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An inequality for local unitary Theta correspondence

Z. GongL. Grenié — 2011

Annales de la faculté des sciences de Toulouse Mathématiques

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 ) 2 n + 2 where n is the dimension of π .

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