Representations of the General Linear Groupoid Over a Non-Archimedean Local Field.

摘要

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.