Fourier inversion for unipotent invariant integrals

Author:
Dan Barbasch

Journal:
Trans. Amer. Math. Soc. **249** (1979), 51-83

MSC:
Primary 22E30; Secondary 10D40

MathSciNet review:
526310

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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****[2]**D. Barbasch,*Fourier inversion formulas of orbital integrals*, thesis, University of Illinois at Urbana-Champaign, 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****[4a]**Harish-Chandra,*A formula for semisimple Lie groups*, Amer. J. Math.**79**(1957), 733–760. MR**0096138****[4b]**Harish-Chandra,*Invariant distributions on Lie algebras*, Amer. J. Math.**86**(1964), 271–309. MR**0161940****[4c]**Harish-Chandra,*Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions*, Acta Math.**113**(1965), 241–318. MR**0219665****[4d]**Harish-Chandra,*Discrete series for semisimple Lie groups. II. Explicit determination of the characters*, Acta Math.**116**(1966), 1–111. MR**0219666****[4e]**Harish-Chandra,*Two theorems on semi-simple Lie groups*, Ann. of Math. (2)**83**(1966), 74–128. MR**0194556****[5]**Sigurđur Helgason,*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455****[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**, 10.1090/S0002-9904-1977-14287-0**[8a]**Bertram Kostant,*The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group*, Amer. J. Math.**81**(1959), 973–1032. MR**0114875****[8b]**Bertram Kostant and Stephen Rallis,*On orbits associated with symmetric spaces*, Bull. Amer. Math. Soc.**75**(1969), 879–883. MR**0257284**, 10.1090/S0002-9904-1969-12337-2**[9]**G. D. Mostow,*Some new decomposition theorems for semi-simple groups*, Mem. Amer. Math. Soc.**No. 14**(1955), 31–54. MR**0069829****[10]**S. Osborne and G. Warner,*Multiplicities of the integrable discrete series: The case of a non-uniform lattice in an***R**-*rank one semi-simple group*(preprint).**[11a]**R. Rao,*Results on even nilpotents*, unpublished.**[11b]**-,*Orbital integrals in reductive Lie groups*, Ann. of Math.**96**(1972), 505-510.**[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****[13]**Hans Samelson,*Notes on Lie algebras*, Van Nostrand Reinhold Mathematical Studies, No. 23, Van Nostrand Reinhold Co., New York- London-Melbourne, 1969. MR**0254112****[14]**Nolan R. Wallach,*Harmonic analysis on homogeneous spaces*, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 19. MR**0498996****[15a]**G. Warner,*Harmonic analysis on semisimple Lie groups,*vols. 1 and 2, Springer-Verlag, Berlin and New York, 1972.**[15b]**-,*The Selberg trace formula for real rank one*(preprint).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
22E30,
10D40

Retrieve articles in all journals with MSC: 22E30, 10D40

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1979-0526310-9

Article copyright:
© Copyright 1979
American Mathematical Society