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Multiplicity one theorems. (English) Zbl 1202.22012

Consider the standard upper left hand corner embedding of \(\mathrm{GL}_n(F)\) in \(\mathrm{GL}_{n+1}(F)\) where \(F\) is a non-Archimedean local field of characteristic \(0\). The main result is the following “multiplicity \(\leq 1\)” theorem and its analogues for orthogonal and unitary groups:
If \(\pi_n, \pi_{n+1}\) are irreducible, admissible representations of \(\mathrm{GL}_n(F), \mathrm{GL}_{n+1}(F)\) respectively, then \(\dim (\operatorname{Hom}_{\mathrm{GL}_n(F)}(\pi_{n+1}|\mathrm{GL}_n(F), \pi_n) ) \leq 1\).
Analogues of this theorem for Archimedean \(F\) were obtained in special cases earlier. “Multiplicity \(\leq 1\)” theorems have several applications to the study of automorphic \(L\)-functions. But a more difficult question is to find when the multiplicity is 1. Recently, the above theorem has been used to deduce the analogous result when \(F\) has positive characteristic.
The transposition map on \(\mathrm{GL}_{n+1}(F)\) is an anti-automorphism of order \(2\) which leaves \(\mathrm{GL}_{n}(F)\) stable. The above theorem is deduced from: “Consider the (adjoint) action of \(\mathrm{GL}_n(F) \times \mathrm{GL}_n(F)\) on \(\mathrm{GL}_{n+1}(F)\) given by \((g_1,g_2)h = g_1hg_2^{-1}\). Then, any invariant distribution on \(\mathrm{GL}_{n+1}(F)\) with respect to this action is also invariant with respect to transposition.”
Employing an old result due to Bernstein, the above theorem also provides an independent proof of Kirillov’s conjecture for non-Archimedean fields of characteristic \(0\). The analogues of the two stated theorems are proved for unitary groups also. Recently, Waldspurger has adapted the proofs to include special orthogonal groups. The proof of the above theorem uses the powerful Bernstein localization principle and a variant of Frobenius reciprocity from 1984. The authors also point out that there is still no simple explanation why the invariant distributions always turn out to be symmetric.

MSC:

22E35 Analysis on \(p\)-adic Lie groups
20G25 Linear algebraic groups over local fields and their integers
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References:

[1] A. Aizenbud and D. Gourevitch, A proof of the multiplicity one conjecture for \(\mathrm{GL}(n)\) in \(\mathrm{GL}(n + 1)\). · Zbl 1185.22006
[2] A. Aizenbud, N. Avni, and D. Gourevitch, Spherical pairs over close local fields. · Zbl 1257.22015
[3] 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
[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
[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
[6] 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, New York: Springer-Verlag, 1984, vol. 1041, pp. 50-102. · Zbl 0541.22009
[7] I. N. Bernvsteuin and A. V. Zelevinskiui, ”Representations of the group \(\mathrm{GL}(n,F),\) where \(F\) is a local non-Archimedean field,” Uspehi Mat. Nauk, vol. 10, iss. 3(189), pp. 5-70, 1976. · Zbl 0348.43007 · doi:10.1070/RM1976v031n03ABEH001532
[8] W. T. Gan, B. H. 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
[9] I. M. Gel\('\)fand and D. A. Kajdan, ”Representations of the group \({ GL}(n,K)\) where \(K\) is a local field,” in Lie Groups and Their Representations, New York: Halsted, 1975, pp. 95-118. · Zbl 0348.22011
[10] B. H. Gross and D. Prasad, ”On the decomposition of a representation of \({ SO}_n\) when restricted to \({ SO}_{n-1}\),” Canad. J. Math., vol. 44, iss. 5, pp. 974-1002, 1992. · Zbl 0787.22018 · doi:10.4153/CJM-1992-060-8
[11] D. Ginzburg, I. Piatetski-Shapiro, and S. Rallis, ”\(L\) functions for the orthogonal group,” Mem. Amer. Math. Soc., vol. 128, iss. 611, p. viii, 1997. · Zbl 0884.11022
[12] B. H. Gross and M. Reeder, ”From Laplace to Langlands via representations of orthogonal groups,” Bull. Amer. Math. Soc., vol. 43, iss. 2, pp. 163-205, 2006. · Zbl 1159.11047 · doi:10.1090/S0273-0979-06-01100-1
[13] J. Hakim, ”Supercuspidal Gelfand pairs,” J. Number Theory, vol. 100, iss. 2, pp. 251-269, 2003. · Zbl 1016.22010 · doi:10.1016/S0022-314X(02)00131-2
[14] Harish-Chandra, Admissible Invariant Distributions on Reductive \(p\)-adic Groups, Providence, RI: Amer. Math. Soc., 1999, vol. 16. · Zbl 0928.22017
[15] H. Jacquet and S. Rallis, ”Uniqueness of linear periods,” Compositio Math., vol. 102, iss. 1, pp. 65-123, 1996. · Zbl 0855.22018
[16] S. Kato, A. Murase, and T. Sugano, ”Whittaker-Shintani functions for orthogonal groups,” Tohoku Math. J., vol. 55, iss. 1, pp. 1-64, 2003. · Zbl 1037.22034 · doi:10.2748/tmj/1113247445
[17] A. Murase and T. Sugano, ”Shintani functions and automorphic \(L\)-functions for \({ GL}(n)\),” Tohoku Math. J., vol. 48, iss. 2, pp. 165-202, 1996. · Zbl 0863.11035 · doi:10.2748/tmj/1178225376
[18] 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
[19] C. Moeglin and J. L. Waldspurger, La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux: le cas général. · Zbl 1276.22007
[20] D. Prasad, ”Trilinear forms for representations of \({ GL}(2)\) and local \(\epsilon\)-factors,” Compositio Math., vol. 75, iss. 1, pp. 1-46, 1990. · Zbl 0731.22013
[21] D. Prasad, ”On the decomposition of a representation of \({ GL}(3)\) restricted to \({ GL}(2)\) over a \(p\)-adic field,” Duke Math. J., vol. 69, iss. 1, pp. 167-177, 1993. · Zbl 0827.22006 · doi:10.1215/S0012-7094-93-06908-6
[22] S. Rallis and G. Schiffmann, Multiplicity one conjectures.
[23] S. Rallis and G. Schiffmann, ”Représentations supercuspidales du groupe métaplectique,” J. Math. Kyoto Univ., vol. 17, iss. 3, pp. 567-603, 1977. · Zbl 0398.22023
[24] T. A. Springer and R. Steinberg, ”Conjugacy classes,” in Seminar on Algebraic Groups and Related Finite Groups, New York: Springer-Verlag, 1970, vol. 131, pp. 167-266. · Zbl 0249.20024
[25] B. Sun, Multiplicity one theorems for Fourier Jacobi models. · Zbl 1280.22022 · doi:10.1353/ajm.2012.0044
[26] B. Sun and C. -B. Zhu, Multiplicity one theorems: the Archimedean case. · Zbl 1239.22014 · doi:10.4007/annals.2012.175.1.2
[27] 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
[28] J. L. Waldspurger, Une variante d’un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann. · Zbl 1308.22008
[29] J. L. Waldspurger, Une formule intégrale reliée a la conjecture locale de Gross-Prasad. · Zbl 1200.22010
[30] J. L. Waldspurger, Une formule intégrale reliée a la conjecture locale de Gross-Prasad, 2ème partie: extension aux représentations tempérées. · Zbl 1290.22012
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