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 End∨Ф. The main result of the work is that End∨Ф 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.
|School Location:||United States -- Connecticut|
|Source:||DAI-B 75/09(E), Dissertation Abstracts International|
|Keywords:||Groupoid, Non-Archimedean Local Field, Representation|
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