Geometry and Topology
Speaker: Vic Snaith (Sheffield)
"The Bernstein centre of smooth representations"
Time: 15:30
Room: MC 107
In the 1980's Bernstein-Zelevinski calculated the centre of the
abelian category of smooth representations on GLnK when K is a local field.
Soon after Deligne generalised this to all reductive algebraic groups G over K.
The centre of a category consists of all families zA∈End(A) as A varies
through all objects such that for any morphism in the category f:A⟶B
we have fzA=zBf. Deligne's answer comes in terms of distributions on G.
Over the last decade or so, I developed the notion of monomial resolutions for such
representations. This amounts to an embedding of the representation category into a
derived category of monomial objects. Using Bruhat's thesis I shall explain how to
interpret the monomial morphisms in terms of spaces of distributions and thereby to
re-derive Deligne's result.
I know to my cost how technical this stuff can get - so I shall try to navigate by means of
conceptual insights. For example, for us topologists, I shall explain how sheaves of distributions
behave in a manner precisely analogous to a famous result of Swan and Serre about
sections of topological vector bundles.