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On an Invariant Bilinear Form on the Space of Automorphic Forms via Asymptotics.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	On an Invariant Bilinear Form on the Space of Automorphic Forms via Asymptotics./
Author:	Wang, Jonathan Peiyu.
Description:	1 online resource (172 pages)
Notes:	Source: Dissertation Abstracts International, Volume: 78-12(E), Section: B.
Contained By:	Dissertation Abstracts International78-12B(E).
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9780355076660

On an Invariant Bilinear Form on the Space of Automorphic Forms via Asymptotics.
Wang, Jonathan Peiyu.

On an Invariant Bilinear Form on the Space of Automorphic Forms via Asymptotics. - 1 online resource (172 pages)

Source: Dissertation Abstracts International, Volume: 78-12(E), Section: B.

Thesis (Ph.D.)

Includes bibliographical references

This thesis concerns the study of a new invariant bilinear form B on the space of automorphic forms of a split reductive group G over a global field. The form B is natural from the viewpoint of the geometric Langlands program.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2018

Mode of access: World Wide Web

ISBN: 9780355076660Subjects--Topical Terms:

527692
Mathematics.
Index Terms--Genre/Form:

554714
Electronic books.

On an Invariant Bilinear Form on the Space of Automorphic Forms via Asymptotics.
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099 $a TUL $f hyy $c available through World Wide Web
100 1 $a Wang, Jonathan Peiyu. $3 1180675
245 1 0 $a On an Invariant Bilinear Form on the Space of Automorphic Forms via Asymptotics.
264 0 $c 2017
300 $a 1 online resource (172 pages)
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
500 $a Source: Dissertation Abstracts International, Volume: 78-12(E), Section: B.
500 $a Adviser: Vladimir G. Drinfeld.
502 $a Thesis (Ph.D.) $c The University of Chicago $d 2017.
504 $a Includes bibliographical references
520 $a This thesis concerns the study of a new invariant bilinear form B on the space of automorphic forms of a split reductive group G over a global field. The form B is natural from the viewpoint of the geometric Langlands program.
520 $a First, we study a certain reductive monoid M associated to a parabolic subgroup P of G. The monoid M is used implicitly in the study of the geometry of Drinfeld's compactifications of the moduli stacks BunP and BunG. We show that M is a retract of the affine closure of the quasi-affine variety G/U, and we relate M to the Vinberg semigroup of G. Second, we define B over a function field using the asymptotics maps defined in Bezrukavnikov-Kazhdan and Sakellaridis-Venkatesh using the geometry of the wonderful compactification of G. We show that B is related to the miraculous duality functor studied by Drinfeld and Gaitsgory through the functions-sheaves dictionary. In the proof, we use the work of Schieder, which concerns the singularities of Drinfeld's compactification of BunG. We then give an alternate definition of B, which extends to number fields, using the constant term operator and the inverse of the standard intertwining operator. The form B defines an invertible operator L from the space of compactly supported automorphic forms to a new space of "pseudo-compactly" supported automorphic forms. We give a formula for L--1 in terms of pseudo-Eisenstein series and constant term operators which suggests that L--1 is an analog of the Aubert-Zelevinsky involution.
520 $a Lastly, we study the Radon transform as an operator R : C+ → C -- from the space of smooth K-finite functions on F n \ {0} with bounded support to the space of smooth K-finite functions on Fn \ {0} supported away from a neighborhood of 0, where F is a (possibly Archimedean) local field. When n = 2, the Radon transform coincides with the standard intertwining operator. We prove that R is an isomorphism and provide explicit formulas for R--1 . These formulas in turn give a formula for B over a number field when G = SL(2).
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2018
538 $a Mode of access: World Wide Web
650 4 $a Mathematics. $3 527692
655 7 $a Electronic books. $2 local $3 554714
690 $a 0405
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a The University of Chicago. $b Mathematics. $3 1180611
773 0 $t Dissertation Abstracts International $g 78-12B(E).
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10266436 $z click for full text (PQDT)