Dissertation/Thesis Abstract

Representations of the General Linear Groupoid Over a Non-Archimedean Local Field
by Frailey, Conor Nelson, Ph.D., Yale University, 2014, 73; 3580686
Abstract (Summary)

We first recall the notion of a groupoid as a certain categorical generalization of a group along with along with attendant topological and representation theoretic notions.

We then recall the work of Joyal and Street on the category of representations of the general linear groupoid over a finite field. We introduce another notion of a general linear groupoid over a finite field and its category of representations and show consistency with those of Joyal and Street.

We then consider a non-Archimedean local field F and construct two notions of the general linear groupoid over F similarly as for finite fields. We then construct representations of categories of those groupoids. We examine a monoidal structure on those categories of representations and exhibit an equivalence of categories.

We next introduce a fiber functor Ф that assigns to each representation those points which are bi-invariant under action by elements of certain subgroups (essentially the general linear groups over the ring of integers). This bi-invariance study is classical to the study of non-Archimedean local fields. We show that this fiber functor is monoidal.

Finally, following the work of Joyal and Street, we introduce an endomorphism coalgebra. EndФ and its "pre-dual" bialgebra EndФ. The main result of the work is that EndФ is commutative.

This can be seen as a sort of dual to the classical result due to Gelfand of the commutativity of the collection of spherical Hecke algebras.

Indexing (document details)
Advisor: Kapranov, Mikhail
School: Yale University
School Location: United States -- Connecticut
Source: DAI-B 75/09(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Groupoid, Non-Archimedean Local Field, Representation
Publication Number: 3580686
ISBN: 978-1-321-05505-4
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