Towards a theta correspondence in families for type II dual pairs

Abstract: This is current work with Gil Moss. The classical local theta correspondence for p-adic reductive dual pairs defines a bijection between prescribed subsets of irreducible smooth complex representations coming from two groups (H,H'), forming a dual pair in a symplectic group. Alberto Mínguez extended this result for type II dual pairs, i.e. when (H,H') is made of general linear groups, to representations with coefficients in an algebraically closed field of characteristic l as long as the characteristic l does not divide the pro-orders of H and H'. For coefficients rings like Z[1/p], we explain how to build a theory in families for type II dual pairs that is compatible with reduction to residue fields of the base coefficient ring, where central to this approach is the integral Bernstein centre. We translate some weaker properties of the classical correspondence, such as compatibility with supercuspidal support, as a morphism between the integral Bernstein centres of H and H' and interpret it for the Weil representation. In general, we only know that this morphism is finite though we may expect it to be surjective. This would result in a closed immersion between the associated affine schemes as well as a correspondence between characters of the Bernstein centre.

number theory

Audience: researchers in the topic

London number theory seminar

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