Duke Mathematical Journal

Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem

Avraham Aizenbud, Dmitry Gourevitch, and Eitan Sayag

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


In the first part of this article, we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field $F$ of characteristic zero. Our main tool is the Luna slice theorem.

In the second part, we apply this technique to symmetric pairs. In particular, we prove that the pairs $({\rm GL}_{n+k}(F),{\rm GL}_n(F) \times {\rm GL}_k(F))$ and $({\rm GL}_n(E),{\rm GL}_n(F))$ are Gelfand pairs for any local field $F$ and its quadratic extension $E$. In the non-Archimedean case, the first result was proved earlier by Jacquet and Rallis [JR] and the second result was proved by Flicker [F].

We also prove that any conjugation-invariant distribution on ${\rm GL}_n(F)$ is invariant with respect to transposition. For non-Archimedean $F$, the latter is a classical theorem of Gelfand and Kazhdan

Article information

Duke Math. J. Volume 149, Number 3 (2009), 509-567.

First available in Project Euclid: 24 August 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 46F10: Operations with distributions
Secondary: 20C99: None of the above, but in this section 20G05: Representation theory 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 14L24: Geometric invariant theory [See also 13A50] 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]


Aizenbud, Avraham; Gourevitch, Dmitry; Sayag, Eitan. Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem. Duke Math. J. 149 (2009), no. 3, 509--567. doi:10.1215/00127094-2009-044. http://projecteuclid.org/euclid.dmj/1251120011.

