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Principal series representations of infinite-dimensional Lie groups, I: Minimal parabolic subgroups

  • Joseph A. WolfEmail author
Chapter
Part of the Progress in Mathematics book series (PM, volume 257)

Abstract

We study the structure of minimal parabolic subgroups of the classical infinite-dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras. This depends on the recently developed structure of parabolic subgroups and subalgebras that are not necessarily direct limits of finite-dimensional parabolics. We then discuss the use of that structure theory for the infinite-dimensional analog of the classical principal series representations. We look at the unitary representation theory of the classical lim-compact groups U(), SO() and Sp() in order to construct the inducing representations, and we indicate some of the analytic considerations in the actual construction of the induced representations.

Keywords

Principle series representation Infinite-dimensional Lie group Minimal parabolic subgroup 

Mathematics Subject Classification

32L25 22E46 32L10 

References

  1. 1.
    A. A. Baranov, Finitary simple Lie algebras, J. Algebra 219 (1999), 299–329.CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    A. A. Baranov and H. Strade, Finitary Lie algebras, J. Algebra 254 (2002), 173–211.CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    E. Dan-Cohen, Borel subalgebras of root-reductive Lie algebras, J. Lie Theory 18 (2008), 215–241.MathSciNetzbMATHGoogle Scholar
  4. 4.
    E. Dan-Cohen and I. Penkov, Parabolic and Levi subalgebras of finitary Lie algebras, Internat. Math. Res. Notices 2010, No. 6, 1062–1101.Google Scholar
  5. 5.
    E. Dan-Cohen and I. Penkov, Levi components of parabolic subalgebras of finitary Lie algebras, Contemporary Math. 557 (2011), 129–149.CrossRefMathSciNetGoogle Scholar
  6. 6.
    E. Dan-Cohen, I. Penkov, and J. A. Wolf, Parabolic subgroups of infinite-dimensional real Lie groups, Contemporary Math. 499 (2009), 47–59.CrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Dawson, G. Ólafsson, and J. A. Wolf, Direct systems of spherical functions and representations, J. Lie Theory 23 (2013), 711–729.MathSciNetzbMATHGoogle Scholar
  8. 8.
    I. Dimitrov and I. Penkov, Weight modules of direct limit Lie algebras, Internat. Math. Res. Notices 1999, No. 5, 223–249.Google Scholar
  9. 9.
    I. Dimitrov and I. Penkov, Borel subalgebras of \(\mathfrak{l}(\infty )\), Resenhas IME-USP 6 (2004), 153–163.MathSciNetzbMATHGoogle Scholar
  10. 10.
    I. Dimitrov and I. Penkov, Locally semisimple and maximal subalgebras of the finitary Lie algebras gl(∞), sl(∞), so(∞), and sp(∞), Journal of Algebra 322 (2009), 2069–2081.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    A. Habib, Direct limits of Zuckerman derived functor modules, J. Lie Theory 11 (2001), 339–353.MathSciNetzbMATHGoogle Scholar
  12. 12.
    A. A. Kirillov, Representations of the infinite-dimensional unitary group, Soviet Math. Dokl. 14 (1973), 1355–1358.zbMATHGoogle Scholar
  13. 13.
    G. W. Mackey, On infinite-dimensional linear spaces, Trans. Amer. Math. Soc. 57 (1945), 155–207.CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    L. Natarajan, Unitary highest weight modules of inductive limit Lie algebras and groups, J. Algebra 167 (1994), 9–28.CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    L. Natarajan, E. Rodríguez-Carrington, and J. A. Wolf, The Bott–Borel–Weil theorem for direct limit groups, Trans. Amer. Math. Soc. 124 (2002), 955–998.Google Scholar
  16. 16.
    G. I. Olshanskii, Unitary representations of infinite-dimensional pairs (G,K) and the formalism of R. Howe, in Representations of Lie Groups and Related Topics, ed. A. Vershik and D. Zhelobenko, Advanced Studies in Contemporary Mathematics 7 (1990), 269–463.Google Scholar
  17. 17.
    I. E. Segal, The structure of a class of representations of the unitary group on a Hilbert space, Proc. Amer. Math. Soc. 8 (1957), 197–203.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    S. Strătilă and D. Voiculescu, Representations of AF–algebras and of the Group U(), Lecture Notes Math. 486, Springer–Verlag, 1975.Google Scholar
  19. 19.
    S. Strătilă and D. Voiculescu, A survey of the representations of the unitary group U(∞), in Spectral Theory, Banach Center Publ., 8, Warsaw, 1982.Google Scholar
  20. 20.
    S. Strătilă and D. Voiculescu, On a class of KMS states for the unitary group U(∞), Math. Ann. 235 (1978), 87–110.CrossRefMathSciNetGoogle Scholar
  21. 21.
    N. Stumme, Automorphisms and conjugacy of compact real forms of the classical infinite-dimensional matrix Lie algebras, Forum Math. 13 (2001), 817–851.CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    D. Voiculescu, Sur les représentations factorielles finies du U(∞) et autres groupes semblables, C. R. Acad. Sci. Paris 279 (1972), 321–323.Google Scholar
  23. 23.
    H. Weyl, The Classical Groups, Their Invariants, and Representations, Princeton Univ. Press, 1946.zbMATHGoogle Scholar
  24. 24.
    J. A. Wolf, Principal series representations of direct limit groups, Compositio Mathematica, 141 (2005), 1504–1530.CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    J. A. Wolf, Principal series representations of direct limit Lie groups, in Mathematisches Forschungsinstitut Oberwolfach Report 51/210, Infinite-dimensional Lie Theory (2010), 2999–3003.Google Scholar
  26. 26.
    J. A. Wolf, Principal series representations of infinite-dimensional Lie groups, II: Construction of induced representations, Contemporary Mathematics, Vol. 598 (2013), 257–280.CrossRefGoogle Scholar
  27. 27.
    J. A. Wolf, Principal series representations of infinite-dimensional Lie groups, III: Function theory on symmetric spaces. In preparation.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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