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Ordinary mod p representations of the metaplectic cover of p-adic SL_2

Citation

Peskin, Laura Rebecca Rynne (2013) Ordinary mod p representations of the metaplectic cover of p-adic SL_2. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:06082013-032145943

Abstract

We classify the genuine ordinary mod p representations of the metaplectic group SL(2,F)-tilde, where F is a p-adic field, and compute its genuine mod p spherical and Iwahori Hecke algebras. The motivation is an interest in a possible correspondence between genuine mod p representations of SL(2,F)-tilde and mod p representations of the dual group PGL(2,F), so we also compare the two Hecke algebras to the mod p spherical and Iwahori Hecke algebras of PGL(2,F). We show that the genuine mod p spherical Hecke algebra of SL(2,F)-tilde is isomorphic to the mod p spherical Hecke algebra of PGL(2,F), and that one can choose an isomorphism which is compatible with a natural, though partial, correspondence of unramified ordinary representations via the Hecke action on their spherical vectors. We then show that the genuine mod p Iwahori Hecke algebra of SL(2,F)-tilde is a subquotient of the mod p Iwahori Hecke algebra of PGL(2,F), but that the two algebras are not isomorphic. This is in contrast to the situation in characteristic 0, where by work of Savin one can recover the local Shimura correspondence for representations generated by their Iwahori fixed vectors from an isomorphism of Iwahori Hecke algebras.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:mod p representation, Hecke algebra, ordinary representation, metaplectic group
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ramakrishnan, Dinakar
Thesis Committee:
  • Ramakrishnan, Dinakar (chair)
  • Mantovan, Elena
  • Flach, Matthias
  • Jorza, Andrei
Defense Date:29 May 2013
Non-Caltech Author Email:laura.peskin (AT) gmail.com
Record Number:CaltechTHESIS:06082013-032145943
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:06082013-032145943
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7875
Collection:CaltechTHESIS
Deposited By: Laura Peskin
Deposited On:08 Jul 2013 17:49
Last Modified:08 Jul 2013 17:49

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