Let ${\rm Sp}(2n,{\bf R}$ be the group preserving a symplectic form on a real vector space of dimension $2n$. Let $O(p,q)$ be the group preserving a symmetric form of signature $p,q$ with $p+q=2n+1$. The dual pair correspondence [1] establishes a bijection between subsets of the irreducible representations of $\widetiel{\rm Sp}(2n,{\bf R})$ and thse of $O(p,q)$. Here $\widetile{\rm Sp}(2n,{\bf R})$ is the metaplectic group, the non-linear connected two-fold cover of ${\rm Sp}(2n,{\bf R})$. We are concerned with the explicit description of this correspondence.
The representations of $\widetile{\rm Sp}(2n,{\bf R})$ which arise in the correspondence are all genuine, i.e. they do not factor to ${\rm Sp}(2n,{\bf R})$. Fix a non-trivial additive character $\psi$ of ${\bf R}$, and $\delta=\pm 1$. The main result is that the duality correspondence determines a bijection between the set $$\widetilde{\rm Sp}(2n,{\bf R})\sphat_{\rm genuine}$$ of (equivalence classes of) genuine irreducible admissible representations of $\widetilde{\rm Sp}(2n,{\bf R})$ and the union $$\bigcup\Sb p+q=2n+1\\ (-1)^q=\delta\endSb {\rm SO}(p,q)\sphat$$ of the irreducible admissible representations of the groups ${\rm SO}(p,q)$.
Note that an irreducible representation $\pi$ of $\widetilde{\rm Sp}(2n,{\bf R}$ lifts to a representation $\pi'$ of precisely one orthogonal group $O(p,q)$ with given discriminant $\delta=(-1)^q$. We also give a relation with the epsilon factor of $\pi'$. Write $V$ for the orthogonal space of signature $p,q$. Let $\delta(V)$ be the discriminant of $V$, and let $\chi_V(x)=(x,\delta(V)(-1)^n)_{\bf R}$. Then $$\epsilon(\pi,\psi)=\epsilon(\pi')\chi_V(-1)^n\kappa({\rm SO}(V))$$ Here $\epsilon(\pi,\psi)$ is the central character invariant of $\pi$ of [3], $\epsilon(\pi')$ is the epsilon factor of $\pi'$, and $\kappa$ is the Kottwitz invariant of ${\rm SO}(V)$ [2].
These results confirm conjectures of Kudla, in the special case of ${\bf R}$, which in turn are generalizations of results of Waldspurger in the case $n=1$ [3].
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