Invariant generalized functions supported on an orbit



We study the space of invariant generalized functions supported on an orbit of the action of a real algebraic group on a real algebraic manifold. This space is equipped with the Bruhat filtration. We study the generating function of the dimensions of the filtras, and give some methods to compute it. To illustrate our methods we compute those generating functions for the adjoint action of GL3(ℂ). Our main tool is the notion of generalized functions on a real algebraic stack, introduced recently in [Sak16].


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  1. [AG08]
    A. Aizenbud and D. Gourevitch, Schwartz functions on Nash manifolds, International Mathematics Research Noties (2008), Art. ID rnm 155, 37.Google Scholar
  2. [AG09a]
    A. Aizenbud and D. Gourevitch, Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet–Rallis’s theorem, Duke Mathematical Journal 149 (2009), 509–567.Google Scholar
  3. [AG09b]
    A. Aizenbud and D. Gourevitch, Multiplicity one theorem for (GLn+1(R), GLn(R)), Selecta Mathematica 15 (2009), 271–294.Google Scholar
  4. [AG10]
    A. Aizenbud and D. Gourevitch, The de-Rham theorem and Shapiro lemma for Schwartz function on Nash manifolds, Israel Journal of Mathematics 177 (2010), 155–188.Google Scholar
  5. [AG13]
    A. Aizenbud and D. Gourevitch, Smooth transfer of Kloosterman integrals (the Archimedean case), American Journal of Mathematics 135 (2013), 143–182.Google Scholar
  6. [AGRS10]
    A. Aizenbud, D. Gourevitch, S. Rallis and G. Schiffmann, Multiplicity one theorems, Annals of Mathematics 172 (2010), 1407–1434.Google Scholar
  7. [AHR]
    J. Alper, J. Hall and D. Rydh, A luna slice theorem for algebraic stacks, arXiv:1504.06467v1.Google Scholar
  8. [AM69]
    M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, MA–London–Don Mills, ON, 1969.Google Scholar
  9. [Bar03]
    E. M. Baruch, A proof of Kirillov’s conjecture, Annals of Mathematics 158 (2003), 207–252.Google Scholar
  10. [Ber84]
    J. N. Bernstein, P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-Archimedean case), in Lie Group Representations, II (College Park, Md., 1982/1983), Lecture Notes in Mathematics, Vol. 1041, Springer, Berlin, 1984, pp. 50–102.Google Scholar
  11. [GK75]
    I. M. Gelfand and D. A. Kazhdan, Representations of the group GL(n,K) where K is a local field, in Lie Groups and their Representations (Proc. Summer School, Bolyai Jénos Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 95–118.Google Scholar
  12. [HC63]
    Harish-Chandra, Invariant eigendistributions on semisimple Lie groups, Bulletin of the American Mathematical Society 69 (1963), 117–123.Google Scholar
  13. [HC65]
    Harish-Chandra, Invariant eigendistributions on a semisimple Lie group, Transactions of the American Mathematical Society 119 (1965), 457–508.Google Scholar
  14. [Hör90]
    L. Hörmander, The analysis of linear partial differential operators. I, second ed., Grundlehren der Mathematischen Wissenschaften, Vol. 256, Springer-Verlag, Berlin, 1990.Google Scholar
  15. [JR96]
    H. Jacquet and S. Rallis, Uniqueness of linear periods, Compositio Mathematica 102 (1996), 65–123.Google Scholar
  16. [KV96]
    J. A. C. Kolk and V. S. Varadarajan, On the transverse symbol of vectorial distributions and some applications to harmonic analysis, Indagationes Mathematicae 7 (1996), 67–96.Google Scholar
  17. [Sak16]
    Y. Sakellaridis, The Schwartz space of a smooth semi-algebraic stack, Selecta Mathematica 22 (2016), 2401–2490.Google Scholar
  18. [Sha74]
    J. A. Shalika, The multiplicity one theorem for GLn, Annals of Mathematics 100 (1974), 171–193.Google Scholar
  19. [SZ12]
    B. Sun and C.-B. Zhu, Multiplicity one theorems: the Archimedean case, Annals of Mathematics 175 (2012), 23–44.Google Scholar

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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceWeizmann Institute of ScienceRehovotIsrael

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