Algebraic geometry in representation theory?

Question

I heard that today algebraic geometry plays some significant role in representation theory, which is a little surprising because when I learnt representation theory it is basically algebra, topology, differential geometry and a little functional analysis.

I am wondering whether someone can tell me how algebraic geometry enters the picture. I do not know much algebraic geometry so I am just looking for some expository, so maybe we can forget the technicalities for now.

Thanks very much!

Answer 1

Many modern representation theorists are interested in "geometric representation theory". One of the goals in this field is to realize a representation (e.g. a representation of a Lie algebra) geometrically. What this means is to realize the underlying vector space as the (co)homology of some algebraic variety and the action (e.g. the action of the Lie algebra) via some geometrically defined operations, such as cup products or convolution. There are several reasons why one would want to do this. One of the most important (in my opinion) is that the geometric approach often yields very nice bases in the representation, e.g., bases whose structure coefficients are positive integers (i.e. when you write the product of two basis elements as a linear combination of the basis elements, the coefficients are positive integers). These bases can be hard to define from a purely algebraic viewpoint.

Answer 2

Maybe the following might interest you: Let $X$ be a smooth projective curve over $\mathbb{C}$ of genus $\geq 2$. It is a classical theorem by Narasimhan and Seshadri that there is an equivalence of categories between the category of stable vector bundles of degree $0$ on $X$ and the category of irreducible unitary finite dimensional complex representations of $\pi_1(X)$. There are also $p$-adic versions due to C. Deninger and A. Werner. Thus, when studying representations of a profinite group, it might be helpful to realize it as the fundamental group of a curve, use the above theorem and classify representations by geometric means. But I'm not really sure if this has happened so far.