Kobayashi, Toshiyuki \(L^ p\)-analysis on homogeneous manifolds of reductive type and representation theory. (English) Zbl 0883.22015 Proc. Japan Acad., Ser. A 73, No. 4, 62-66 (1997). \(L^p\)-analysis on homogeneous manifolds of reductive type and representation theory are discussed. The main results are: Let \(G\) be a real reductive linear Lie group, \(K\) a maximal compact subgroup of \(G\), and \(\theta\) the corresponding Cartan involution, \(H\) a closed \(\theta\)-stable subgroup of \(G\) with finitely many connected components. There are four sections in this paper: (1) Invariant measures on homogeneous manifolds of reductive type. (2) Irreducible representations in \(L^p(G/H)\). (3) Holomorphic discrete series representations. (4) Some valuable examples are given. Reviewer: Su Weiyi (Nanjing) Cited in 1 Document MSC: 22E46 Semisimple Lie groups and their representations 43A85 Harmonic analysis on homogeneous spaces 22E15 General properties and structure of real Lie groups 22E30 Analysis on real and complex Lie groups Keywords:invariant measures; irreducible representations; homogeneous manifolds; real reductive linear Lie group PDFBibTeX XMLCite \textit{T. Kobayashi}, Proc. Japan Acad., Ser. A 73, No. 4, 62--66 (1997; Zbl 0883.22015) Full Text: DOI References: [1] M. Flensted-Jensen : Discrete series for semisim-ple symmetric spaces. Ann. of Math., 111, 253-311 (1980). JSTOR: · Zbl 0462.22006 [2] M. Flensted-Jensen: Analysis on Non-Riemannian Symmetric Spaces. Conf. Board, 61, A. M. S. Providence R.I., ISBN 0-8218-0711-0 (1986). · Zbl 0589.43008 [3] A. Knapp and D. Vogan : Cohomological Induction and Unitary Representations. Princeton University Press, Princeton, ISBN 0-691-03756-6 (1995). · Zbl 0863.22011 [4] T. Kobayashi: Singular Unitary Representations and Discrete Series for Indefinite Stiefel Manifolds U(p, q ;F) /U(p - m, q;F). vol. 462, Memoirs of A. M. S., Providence R. I., ISBN 0-8218-2524-0 (1992). · Zbl 0752.22007 [5] T. Kobayashi: Discrete decomposability of the restriction of Aq(/D with respect to reductive subgroups and its applications. Invent. Math. 117, 181-205 (1994); Part II (preprint); Part III (to appear in Invent Math). · Zbl 0826.22015 [6] T. Kobayashi: The Restriction of A (X) to reductive subgroups. I, II. Proc. Japan Acad., 69A, 262-267 (1993); 71, 24-26 (1995). · Zbl 0826.22014 [7] T. Kobayashi: Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory. Sugaku Exposition of Math. Soc. Japan, 46, 124-143 (1994); English translation, Sugaku Exposition of Amer. Math. Soc. Translation Series, ser. 2 · Zbl 0895.22006 [8] T. Kobayashi: Invariant measure on homogeneous manifolds of reductive type. Jour, reine angew. Math, (to appear). · Zbl 0881.22013 [9] T. Kobayashi: Discrete series representations for the orbit spaces arizing from two involutions of real reductive Lie groups. J. Funct. Anal, (to appear). · Zbl 0937.22008 [10] T. Matsuki: Double coset decompositions of algebraic groups arising from two involutions I. J. Algebra, 175, 865-925 (1995). · Zbl 0831.22002 [11] T. Matsuki: Double coset decompositions of reductive Lie groups arising from two involutions (1995) (preprint). · Zbl 0887.22009 [12] T. Matsuki and T. Oshima: A description of discrete series for semisimple symmetric spaces. Advanced Studies in Pure Math, 4, 331-390 (1984). · Zbl 0577.22012 [13] G. Olafsson and B. 0rsted : The holomorphic discrete series of an affine symmetric space. I, J. Funct. Anal., 81, 126-159 (1988). · Zbl 0678.22008 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.