, No. 23, Van Nostrand Reinhold Co., New York, 1969. MR 0254112
(40 #7322)
 [14]
Nolan
R. Wallach, Harmonic analysis on homogeneous spaces, Marcel
Dekker Inc., New York, 1973. Pure and Applied Mathematics, No. 19. MR 0498996
(58 #16978)
 [15a]
G. Warner, Harmonic analysis on semisimple Lie groups, vols. 1 and 2, SpringerVerlag, Berlin and New York, 1972.
 [15b]
, The Selberg trace formula for real rank one (preprint).
 [1]
 S. Araki, On root structure and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka Univ. 13 (1962), 134. MR 0153782 (27:3743)
 [2]
 D. Barbasch, Fourier inversion formulas of orbital integrals, thesis, University of Illinois at UrbanaChampaign, 1976.
 [3]
 A. Borel, Seminar on algebraic groups and related finite groups, Lecture Notes in Math., vol. 131, SpringerVerlag, Berlin and New York, 1970. MR 0258838 (41:3484)
 [4a]
 HarishChandra, A formula for semisimple Lie groups, Amer. J. Math. 79 (1957), 733760. MR 0096138 (20:2633)
 [4b]
 , Invariant distributions on Lie algebras, Amer. J. Math. 86 (1964), 271309. MR 0161940 (28:5144)
 [4c]
 , Discrete series for semisimple Lie groups. I, Acta Math. 113 (1965), 241318. MR 0219665 (36:2744)
 [4d]
 , Discrete series for semisimple Lie groups. II, Acta Math. 116 (1966), 1111. MR 0219666 (36:2745)
 [4e]
 , Two theorems on semisimple Lie groups, Ann. of Math. (2) 83 (1966), 74128. MR 0194556 (33:2766)
 [5]
 S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. MR 0145455 (26:2986)
 [6]
 B. Herb, Fourier inversion of invariant integrals on semisimple real Lie groups (preprint).
 [7]
 B. Herb and P. Sally, Singular invariant eigendistributions as characters, Bull. Amer. Math. Soc. 83 (1977), 252254. MR 0480875 (58:1024)
 [8a]
 B. Kostant, The principal threedimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 9731032. MR 0114875 (22:5693)
 [8b]
 B. Kostant and S. Rallis, On orbits associated with symmetric spaces, Bull. Amer. Math. Soc. 75 (1969), 884887. MR 0257284 (41:1935)
 [9]
 G. Mostow, Some new decomposition theorems for semisimple Lie groups, Mem. Amer. Math. Soc., no. 14 (1955), 3154. MR 0069829 (16:1087g)
 [10]
 S. Osborne and G. Warner, Multiplicities of the integrable discrete series: The case of a nonuniform lattice in an Rrank one semisimple group (preprint).
 [11a]
 R. Rao, Results on even nilpotents, unpublished.
 [11b]
 , Orbital integrals in reductive Lie groups, Ann. of Math. 96 (1972), 505510.
 [12]
 P. Sally, Jr. and G. Warner, The Fourier transform on semisimple Lie groups of real rank one, Acta Math. 131 (1973), 126. MR 0450461 (56:8755)
 [13]
 H. Samelson, Notes on Lie algebras, Van Nostrand, Princeton, N. J., 1969. MR 0254112 (40:7322)
 [14]
 N. Wallach, Harmonic analysis on homogeneous spaces, Dekker, New York, 1973. MR 0498996 (58:16978)
 [15a]
 G. Warner, Harmonic analysis on semisimple Lie groups, vols. 1 and 2, SpringerVerlag, Berlin and New York, 1972.
 [15b]
 , The Selberg trace formula for real rank one (preprint).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905263109
PII:
S 00029947(1979)05263109
Article copyright:
© Copyright 1979 American Mathematical Society
