Abstract
1. In these lectures it will be my aim to discuss some aspects of the problem of obtaining an explicit Plancherel formula for a connected real semisimple Lie group with finite center, and the close connection of this problem with the theory of eigenfunction expansions on the group. The central results are those of Harish Chandra, and it is impossible to give anything more than a partial outline of his monumental work that began in the early ' 50' s and has spanned almost three decades.
For a given locally compact group which is separable and unimodular, the fundamental problem is that of decomposing its regular representation into irreducible constituents. If the group is commutative or compact this is quite classical. However, apart from some general existence theorems (see for instance Segal [l]), there is no systematic development of harmonic analysis on general locally compact groups. The category of locally compact groups (even separable and unimodular) is so extensive and the structure of its individual members so varied that it has so far proved impossible to develop analysis on these groups beyond a few general theorems. For Lie groups the situation is much better, and among these the semisimple groups (both real and p-adic) occupy a central position. We know their structure in great detail and are able to use this knowledge in formulating and solving the questions of harmonic analysis in a significant manner.
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References
M. Eguchi [l] Asymptotic exapnsions of Eisenstein integrals and Fourier transform on symmetric spaces, Jour. of Functional Analysis 34 (1979), 167–216.
L. Ehrenpreis and F. I. Mautner [l] Some properties of the Fourier transform on semisimple Lie groups, I, Ann. Math. 61 (1955), 406–439; II, Trans. Amer. Math. Soc. 84 (1957), 1–55; III, Trans. Amer. Math. Soc. 90 (1959), 431–484.
T. J. Enright [l] On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions, and multiplicity formulae, Ann. Math. 110 (1979), 1–82.
T. J. Enright and N. R. Wallach [l] The fundamental series of semisimple Lie algebras and semisimple Lie groups (preprint).
I. M. Gel'fand and M. A. Naǐmark [1] Unitäre Darstellungen der Klassischen Gruppen, Akademic-Verlag, Berlin 1957
S. G. Gindikin and F. I. Karpelevič [l] Plancherel measure for symmetric Riemannian spaces of non positive curvature, Dok. Akad. Nauk. SSSR. 145(1962), 252–255.
Harish Chandra [l] Spherical functions on a semisimple Lie group, I. Amer. Jour. Math. 80 (1958)., 241–310.
Spherical functions on a semisimple Lie group II, Amer. Jour. Math. 80 (1958), 553–613.
Harmonic Analysis on real reductive groups I. The theory of the constant term. Jour, of Functional Analysis 19 (1975), 104–204.
Harmonic Analysis on real reductive groups II. Wave packets in the Schwartz space, Inv. Math. 36 (1976), 1–55.
Harmonic Analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula, Ann. Math. 104 (1976), 117–201.
Discrete series for semisimple Lie groups, I, Acta Math. 113 (1965), 251–318.
Discrete series for semisimple Lie groups, II, Acta Math. 116 (1966), 2–111.
Harmonic Analysis on semisimple Lie groups, Bull. AMS 76 (1970), 529–551.
The Plancherel formula for complex semisimple Lie groups, Trans. AMS 76 (1954), 485–528.
Plancherel formula for the 2×2 real unimodular group, Proc. Nat. Acad. Sci. USA 38 (1952), 337–342.
Some results on differential equations and their applications, Proc. Nat. Acad. Sci. USA 45 (1959), 1763–1764.
Two theorems on semisimple Lie groups, Ann. Math. 83 (1966), 74–128.
A.W. Knapp [1] Commutativity of Intertwining operators II, Bull. AMS 82 (1976), 271–273.
B. Kostant [1] On the existence and irreducibility of certain series of representations, Lie groups and their representations, Edited by I. M. Gel'fand, Halsted Press, 1975.
S. Lang [1] “SL2 (R)”, Addison-Wesley, Reading, Mass., 1975.
K. R. Barthasarathy, R. Ranga Rao, and V. S. Varadarajan [1] Representations of complex semisimple Lie groups and Lie algebras, Ann. Math. 85. (1967), 383–429.
I. E. Segal [1] An extension of Plancherel's formula to separable unimodular groups, Ann. Math. 52 (1950), 272–292.
P. C. Trombi [1] Asymptotic expansions of matrix coefficients: the real rank one case, Jour, of Functional Analysis 30 (1978), 83–105.
Harmonic analysis of CP(G:F) (l ≤p < 2), (preprint).
Invariant harmonic analysis on split rank one groups with applications, (preprint).
P. C. Trombi and V. S. Varadarajan [1] Spherical transforms on semisimple Lie groups, Ann. Math 94 (1971), 246–303.
V. S. Varadarajan [1] Harmonic Analysis on real reductive groups, Lecture Notes in Mathematics #576, Springer Verlag, 1977.
Lie groups, Lie Algebras, and their representations, Prentice Hall, 1974.
Infinitesimal theory of representations of semisimple Lie groups, Lectures given at the Nato Advanced Study Institute at Liege, Belgium on Representations of Lie groups and Harmonic Analysis, 1977.
N. R. Wallach [1] Cyclic vectors and irreducibility for principal series of representations, Trans. AMS 158 (1971), 107–112.
G. Warner [I] Harmonic Analysis on semisimple Lie groups, I, II. Springer Verlag, 1972.
D. P. Zhelebenko [1] The analysis of irreducibility in the class of elementary representations of a complex semisimple Lie group, Math - USSR Izvestra 2 (1968), 105–128.
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Varadarajan, V. (2010). Eigenfunction Expansions On Semisimple Lie Groups. In: Talamanca, A.F. (eds) Harmonic Analysis and Group Representation. C.I.M.E. Summer Schools, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11117-4_6
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