BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Justin Trias (Imperial) DTSTART:20230516T130000Z DTEND:20230516T140000Z DTSTAMP:20260421T153819Z UID:UEAPS/7 DESCRIPTION:Title: To wards a theta correspondence in families for type II dual pairs\nby Ju stin Trias (Imperial) as part of ANTLR seminar\n\nLecture held in QUEENS 1 .03.\n\nAbstract\nThe classical local theta correspondence for p-adic redu ctive dual pairs defines a bijection between prescribed subsets of irreduc ible smooth complex representations coming from two groups (H\,H')\, formi ng a dual pair in a symplectic group. Alberto Mínguez extended this resul t for type II dual pairs\, i.e. when (H\,H') is made of general linear gro ups\, to representations with coefficients in an algebraically closed fiel d 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 ho w 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 cent ral to this approach is the integral Bernstein centre. We translate some w eaker properties of the classical correspondence\, such as compatibility w ith supercuspidal support\, as a morphism between the integral Bernstein c entres of H and H' and interpret it for the Weil representation. In genera l\, we only know that this morphism is finite though we may expect it to b e surjective. This would result in a closed immersion between the associat ed affine schemes as well as a correspondence between characters of the Be rnstein centre. This is current work with Gil Moss.\n LOCATION:https://researchseminars.org/talk/UEAPS/7/ END:VEVENT END:VCALENDAR