An inequality for local unitary Theta correspondence
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 1, pp. 167-202.

Étant donnée une représentation π d’un groupe unitaire local G et un autre groupe unitaire local H, on sait que soit la correspondance Theta fournit une représentation θ H (π) de H soit on pose θ H (π)=0. Si on fixe G et on laisse H varier dans une tour de Witt, une question naturelle est  : pour quels H a-t-on θ H (π)0  ? Pour chaque dimension m il y a exactement deux classes d’équivalence d’espaces unitaires que nous dénotons H m ± . Pour ε{0;1}, dénotons m ε ± (π) le plus petit m de la parité de ε tel que θ H m ± (π)0, alors nous montrons que m ε + (π)+m ε - (π)2n+2n est la dimension de π.

Given a representation π of a local unitary group G and another local unitary group H, either the Theta correspondence provides a representation θ H (π) of H or we set θ H (π)=0. If G is fixed and H varies in a Witt tower, a natural question is: for which H is θ H (π)0 ? For given dimension m there are exactly two isometry classes of unitary spaces that we denote H m ± . For ε{0,1} let us denote m ε ± (π) the minimal m of the same parity of ε such that θ H m ± (π)0, then we prove that m ε + (π)+m ε - (π)2n+2 where n is the dimension of π.

DOI : 10.5802/afst.1289
Gong, Z. 1 ; Grenié, L. 2

1 Lycée annexe à l’Université Fudan, N.383 Rue Guo Quan, Shanghai, Chine
2 Università degli Studi di Bergamo, viale Marconi 5, 24044 Dalmine (BG), Italy
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Gong, Z.; Grenié, L. An inequality for local unitary Theta correspondence. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 20 (2011) no. 1, pp. 167-202. doi : 10.5802/afst.1289. http://www.numdam.org/articles/10.5802/afst.1289/

[Har97] Harris (M.).— L-functions and periods of polarized regular motives, Journal für die reine und angewandte Mathematik 483, p. 75-161 (1997). | MR | Zbl

[Har07] —–, Cohomological automorphic forms on unitary groups. II. Period relations and values of L-functions, Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 12, World Sci. Publ., Hackensack, NJ, p. 89-149 (2007). MR MR2401812 | MR

[HKS96] Harris (M.), Kudla (S. S.), and Sweet (W. J.).— Theta dichotomy for unitary groups, J. Amer. Math. Soc. 9, no. 4, p. 941-1004 (1996). MR MR1327161 (96m:11041) | MR | Zbl

[How] Howe (R.).— L 2 duality for stable reductive dual pairs, preprint.

[KR05] Kudla (S. S.) and Rallis (S.).— On first occurrence in the local theta correspondence, Automorphic representations, L-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, p. 273-308 (2005). MR MR2192827 (2007d:22028) | MR | Zbl

[KS97] Kudla (S. S.) and Sweet Jr. (W. J.).— Degenerate principal series representations for U(n;n), Israel J. Math. 98, p. 253-306 (1997). MR MR1459856 (98h:22021) | MR | Zbl

[Kud86] Kudla (S. S.).— On the local theta-correspondence, Invent. Math. 83, no. 2, p. 229-255 (1986). MR MR818351 (87e:22037) | MR | Zbl

[Kud94] —–, Splitting metaplectic covers of dual reductive pairs, Israel Journal of Mathematics 87, p. 361-401 (1994). | MR

[Kud96] —–, Notes on the local theta correspondence, Available on Kudla’s home page, http://www.math.utoronto.ca/ssk/castle.pdf, (1996).

[Li92] Li (J.S.).— Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew. Math. 428, p. 177-217 (1992). MR MR1166512 (93e:11067) | MR | Zbl

[LR05] Lapid (E. M.) and Rallis (S.).— On the local factors of representations of classical groups, Automorphic representations, L-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, p. 309-359 (2005). MR MR2192828 (2006j:11071) | MR | Zbl

[MVW87] Moeglin (C.), Vigneras (M.-F.) and Waldspurger (J.-L.).— Correspondance de Howe sur un corps p-adique, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, (1987). | MR

[Ral84] Rallis (S.).— On the Howe duality conjecture, Compositio Math. 51, no. 3, p. 333-399 (1984). MR MR743016 (85g:22034) | Numdam | MR | Zbl

[Rao93] Ranga Rao (R.).— On some explicit formulas in the theory of the Weil representation, Pacific J. Math. 157 (1993), p. 335-371. | MR | Zbl

[Wal90] Waldspurger (J.-L.).— Démonstration d’une conjecture de dualité de Howe dans le cas p-adique, p2, Festscrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday (Stephen Gelbart, Roger Howe, and P. Sarnak, eds.), Israel Mathematical Conference Proceedings, vol. 2, The Weizmann science press of Israel, 1990, p. 267-324. MR MR1159105 (93h:22035) | MR | Zbl

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