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Joseph Bernstein (TAU): Groups, Groupoids, Stacks and Representation Theory.

A talk at the conference on
Representation Theory and Algebraic Geometry on the occasion of Joseph Bernstein’s 80th birthday, held at the Weizmann Institute, Israel in May 2025.
See https://sites.google.com/view/gelbart....

Link to the slides:
https://www.dropbox.com/scl/fi/xry7kj...

Abstract:
In my talk I will discuss an important basic question in Representation Theory that informally might be stated as follows: What is a representation of a group ?
So far this is quite informal ”philosophical question”. To be more specific, given some group G (in fact a group object G of some type) we would like to give a technical definition that describes a category Rep(G) that we think corresponds to our intuitive understanding of the category of representations. Note, that there might be different definitions if we want to describe representations of different types (continuous, algebraic and so on). It turns out that this is a subtle question that requires a careful analysis. One of the reasons is that there are several approaches to this question and some of them give ”wrong” answers.
Moreover, we will see that the large chunk of mathematical literature is devoted to the study of these ”wrong” categories. This makes many results and methods unnecessary complicated.
Let G be an abstract group and k some field. A representation of the group G over the field k is usually defined as a pair (π, V ) where V is a vector space over k and π is a morphism π : G → Aut(V ). One of basic problems in Representation Theory is the study of the category Rep(G) of such representations.
In my talk I will explain that there is another way to describe this category. Namely, the category Rep(G) is naturally equivalent to the category of sheaves Sh(B(G)) on some ”geometric” object – the basic groupoid BG of the group G.
Thus we have two equivalent definitions of representations – the standard one and a categorical definition in terms of groupoids. I will explain that more sophisticated categorical description is more ”correct” one.
The gap between these two definitions becomes much more profound when we move from the category of sets to other categories (more precisely – sites). In this case the role of groupoids are played by stacks. So I propose to define the category of representations of a group G as the category of sheaves on the basic stack BG.
I will discuss how these things play out in the important case when G is an algebraic group over a local or a finite field.