Duke Mathematical Journal

Wave front sets of reductive Lie group representations

Benjamin Harris, Hongyu He, and Gestur Ólafsson

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

If G is a Lie group, HG is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξig to be in the wave front set of IndHGτ. In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a real, reductive algebraic group and π is a unitary representation of G that is weakly contained in the regular representation, then the authors give a geometric description of WF(π) in terms of the direct integral decomposition of π into irreducibles. Special cases of this result were previously obtained by Kashiwara–Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.

Article information

Source
Duke Math. J., Volume 165, Number 5 (2016), 793-846.

Dates
Received: 9 November 2013
Revised: 24 March 2015
First available in Project Euclid: 14 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1452780579

Digital Object Identifier
doi:10.1215/00127094-3167168

Mathematical Reviews number (MathSciNet)
MR3482333

Zentralblatt MATH identifier
1341.22008

Subjects
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 43A85: Analysis on homogeneous spaces

Keywords
wave front set singular spectrum analytic wave front set reductive Lie group induced representation tempered representation branching problem discrete series reductive homogeneous space

Citation

Harris, Benjamin; He, Hongyu; Ólafsson, Gestur. Wave front sets of reductive Lie group representations. Duke Math. J. 165 (2016), no. 5, 793--846. doi:10.1215/00127094-3167168. https://projecteuclid.org/euclid.dmj/1452780579


