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Multiplicity one theorems: the Archimedean case. (English) Zbl 1239.22014
Multiplicity theorems on the irreducible representations of classical Lie groups and their subgroups are presented. One denotes by $$G$$ the classical Lie groups $\mathrm {GL}_{n+1}(\mathbb R), \mathrm {GL}_{n+1}(\mathbb C), \mathrm {U}(p,q+1), \mathrm {O}(p,q+1), \mathrm {O}_{n+1}(\mathbb C), \mathrm {SO}(p,q+1), \mathrm {SO}_{n+1}(\mathbb C),$ and by $$G'$$ respectively their subgroups $\mathrm {GL}_n(\mathbb R), \mathrm {GL}_n(\mathbb C), \mathrm {U}(p,q), \mathrm {O}(p,q), \mathrm {O}_n(\mathbb C), \mathrm {SO}(p,q), \mathrm {SO}_n(\mathbb C)$ embedded in $$G$$ in the standard way. Then, it is proven that every irreducible Casselman-Wallach representation of $$G'$$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $$G$$. The main technical result of this paper is: There exists a real algebraic anti-automorphism $$\sigma$$ on $$G$$ preserving $$G'$$ with the following property: every generalized function on $$G$$ which is invariant under the adjoint action of $$G'$$ is automatically $$\sigma$$-invariant. Combined with a version of the Gelfand-Kazhdan criterion, this result implies the above property on the multiplicity of the irreducible Casselman-Wallach representations of the respective classical groups. Similar results are proved for Jacobi groups with their respective subgroups.

##### MSC:
 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods 2.2e+31 Analysis on real and complex Lie groups
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##### References:
 [1] A. Aizenbud and D. Gourevitch, ”Schwartz functions on Nash manifolds,” Int. Math. Res. Not., vol. 2008, iss. 5, p. I, 2008. · Zbl 1161.58002 · doi:10.1093/imrn/rnm155 · arxiv:0704.2891 [2] A. Aizenbud and D. Gourevitch, ”Multiplicity one theorem for $$({ GL}_{n+1}(\mathbb R),{ GL}_n(\mathbb R))$$,” Selecta Math., vol. 15, iss. 2, pp. 271-294, 2009. · Zbl 1185.22006 · doi:10.1007/s00029-009-0544-7 · arxiv:0808.2729 [3] A. Aizenbud, D. Gourevitch, S. Rallis, and G. Schiffmann, ”Multiplicity one theorems,” Ann. of Math., vol. 172, iss. 2, pp. 1407-1434, 2010. · Zbl 1202.22012 · doi:10.4007/annals.2010.172.1413 · pjm.math.berkeley.edu [4] A. Aizenbud, D. Gourevitch, and E. Sayag, ”$$({ GL}_{n+1}(F),{ GL}_n(F))$$ is a Gelfand pair for any local field $$F$$,” Compos. Math., vol. 144, iss. 6, pp. 1504-1524, 2008. · Zbl 1157.22004 · doi:10.1112/S0010437X08003746 · arxiv:0709.1273 [5] A. Aizenbud, D. Gourevitch, and E. Sayag, ”$$({ O}(V\oplus F),{ O}(V))$$ is a Gelfand pair for any quadratic space $$V$$ over a local field $$F$$,” Math. Z., vol. 261, iss. 2, pp. 239-244, 2009. · Zbl 1179.22017 · doi:10.1007/s00209-008-0318-5 · arxiv:0711.1471 [6] A. Aizenbud and D. Gourevitch, ”Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis’s theorem, with an appendix by the authors and E. Sayag,” Duke Math. J., vol. 149, iss. 3, pp. 509-567, 2009. · Zbl 1221.22018 · doi:10.1215/00127094-2009-044 · arxiv:0812.5063 [7] J. Bernstein and B. Krotz, Smooth Fréchet globalizations of Harish-Chandra modules. [8] W. Casselman, ”Canonical extensions of Harish-Chandra modules to representations of $$G$$,” Canad. J. Math., vol. 41, iss. 3, pp. 385-438, 1989. · Zbl 0702.22016 · doi:10.4153/CJM-1989-019-5 [9] D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, New York: Van Nostrand Reinhold Co., 1993. · Zbl 0972.17008 [10] G. van Dijk, ”$$({ U}(p,q), { U}(p-1,q))$$ is a generalized Gelfand pair,” Math. Z., vol. 261, iss. 3, pp. 525-529, 2009. · Zbl 1158.22010 · doi:10.1007/s00209-008-0335-4 [11] G. van Dijk, ”Multiplicity free subgroups of semi-direct products,” Indag. Math., vol. 20, iss. 1, pp. 49-56, 2009. · Zbl 1182.22007 · doi:10.1016/S0019-3577(09)80002-5 [12] W. T. Gan, B. Gross, and D. Prasad, Symplectic local root numbers, central critical $$L$$-values, and restriction problems in the representation theory of classical groups. · Zbl 1280.22019 · arxiv:0909.2999 [13] D. Jiang, B. Sun, and C. Zhu, ”Uniqueness of Bessel models: the Archimedean case,” Geom. Funct. Anal., vol. 20, iss. 3, pp. 690-709, 2010. · Zbl 1200.22008 · doi:10.1007/s00039-010-0077-4 · arxiv:0908.1728 [14] D. Jiang, B. Sun, and C. Zhu, ”Uniqueness of Ginzburg-Rallis models: the Archimedean case,” Trans. Amer. Math. Soc., vol. 363, iss. 5, pp. 2763-2802, 2011. · Zbl 1217.22011 · doi:10.1090/S0002-9947-2010-05285-7 · arxiv:0903.1411 [15] C. Moeglin, M. Vignéras, and J. Waldspurger, Correspondances de Howe sur un Corps $$p$$-Adique, New York: Springer-Verlag, 1987, vol. 1291. · Zbl 0642.22002 · doi:10.1007/BFb0082712 [16] D. Prasad, ”Some applications of seesaw duality to branching laws,” Math. Ann., vol. 304, iss. 1, pp. 1-20, 1996. · Zbl 0838.22005 · doi:10.1007/BF01446282 · eudml:165392 [17] B. Sun, Multiplicity one theorems for Fourier-Jacobi models. · Zbl 1280.22022 · doi:10.1353/ajm.2012.0044 · arxiv:0903.1417 [18] B. Sun, On representations of real Jacobi groups. · Zbl 1241.22010 · doi:10.1007/s11425-011-4333-3 · arxiv:1004.5508 [19] B. Sun and C. -B. Zhu, ”A general form of Gelfand-Kazhdan criterion,” Manuscripta Math., vol. 136, pp. 185-197, 2011. · Zbl 1229.22014 · doi:10.1007/s00229-011-0437-x · arxiv:0903.1409 [20] B. Sun and C. -B. Zhu, Fourier transform and rigidity of certain distributions. · Zbl 1270.42028 · doi:10.1142/S0129167X12501297 · arxiv:1010.2342 [21] M. Shiota, Nash Manifolds, New York: Springer-Verlag, 1987, vol. 1269. · Zbl 0629.58002 · doi:10.1007/BFb0078571 [22] J. -L. Waldspurger, Une variante d’un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann. · Zbl 1308.22008 [23] N. R. Wallach, Real Reductive Groups. I, Boston, MA: Academic Press, 1988, vol. 132. · Zbl 0666.22002 [24] N. R. Wallach, Real Reductive Groups. II, Boston, MA: Academic Press, 1992. · Zbl 0785.22001
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