Advanced Search

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 $\pi$.