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SUMMARY:Finn McGlade (UCSD)
DTSTART:20221103T210000Z
DTEND:20221103T220000Z
DTSTAMP:20260423T022808Z
UID:UCSD_NTS/75
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCSD_NTS/75/
 ">Fourier coefficients on quaternionic U(2\,n)</a>\nby Finn McGlade (UCSD)
  as part of UCSD number theory seminar\n\nLecture held in APM 6402 and onl
 ine.\n\nAbstract\nLet $E/\\mathbb{Q}$ be an imaginary quadratic extension 
 and suppose $G$ is the unitary group attached to hermitian space over $E$ 
 of signature $(2\,n)$. The symmetric domain $X$ attached to $G$ is a quate
 rnionic Kahler manifold. In the late nineties N. Wallach studied harmonic 
 analysis on $X$ in the context of this quaternionic structure. He establis
 hed a multiplicity one theorem for spaces of generalized Whittaker periods
  appearing in the cohomology of certain $G$-bundles on $X$. \n\nWe prove a
 n analogous multiplicity one statement for some degenerate generalized Whi
 ttaker periods and give explicit formulas for these periods in terms of mo
 dified K-Bessel functions. Our results give a refinement of the Fourier ex
 pansion of certain automorphic forms on $G$ which are quaternionic discret
 e series at infinity. As an application\, we study the Fourier expansion o
 f cusp forms on $G$ which arise as theta lifts of holomorphic modular form
 s on quasi-split $\\mathrm{U}(1\,1)$. We show that these cusp forms can be
  normalized so that all their Fourier coefficients are algebraic numbers. 
 (joint with Anton Hilado and Pan Yan)\n\npre-talk at 1:20pm\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/75/
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