BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:David Vogan (MIT) DTSTART:20230112T153000Z DTEND:20230112T170000Z DTSTAMP:20260422T151157Z UID:atlas/51 DESCRIPTION:Title: C omputing unitary duals\, I: cohomological induction\nby David Vogan (M IT) as part of Real reductive groups/atlas\n\n\nAbstract\nWe are planning to conclude this seminar (for now?) with three talks January 12\, 19\, and 26\, 2023. Topic is progress and plans for the original goal of the atlas project: to make software that can describe the unitary dual of any real reductive group G(R).\n\nThere are two fundamental classical techniques to construct unitary representations. The first (due to Israel Gelfand and h is collaborators) is real parabolic induction. A theorem in Knapp's "Overv iew" book gives a very simple way to identify most of the unitary represen tations that can be obtained in this way: they are the ones with non-real infinitesimal character. In light of that theorem\, one can study _only_ r epresentations with REAL infinitesimal character.\n\nThe second classical technique (due to Gregg Zuckerman and those who stole from him) is cohomol ogical parabolic induction. The analogue of the theorem in Knapp's book wo uld say that any unitary representation with non-imaginary infinitesimal c haracter can be obtained by cohomological induction. THIS IS NOT TRUE\, bu t it is nearly true.\n\nWhat's actually true is that any unitary represent ation for which the real part of the infinitesimal character is LARGE ENOU GH can be obtained by cohomological induction. The question of what "large enough" means is best expressed in terms of "nonunitarity certificates. T oday I will state these results with some care\, and start to look at nonu nitarity certificates.\n LOCATION:https://researchseminars.org/talk/atlas/51/ END:VEVENT END:VCALENDAR