BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:David Vogan (MIT) DTSTART:20230119T153000Z DTEND:20230119T170000Z DTSTAMP:20260422T150132Z UID:atlas/52 DESCRIPTION:Title: C omputing unitary duals\, II: nonunitarity certificates\nby David Vogan (MIT) as part of Real reductive groups/atlas\n\n\nAbstract\nLast week I s tated the Langlands classification in terms of atlas parameters. I showed how an atlas parameter defines a real parabolic P^u = M^u N^u\, and stated the theorem from Knapp's "Overview" that the corresponding representation pi_G is unitary if and only if it's unitarily induced from pi_{M^u}. Clas sifying the unitary dual therefore reduces to the case M^u = G\, which is equivalent to REAL INFINITESIMAL CHARACTER.\n\nNext\, I showed how to atta ch to an atlas parameter p a theta-stable parabolic q = l+u\, and a parame ter pL for L\; and sketched how the G-rep pi attached to p is cohomologica lly induced from the L-rep piL attached to pL.\n\nTopic for today is to un derstand how the correspondence piL --> pi is related to unitarity.\n\nApp arently unrelated\, but actually deeply connected: if pi is an irreducible representation of a real reductive G(R)\, we get a multiplicity function m_{pi}: K^ --> N\, m_{pi}(mu) = multiplicity of mu in pi|_K. The atlas so ftware computes this function (up to K-types of any specified height). If pi admits an invariant Hermitian form\, then the form has a signature: m_p i is the sum of two N-valued functions p_{pi} and q_{pi}. We arrange the f orm to be positive definite on some lowest K-type mu_0\, so that p_{pi}(mu _0) = 1 and q_{pi}(\\mu_0) = 0.\n\nA NONUNITARITY CERTIFICATE for pi is a particular K-type mu' with the property that q_{pi}(mu') > 0.\n\nFor each K-type mu there is a FINITE set {mu'_0\,...\,mu'_r} of K-types so that ANY NONUNITARY REP OF LKT \\mu MUST HAVE A NONUNITARITY CERTIFICATE IN {mu'_i }. Such a set is called a NONUNITARITY CERTIFICATE FOR THE LOWEST K-TYPE m u. I will explain how atlas can hope to compute this finite set (it doesn' t yet!) and how knowledge of such sets can help to compute unitary duals.\ n LOCATION:https://researchseminars.org/talk/atlas/52/ END:VEVENT END:VCALENDAR