Export citation


  • A. Aizenbud, A partial analog of integrability theorem for distributions on p-adic spaces and applications, preprint,\arxiv0811.2768v1[math.RT]
  • A. Aizenbud and D. Gourevitch, Schwartz functions on Nash manifolds, Int. Math. Res. Not. 2008, no. 5, Art. ID rnm155,
  • —, De-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds, preprint, to appear in the Israel J. Math.,\arxiv0802.3305v2[math.AG]
  • —, Some regular symmetric pairs, preprint, to appear in Trans. Amer. Math. Soc.,\arxiv0805.2504v1[math.RT]
  • A. Aizenbud, D. Gourevitch, S. Rallis, and G. Schiffmann, Multiplicity one theorems, preprint, to appear in Ann. of Math.,\arxiv0709.4215v1[math.RT]
  • A. Aizenbud, D. Gourevitch, and E. Sayag, $(\rm GL_n+1(F),\rm GL_n(F))$ is a Gelfand pair for any local field $F$, Compos. Math. 144 (2008), 1504--1524.
  • —, $(\rm O(V \oplus F), \rm O(V))$ is a Gelfand pair for any quadratic space $V$ over a local field $F$, Math. Z. 261 (2009), 239--244.
  • A. Aizenbud and E. Sayag, Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of $(\rm GL_2n(\mathbbR), Sp_2n(\mathbbR))$, preprint,\arxiv0810.1853v1[math.RT]
  • E. M. Baruch, A proof of Kirillov's conjecture, Ann. of Math. (2) 158 (2003), 207--252.
  • J. N. Bernstein, ``$P$-invariant distributions on $\rm GL(N)$ and the classification of unitary representations of $\rm GL(N)$ (non-Archimedean case)'' in Lie Group Representations, II (College Park, Md., 1982/1983), Lecture Notes in Math. 1041, Springer, Berlin, 1984, 50--102.
  • I. N. Bernšteĭn [j. n. bernstein] and A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31, no. 3 (1976), 5--70.; English translation in Russian Math. Surveys 31, no. 3 (1976), 1--68.
  • D. Birkes, Orbits of linear algebraic groups, Ann. of Math. (2) 93 (1971), 459--475.
  • J. Bochnak, M. Coste, and M.-F. Roy, Real Algebraic Geometry, Ergeb. Math. Grenzgeb (3) 36, Springer, Berlin, 1998.
  • E. P. H. Bosman and G. Van Dijk, A new class of Gelfand pairs, Geom. Dedicata 50 (1994), 261--282.
  • J.-M. DréZet, ``Luna's slice theorem and applications'' in Algebraic Group Actions and Quotients (Wykno, Poland, 2000), Hindawi, Cairo, 2004, 39--89.
  • Y. Z. Flicker, On distinguished representations, J. Reine Angew. Math. 418 (1991), 139--172.
  • S. S. Gelbart, Weil's Representation and the Spectrum of the Metaplectic Group, Lecture Notes in Math. 530, Springer, Berlin, 1976.
  • I. M. Gelfand and D. A. Kajdan [Kazhdan], ``Representations of the group $\rm GL(n,K)$ where $K$ is a local field'' in Lie Groups and Their Representations (Budapest, 1971), Halsted, New York, 1975, 95--118.
  • B. H. Gross, Some applications of Gelfand pairs to number theory, Bull. Amer. Math. Soc. (N.S.) 24 (1991), 277--301.
  • Harish-Chandra, Admissible Invariant Distributions on Reductive $p$-adic Groups, Univ. Lecture Ser. 16, Amer. Math. Soc., Providence, 1999.
  • N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, 10, Interscience, New York, 1962.
  • H. Jacquet and S. Rallis, Uniqueness of linear periods, Compositio Math. 102 (1996), 65--123.
  • B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753--809.
  • D. Luna, ``Slices étales'' in Sur les groupes algebriques, Bull. Soc. Math. France Mem. Soc. Math. France, 33, Montrouge, 1973, 81--105.
  • —, Sur certaines opérations différentiables des groupes de Lie, Amer. J. Math. 97 (1975), 172--181.
  • D. Mumford, The Red Book of Varieties and Schemes, 2nd ed., Lecture Notes in Math. 1358, Springer, New York, 1999.
  • D. Prasad, Trilinear forms for representations of $\rm GL(2)$ and local $\epsilon$-factors, Compositio Math. 75 (1990), 1--46.
  • C. Rader and S. Rallis, Spherical characters on p-adic symmetric spaces, Amer. J. Math. 118 (1996), 91--178.
  • S. Rallis and G. Schiffmann, Automorphic forms constructed from the Weil representation: Holomorphic case, Amer. J. Math. 100 (1978), 1049--1122.
  • —, Multiplicity one conjectures, preprint,\arxiv0705.2168v1[math.RT]
  • W. Rudin, Functional Analysis, McGraw-Hill Ser. Higher Math., McGraw-Hill, New York, 1973.
  • E. Sayag, $(\rm GL_2n(\mathbbC), Sp_2n(\mathbbC))$ is a Gelfand pair, preprint,\arxiv0805.2625v1[math.RT]
  • —, Regularity of invariant distributions on nice symmetric spaces and Gelfand property of symmetric pairs, preprint.
  • J.-P. Serre, Lie Algebras and Lie Groups, Lecture Notes in Math. 1500, Springer, New York, 1964.
  • J. A. Shalika, The multiplicity one theorem for $\rm GL_n$, Ann. of Math. (2) 100, 1974, 171--193.
  • M. Shiota, Nash Manifolds, Lecture Notes in Math. 1269, Springer, Berlin, 1987.
  • E. G. F. Thomas, ``The theorem of Bochner-Schwartz-Godement for generalised Gelfand pairs'' in Functional Analysis: Surveys and Recent Results III, (Paderborn, Germany, 1983), North-Holland Math. Stud. 90, North-Holland, Amsterdam, 1984.
  • G. Van Dijk, On a class of generalized Gelfand pairs, Math. Z. 193 (1986), 581--593.
  • G. Van Dijk and M. Poel, The irreducible unitary $\rm GL(n-1(\R))$-spherical representations of $\rm SL\, (n,\R)$, Compositio Math. 73 (1990), 1--30.
  • N. R. Wallach, Real Reductive Groups, I, Pure Appl. Math. 132, Academic Press, Boston, 1988.
  • —, Real Reductive Groups, II, Pure Appl. Math. 132-II, Academic Press, Boston, 1992.
  • O. Yakimova, Gelfand pairs, Ph.D. dissertation, Universität Bonn, Bonn, Germany, 2004, Bonner Math. Schriften 374, Universität Bonn, Mathematisches Institut, Bonn, 2005.