Euler Poincare Characteristic for the Oscillator Representation

Suppose $(G,G')$ is a dual pair of subgroups of a metaplectic group. The dual pair correspondence is a bijection between (subsets of the) irreducible representations of $G$ and $G'$, defined by the non-vanishing of Hom$(ω,π\timesπ')$, where $ω$ is the oscillator representation. Alternatively one considers Hom$_G(ω,π)$ as a $G'$-module. It is fruitful to replace Hom with Ext$^i$, and general considerations suggest that the Euler-Poincare characteristic EP$(ω,π)$, the alternating sum of Ext$^i(ω,π)$, will be a more elementary object. We restrict to the case of $p$-adic groups, and prove that EP$(ω,π)$ is a well defined element of the Grothendieck group of finite length representations of $G'$, and show that it is indeed more elementary than Hom$(ω,π)$. We expect that computation of EP, together with vanishing results for higher Ext groups, will be a useful tool in computing the dual pair correspondence, and will help to elucidate the structure of Hom$(ω,π)$.

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