An inequality for local unitary Theta correspondence

Z. Gong; L. Grenié

doi:10.5802/afst.1289

Z. Gong ¹; L. Grenié 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

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 1, pp. 167-202.

Abstract
Abstract

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 $θ_{H} (π) \neq 0$ ? For given dimension $m$ there are exactly two isometry classes of unitary spaces that we denote $H_{m}^{\pm}$ . For $ε \in {0, 1}$ let us denote $m_{ε}^{\pm} (π)$ the minimal $m$ of the same parity of $ε$ such that $θ_{H_{m}^{\pm}} (π) \neq 0$ , then we prove that $m_{ε}^{+} (π) + m_{ε}^{-} (π) \geq 2 n + 2$ where $n$ is the dimension of $π$ .

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 $θ_{H} (π) \neq 0$ ? Pour chaque dimension $m$ il y a exactement deux classes d’équivalence d’espaces unitaires que nous dénotons $H_{m}^{\pm}$ . Pour $ε \in {0; 1}$ , dénotons $m_{ε}^{\pm} (π)$ le plus petit $m$ de la parité de $ε$ tel que $θ_{H_{m}^{\pm}} (π) \neq 0$ , alors nous montrons que $m_{ε}^{+} (π) + m_{ε}^{-} (π) \geq 2 n + 2$ où $n$ est la dimension de $π$ .

MR

DOI: 10.5802/afst.1289

Author's affiliations:

Z. Gong ¹; L. Grenié 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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     title = {An inequality for local unitary {Theta} correspondence},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {167--202},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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     year = {2011},
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     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1289/}
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Z. Gong; L. Grenié. An inequality for local unitary Theta correspondence. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 1, pp. 167-202. doi : 10.5802/afst.1289. https://afst.centre-mersenne.org/articles/10.5802/afst.1289/

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