×

zbMATH — the first resource for mathematics

Invariant analytic domains in complex semisimple groups. (English) Zbl 0867.22004
Let \(H_\mathbb{R}\) be a real form of a complex, connected and simply connected, semisimple algebraic group \(H\) with complex conjugation \(\vartheta\). Let \(\Omega\) be the open dense subset of elements \(x\in H\) such that \(x\vartheta (x^{-1})\) is regular semisimple. The paper provides a study of the connected components of \(\Omega\), which are complex analytic domains, and cross-sections of the \(H_\mathbb{R} \times H_\mathbb{R}\) action on these components. This study is motivated by the Gelfand-Gindikin program to construct explicit unitary representations of \(H_\mathbb{R}\) related to complex analytic objects.
Reviewer: G.Roos (Poitiers)

MSC:
22E10 General properties and structure of complex Lie groups
22E46 Semisimple Lie groups and their representations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. Borel,Linear Algebraic Groups, Second Enlarged Edition, Springer-Verlag, New York, 1991. · Zbl 0726.20030
[2] R. J. Bremigan,Quotients for algebraic group actions over non-algebraically closed fields, J. reine angew. Math.453 (1994), 21–47. · Zbl 0808.14040 · doi:10.1515/crll.1994.453.21
[3] T. Bröcker & T. tom Dieck,Representations of Compact Lie Groups, Springer-Verlag, New York, 1985. · Zbl 0581.22009
[4] I. M. Gel’fand & S. G. Gindikin,Complex manifolds whose skeletons are semisimple real Lie groups, and analytic discrete series of representations, Funkcional Anal. i. Priložen11 (1977), 19–27, English translation in Funct. Anal. Appl. (1978), 258–265.
[5] J. Hilbert & K.-H. Neeb,Lie Semigroups and Their Applications, Lecture Notes in Mathematics, Vol. 1552, Springer-Verlag, Berlin-Heidelberg-New York, 1993.
[6] J. E. Humphreys,Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972. · Zbl 0254.17004
[7] –,Linear Algebraic Groups, Corrected Second Printing, Springer-Verlag, New York, 1981.
[8] –,Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990. · Zbl 0725.20028
[9] –,Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs, Vol. 43, American Mathematical Society, Providence, Rhode Island, 1995. · Zbl 0834.20048
[10] J. D. Lawson,Semigroups of Ol’shanskiî type, Semigroups in Algebra, Geometry and Analysis, Walter de Gruyter, Berlin, 1995, pp. 121–157. · Zbl 0842.22003
[11] T. Matsuki,Double coset decompositions of algebraic groups arising from two involutions I, J. Algebra175 (1995), 865–925. · Zbl 0831.22002 · doi:10.1006/jabr.1995.1218
[12] T. Matsuki,Double coset decompositions of reductive Lie groups arising from two involutions, preprint. · Zbl 0887.22009
[13] G. I. Ol’shanskii,Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funct. Anal. Appl.15 (1982), 275–285. · Zbl 0503.22011 · doi:10.1007/BF01106156
[14] T. Oshima & T. Matsuki,Orbits on affine symmetric spaces under the action of the isotropy subgroups, J. Math. Soc. Japan32 (1980), 399–414. · Zbl 0451.53039 · doi:10.2969/jmsj/03220399
[15] R. W. Richardson & P. J. Slodowy,Minimum vectors for real reductive algebraic groups, J. London Math. Soc. (2)42 (1990), 409–429. · Zbl 0675.14020 · doi:10.1112/jlms/s2-42.3.409
[16] L. P. Rothschild,Orbits in a real reductive Lie algebra, Trans. Amer. Math. Soc.168 (1972), 403–421. · Zbl 0222.17009 · doi:10.1090/S0002-9947-1972-0349778-3
[17] K. H. Hofmann, J. D. Lawson, E. B. Vinberg (eds.),Semigroups in Algebra, Geometry, and Analysis, De Gruyter Expositions in Mathematics, Vol. 20, Walter de Gruyter, Berlin, 1995. · Zbl 0829.00020
[18] I. Satake,Classification Theory of Semi-Simple Algebraic Groups, Marcel Dekker, New York, 1971. · Zbl 0226.20037
[19] T. A. Springer,Linear Algebraic Groups, Birkhäuser, Boston, 1981. · Zbl 0453.14022
[20] R. Steinberg,Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc.80 (1968). · Zbl 0164.02902
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.