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Passage from compact Lie groups to complex reductive groups November 25, 2010

Posted by Joel Kamnitzer in Algebraic Geometry, representation theory, things I don't understand.
25 comments

Once again, I’m preparing to teach a class and needing some advice concerning an important point. I’m teaching a course of representation theory as a followup to an excellent course on compact Lie groups, taught this semester by Eckhard Meinrenken. In my class, I would like to explain transition from compact Lie groups to complex reductive groups, as a first step towards the Borel-Weil theorem.

A priori, compact connected Lie groups and complex reductive groups, seem to have little in common and live in different worlds. However, there is a 1-1 correspondence between these objects — for example U(n) and GL_n(\mathbb{C}) are related by this correspondence. Surprisingly, it is not that easy to realize this correspondence.

Let us imagine that we start with a compact connected Lie group K and want to find the corresponding complex algebraic group G. I will call this process complexification.

One approach to complexification is to first show that K is in fact the real points of a real reductive algebraic group. For any particular K this is obvious — for example S^1 = U(1) is described by the equation x^2 + y^2 = 1. But one might wonder how to prove this without invoking the classification of compact Lie groups. I believe that one way to do this is to consider the category of smooth finite-dimensional representation of the group and then applying a Tannakian reconstruction to produce an algebraic group. This is a pretty argument, but perhaps not the best one to explain in a first course. A slightly more explicit version would be to simply define G to be Spec (\oplus_{V} V \otimes V^*) where V ranges over the irreducible complex representations of K (the Hopf algebra structure here is slightly subtle).

In fact, not only is every compact Lie group real algebraic, but every smooth map of compact Lie groups is actually algebraic. So the
the category of compact Lie groups embeds into the category of real algebraic groups. For a precise statement along these lines, see this very well written
MO answer by BCnrd.

A different approach to complexification is pursued in
Allen Knutson’s notes and in Sepanski’s book. Here the complexification of K is defined to be any G such that there is an embedding K \subset G(\mathbb{C}) , such that on Lie algebras \mathfrak{g} = \mathfrak{k} \otimes_{\mathbb{R}} \mathbb{C} . (Actually, this is Knutson’s definition, in Sepanski’s definition we first embed K into U(n) .) This definition is more hands-on, but it is not very obvious why such G is unique, without some structural theorems describing the different groups G with Lie algebra \mathfrak{g} .

At the moment, I don’t have any definite opinion on which approach is more mathematically/pedagogically sound. I just wanted to point out something which I have accepted all my mathematical life, but which is still somewhat mysterious to me. Can anyone suggest any more a priori reasons for complexification?

Gender and language in letters of recommendation November 12, 2010

Posted by Ben Webster in jobs.
5 comments

I don’t have time to write about this properly, since I’m going up to Seattle for the Cascade Topology Seminar in a few hours, and then going to Japan for two weeks, but this seemed too important to not point out: new research suggests that people write recommendation letters for women and for men in different ways, which act to the detriment of women.

Essentially, letters for women tend to play up their “nice” side and those for men their “active” side. Interestingly, the effect didn’t seem to depend on the gender of the recommender, and men who received “nice” recommendation letters rather than “active” ones also had more trouble getting hired, controlling for other factors.

Just a reminder that we all have to be really self-aware about how we write recommendation letters.

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