Export citation

References

  • [1] J. Adams, M. van Leuwen, P. Trapa, and D. A. Vogan, Jr., Unitary representations of real reductive groups, preprint, arXiv:1212.2192v4 [math.RT].
  • [2] D. Barbasch and D. A. Vogan, Jr., The local structure of characters, J. Funct. Anal. 37 (1980), 27–55.
  • [3] Y. Benoist and T. Kobayashi, Temperedness of reductive homogeneous spaces, to appear in J. Eur. Math. Soc. (JEMS), preprint, arXiv:1211.1203v2 [math.RT].
  • [4] J. M. Bony, “Équivalence des diverses notions de spectre singulier analytique” in Séminaire Goulaouic-Schwartz (1976/1977), Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 3, Centre Math., École Polytech., Palaiseau, 1977, 1–12.
  • [5] A. Cannas da Silva, Lecture on Symplectic Geometry, Lecture Notes in Math. 1764, Springer, Berlin, 2001.
  • [6] A. Córdoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3 (1978), 979–1005.
  • [7] J. Dixmier, $C^{*}$-Algebras, North-Holland Math. Library 15, North-Holland, Amsterdam, 1977.
  • [8] M. Duflo, Fundamental series representations of a semisimple Lie group, Funct. Anal. Appl. 4 (1970), 122–126.
  • [9] G. B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, Princeton Univ. Press, Princeton, 1989.
  • [10] F. G. Friedlander, Introduction to the Theory of Distributions, with additional material by M. Joshi, Cambridge Univ. Press, Cambridge, 1998.
  • [11] H. Fujiwara, Sur les restrictions des représentations unitaires des groupes de Lie résolubles exponentiels, Invent. Math. 104 (1991), 647–654.
  • [12] S. Hansen, J. Hilgert, and S. Keliny, Asymptotic $K$-support and restrictions of representations, Represent. Theory 13 (2009), 460–469.
  • [13] Harish-Chandra, Representations of semisimple Lie groups, III, Trans. Amer. Math. Soc. 76, no. 2 (1954), 234–253.
  • [14] Harish-Chandra, Fourier transforms on a semisimple Lie algebra, I, Amer. J. Math. 79 (1957), 193–257.
  • [15] Harish-Chandra, Spherical functions on a semisimple Lie group, I, Amer. J. Math. 80 (1958), 241–310.
  • [16] Harish-Chandra, Spherical functions on a semisimple Lie group, II, Amer. J. Math. 80 (1958), 553–613.
  • [17] Harish-Chandra, Some results on an invariant integral on a semisimple Lie algebra, Ann. of Math. (2) 80 (1964), 551–593.
  • [18] Harish-Chandra, Discrete series for semisimple Lie groups, I: Construction of invariant eigendistributions, Acta Math. 113 (1965), 241–318.
  • [19] H. He, Generalized matrix coefficients for unitary representations, J. Ramanujan Math. Soc. 29 (2014), 253–272.
  • [20] H. Hecht and W. Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), 129–154.
  • [21] S. Helgason, A duality for symmetric spaces with applications to group representations, Adv. Math. 5 (1970), 1–154.
  • [22] S. Helgason, “Functions on symmetric spaces” in Harmonic Analysis on Homogeneous Spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972), Amer. Math. Soc., Providence, 1973, 101–146.
  • [23] L. Hörmander, Fourier integral operators, I, Acta Math. 127 (1971), 79–183.
  • [24] L. Hörmander, The Analysis of Linear Partial Differential Operators, I, Grundlehren Math. Wiss. 256, Springer, Berlin, 1983.
  • [25] L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Grundlehren Math. Wiss. 274, Springer, Berlin, 1985.
  • [26] R. Howe, “Wave front sets of representations of Lie groups” in Automorphic Forms, Representation Theory and Arithmetic (Bombay 1979), Tata Inst. Fund. Res. Stud. Math. 10, Tata Inst. Fund. Res., Bombay, 1981, 117–140.
  • [27] D. Iagolnitzer, “Appendix: Microlocal essential support of a distribution and decomposition theorems—an introduction” in Hyperfunctions and Theoretical Physics, Lecture Notes in Math. 449, Springer, Berlin, 1975, 121–132.
  • [28] M. Kashiwara and M. Vergne, “$K$-types and singular spectrum” in Noncommutative Harmonic Analysis (Proc. Third Colloq., Marseilles-Luminy, 1978), Lecture Notes in Math. 728, Springer, Berlin, 1979, 177–200.
  • [29] A. W. Knapp, Representation Theory of Semisimple Groups, Princeton Math. Ser. 36, Princeton Univ. Press, Princeton, 1986.
  • [30] A. W. Knapp and G. J. Zuckerman, Classification of irreducible tempered representations of semisimple groups, Ann. of Math. (2) 118 (1982), 389–455.
  • [31] T. Kobayashi, Discrete decomposability of the restriction of $A_{{q}}(\lambda)$ with respect to reductive subgroups, II: Micro-local analysis and asymptotic K-support, Ann. of Math. (2) 147 (1998), 709–729.
  • [32] T. Kobayashi, Discrete decomposability of the restriction of $A_{{q}}(\lambda)$ with respect to reductive subgroups, III: Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), 229–256.
  • [33] T. Kobayashi, Discrete series representations for the orbit space arising from two involutions of real reductive Lie groups, J. Funct. Anal. 152 (1998), 100–135.
  • [34] T. Kobayashi, “Branching problems of Zuckerman derived functor modules” in Representation Theory and Mathematical Physics, Contemp. Math. 557, Amer. Math. Soc., Providence, 2011, 23–40.
  • [35] B. Kostant, On the conjugacy of real Cartan subalgebras, I, Proc. Natl. Acad. Sci. USA 41 (1955), 967–970.
  • [36] R. P. Langlands, “On the classification of irreducible representations of real algebraic groups” in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Math. Surveys Monogr. 31, Amer. Math. Soc., Providence, 1989, 101–170.
  • [37] M. Morimoto, An Introduction to Sato’s Hyperfunctions, Transl. Math. Monogr. 129, Amer. Math. Soc., Providence, 1993.
  • [38] G. Ólafsson and B. Ørsted, “Generalizations of the Bargmann transform” in Lie Theory and Its Applications to Physics (Clausthal, 1995), World Sci., River Edge, N.J., 1996, 3–14.
  • [39] G. Ólafsson and H. Schlichtkrull, “Representation theory, Radon transform and the heat equation on a Riemannian symmetric space” in Group Representations, Ergodic Theory, and Mathematical Physics, Contemp. Math. 449, Amer. Math. Soc., Providence, 2008, 315–344.
  • [40] B. Ørsted and J. Vargas, Restriction of square integrable representations: discrete spectrum, Duke Math. J. 123 (2004), 609–633.
  • [41] J. Repka, Tensor products of holomorphic discrete series representations, Canad. J. Math. 31 (1979), 836–844.
  • [42] W. Rossmann, Kirillov’s character formula for reductive Lie groups, Invent. Math. 48 (1978), 207–220.
  • [43] W. Rossmann, Limit characters of reductive Lie groups, Invent. Math. 61 (1980), 53–66.
  • [44] W. Rossmann, Limit orbits in reductive Lie algebras, Duke Math. J. 49 (1982), 215–229.
  • [45] W. Rossmann, Picard-Lefschetz theory and characters of a semisimple Lie group, Invent. Math. 121 (1995), 579–611.
  • [46] M. Sato, “Regularity of hyperfunction solutions of partial differential equations” in Actes du Congrès International des Mathématiciens (Nice, 1970), II, Gauthier-Villars, Paris, 1971, 785–794.
  • [47] M. Sato, T. Kawai, and M. Kashiwara, “Microfunctions and pseudo-differential equations” in Hyperfunctions and Pseudo-Differential Equations (Proc. Conf., Katata, 1971), Lecture Notes in Math. 287, Springer, Berlin, 1973, 265–529.
  • [48] W. Schmid and K. Vilonen, Characteristic cycles and wave front cycles of representations of reductive Lie groups, Ann. of Math. (2) 151 (2000), 1071–1118.
  • [49] M. Sugiura, Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, J. Math. Soc. Japan 11 (1959), 374–434.
  • [50] D. A. Vogan, Jr., Representations of Real Reductive Lie Groups, Progr. Math. 15, Birkhäuser, Boston, 1981.
  • [51] D. A. Vogan, Jr., “Associated varieties and unipotent representations” in Harmonic Analysis on Reductive Groups (Brunswick, ME, 1989), Progr. Math. 101, Birkhäuser, Boston, 1991, 315–388.