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SUMMARY:Sandeep Varma (Tata Institute of Fundamental Research)
DTSTART:20220406T023000Z
DTEND:20220406T033000Z
DTSTAMP:20260423T005837Z
UID:POINTS/29
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/POINTS/29/">
 On residues of certain intertwining operators</a>\nby Sandeep Varma (Tata 
 Institute of Fundamental Research) as part of POINTS - Peking Online Inter
 national Number Theory Seminar\n\n\nAbstract\nLet $G$ be a connected reduc
 tive group over a finite extension $F$ of $\\mathbb{Q}_p$. Let $P = MN$ be
  a Levi decomposition of a maximal parabolic subgroup of $G$\, and $\\pi$ 
 an irreducible unitary supercuspidal representation of $M(F)$. One can the
 n consider the representation $Ind_{P(F)}^{G(F)} \\pi$ (normalized parabol
 ic induction). Assume that $P$ is conjugate to an opposite by an element $
 w_0 \\in G(F)$ that normalizes $M$\, and which fixes the isomorphism class
  of $\\pi$ (i.e.\, $\\pi \\cong \\\,^{w_0}\\pi$). Then\, by the work of Ha
 rish-Chandra\, $Ind_{P(F)}^{G(F)} \\pi$ is irreducible if and only if a ce
 rtain family $A(s\, \\pi\, w_0)$ of so called intertwining operators has a
  pole at $s = 0$. In this case\, after making certain choices\, the residu
 e of $A(s\, \\pi\, w_0)$ at $s = 0$ can be captured by a scalar $R(\\tilde
  \\pi) \\in \\mathbb{C}$\, which has a conjectural expression in terms of 
 some gamma factors related to Shahidi's local coefficients\, as described 
 by Arthur's local intertwining relation.\n\nFollowing a program pioneered 
 by Freydoon Shahidi\, and furthered by him as well as David Goldberg\, Ste
 ven Spallone\, Wen-Wei Li\, Li Cai\, Bin Xu\, Xiaoxiang Yu etc.\, one seek
 s to:\n\n(a) get explicit expressions to describe $R(\\tilde \\pi)$ \; and
 \n(b) interpret the resulting expression for $R(\\tilde \\pi)$ suitably\, 
 using the theory of endoscopy when applicable.\n\nSo far\, these questions
  have been studied mostly for classical (including unitary) groups\, or in
  some simple situations. We will discuss (a) above in a non-classical and 
 slightly "less simple" situation\, in the cases where $G$ is an almost sim
 ple group whose absolute root system is of exceptional type or of type $B_
 n$ with $n \\geq 3$ or $D_n$ with $n \\geq 4$\, and where $P$ is a "Heisen
 berg parabolic subgroup". We will then comment on what we can say of (b) a
 bove in the $G_2$\, $B_3$ and $D_4$ cases. Though the reducibility results
  and the $R(\\tilde \\pi)$ values are more or less already known in these 
 cases by the Langlands-Shahidi method and related results (e.g.\, the work
  of Henniart and Lomeli and Caihua Luo in the case of $D_4$)\, our investi
 gations also suggest the existence of harmonic analytic expressions for ce
 rtain gamma values\, which in some cases just amount to the formal degree 
 conjecture of Ichino\, Ikeda and Hiraga\, but in other cases seem slightly
  unwieldy and perhaps intriguing.\n\nZoom link: https://us02web.zoom.us/j/
 81501452154?pwd=OWVtRmU0bFpoMEY3OUxrVW04STFJQT09\n\nZoom number: 815 0145 
 2154\n\nZoom password: 363804\n
LOCATION:https://researchseminars.org/talk/POINTS/29/
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