Theta liftovi ireducibilnih reprezentacija metaplektičke grupe

doctoral thesis

• Title (english): Theta lifts of irreducible representations of the metaplectic group
• Author: Petar Bakić
• Mentor: Marcela Hanzer (mentor)
• Mentor: Neven Grbac (komentor)
• Committee member: Goran Muić (predsjednik povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Neven Grbac (član povjerenstva)
• Committee member: Ivan Matić (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-10-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

U ovoj disertaciji proučavaju se theta liftovi generičkih reprezentacija simplektičke, odnosno metaplektičke grupe definirane nad lokalnim nearhimedskim poljem F karakteristike 0. U radu odgovaramo na osnovna pitanja vezana uz theta korespondenciju: kada su promatrani liftovi različiti od nul-reprezentacije, te kako izgledaju ne-nul liftovi. Za potpuni odgovor na prvo pitanje koristimo Kudlinu filtraciju i rezultat G. Muića o ireducibilnosti standardnih modula generičkih reprezentacija (tzv. Casselman-Shahidijeva slutnja). Također se oslanjamo na rezultate o liftovima temperiranih reprezentacija koje su u svojem radu ustanovili H. Atobe i W. T. Gan. Kao odgovor na drugo pitanje dajemo eksplicitni opis svih ne-nul liftova u terminima Langlandsove klasifikacije. Ovaj dio rada zasniva se na detaljnoj analizi mogućih subkvocijenata tzv. velikih theta liftova. Osim već spomenutih rezultata, ovdje se koristimo osnovnim činjenicama o ireducibilnosti induciranih reprezentacija opće linearne grupe koje je dokazao A. Zelevinsky, te Muićevim rezultatima o liftovima reprezentacija diskretne serije. Koristeći J.-S. Lijeve rezultate o unitarizabilnosti liftova unitarnih reprezentacija u stabilnom rangu, dolazimo do važne posljedice naših rezultata: opis viših liftova omogućuje nam eksplicitnu konstrukciju velikog broja unitarnih reprezentacija.

Abstract (english)

In this thesis we study the theta lifts of generic representations of the symplectic group and its double cover, the metaplectic group, defined over a local non-archimedean field F of characteristic 0. We consider the two basic problems regarding the local theta correspondence: first, determining when the lifts in question are non-zero, and secondly, explicitly describing those lifts. To obtain a full answer to the first question, we use Kudla’s filtration and a result of G. Muić on the irreducibility of standard modules of generic representation (the socalled Casselman-Shahidi conjecture). We also rely on the results on the lifts of tempered representations established by H. Atobe and W. T. Gan. Our answer to the second question includes an explicit description of all the non-zero lifts in terms of the Langlands classification. This part of our work is based on a detailed analysis of the possible subquotients of big theta lifts. Apart from the abovementioned results, we use the basic facts regarding the irreducibility of induced representations of the general linear group (established by A. Zelevinsky), as well as Muić’s results on the lifts of discrete series representations. Using the fact that theta correspondence in the stable range preserves unitarity (established by J.-S. Li), we arrive at an important consequence of our results: our description of theta lifts yields a method for constructing a large variety of unitary representations.

• Keywords: teorija reprezentacija p-adskih grupa; theta korespondencija; theta liftovi; generičke reprezentacije; metaplektička grupa
• Keywords (english): representation theory of p-adic groups; theta correspondence; theta lifts; generic representations; metaplectic group
• Language: croatian
• URN:NBN: urn:nbn:hr:217:147383
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: x, 125 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-03-07 09:32:59