## Duke Mathematical Journal

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

#### Abstract

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

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

Dates
First available in Project Euclid: 24 August 2009

https://projecteuclid.org/euclid.dmj/1251120011

Digital Object Identifier
doi:10.1215/00127094-2009-044

Mathematical Reviews number (MathSciNet)
MR2553879

Zentralblatt MATH identifier
1221.22018

#### Citation

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. https://projecteuclid.org/euclid.dmj/1251120011

#### References

• 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.