Euler Poincare Characteristic for the Oscillator Representation

TL;DR

This paper introduces the Euler-Poincare characteristic for oscillator representations in the context of dual pairs of p-adic groups, providing a more elementary invariant that could aid in understanding the dual pair correspondence.

Contribution

It proves that the Euler-Poincare characteristic is well-defined and resides in the Grothendieck group, offering a new tool for analyzing dual pair correspondences in p-adic groups.

Findings

EP is a well-defined element of the Grothendieck group.

EP is more elementary than Hom$(\omega,\pi)$.

Potential for EP to facilitate computation of dual pair correspondence.

Abstract

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$(\omega,\pi\times\pi')$, where $\omega$ is the oscillator representation. Alternatively one considers Hom$_G(\omega,\pi)$ as a $G'$-module. It is fruitful to replace Hom with Ext$^i$, and general considerations suggest that the Euler-Poincare characteristic EP$(\omega,\pi)$, the alternating sum of Ext$^i(\omega,\pi)$, will be a more elementary object. We restrict to the case of $p$-adic groups, and prove that EP$(\omega,\pi)$ 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$(\omega,\pi)$. 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$(\omega,\pi)$.