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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.

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.

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.

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
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     author = {Z. Gong and L. Greni\'e},
     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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     volume = {Ser. 6, 20},
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     mrnumber = {2830396},
     language = {en},
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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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