An inequality for local unitary Theta correspondence

Gong, Z.; Grenié, L.

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

Étant donnée une représentation $\pi$ d’un groupe unitaire local $G$ et un autre groupe unitaire local $H$, on sait que soit la correspondance Theta fournit une représentation ${\theta }_H\left(\pi \right)$ de $H$ soit on pose ${\theta }_H\left(\pi \right)=0$. Si on fixe $G$ et on laisse $H$ varier dans une tour de Witt, une question naturelle est  : pour quels $H$ a-t-on ${\theta }_H\left(\pi \right)\ne 0$  ? Pour chaque dimension $m$ il y a exactement deux classes d’équivalence d’espaces unitaires que nous dénotons $H_m^{\pm }$. Pour $\epsilon \in \{0;1\}$, dénotons $m_{\epsilon }^{\pm }\left(\pi \right)$ le plus petit $m$ de la parité de $\epsilon$ tel que ${\theta }_{H_m^{\pm }}\left(\pi \right)\ne 0$, alors nous montrons que $m_{\epsilon }^{+}\left(\pi \right)+m_{\epsilon }^{-}\left(\pi \right)\ge 2n+2$ où $n$ est la dimension de $\pi$.

Given a representation $\pi$ of a local unitary group $G$ and another local unitary group $H$, either the Theta correspondence provides a representation ${\theta }_H\left(\pi \right)$ of $H$ or we set ${\theta }_H\left(\pi \right)=0$. If $G$ is fixed and $H$ varies in a Witt tower, a natural question is: for which $H$ is ${\theta }_H\left(\pi \right)\ne 0$ ? For given dimension $m$ there are exactly two isometry classes of unitary spaces that we denote $H_m^{\pm }$. For $\epsilon \in \{0,1\}$ let us denote $m_{\epsilon }^{\pm }\left(\pi \right)$ the minimal $m$ of the same parity of $\epsilon$ such that ${\theta }_{H_m^{\pm }}\left(\pi \right)\ne 0$, then we prove that $m_{\epsilon }^{+}\left(\pi \right)+m_{\epsilon }^{-}\left(\pi \right)\ge 2n+2$ where $n$ is the dimension of $\pi$.

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