Representation theory course

Well, like David, I am teaching a course this semester and writing up notes.

My course is on representation theory. More specifically, I hope to cover the basics of the representation theory of complex reductive groups, including the Borel-Weil theorem. In my class, I have started from the theory of compact groups, for two reasons. First, that is the way, I learned the subject from my advisor Allen during a couple of great courses. Second, I am following up on a course last semester taught by Eckhard Meinrenken on compact groups.

Feel free to take a look at the notes on the course webpage and give me any feedback.

Very soon, I will reach the difficult task of explaining complexification of compact groups. As I complained about in my previous post, I don’t feel that this topic is covered properly in any source, so I am bit struggling with it. Anyway, the answers to that post did help me out, so we will see what happens.

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The decline of quantum algebra (QA)

I was browsing through different category listings on the arXiv today and noting the changes in numbers of papers over the years. As you might expect, there are more and more papers being posted to the arXiv every year. However, one category defies this trends: QA (quantum algebra).

There are actually less papers being posted to QA in the past three years (2010 317, 2009 308, 2008 323), than there were in the late 90s (1998 364, 1997 434, 1996 395). By contrast, there are about 4 times as many AG papers in the past few years compared to the late 90s, about 10 times as many RT papers, and about 5 times as many GT papers.

What do you make of this? Does it represent a trend in the kind of math that people are doing? Or are people just classifying their work differently?

It would be interesting to see if one can use this arXiv category to get a sense of which fields are becoming more and less popular over time.

Passage from compact Lie groups to complex reductive groups

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?

A (partial) explanation of the fundamental lemma and Ngo’s proof

I would like to take Ben up on his challenge (especially since he seems to have solved the problem that I’ve been working on for the past four years) and try to explain something about the Fundamental Lemma and Ngo’s proof.  In doing so, I am aided by a two expository talks I’ve been to on the subject — by Laumon last year and by Arthur this week.

Before I begin, I should say that I am not an expert in this subject, so please don’t take what I write here too seriously and feel free to correct me in the comments.  Fortunately for me, even though the Fundamental Lemma is a statement about p-adic harmonic analysis, its proof involves objects that are much more familiar to me (and to Ben).  As we shall see, it involves understanding the summands occurring in a particular application of the decomposition theorem in perverse sheaves and then applying trace of Frobenius (stay tuned until the end for that!).

First of all I should begin with the notion of “endoscopy”.  Let G, G' be two reductive groups and let \hat{G}, \hat{G}' be there Langlands duals.  Then G' is called an endoscopic group for G if \hat{G}' is the fixed point subgroup of an automorphism of \hat{G} .  A good example of this is to take G = GL_{2n} , G' = SO_{2n+1} .  At first glance these groups having nothing to do with each other, but you can see they are endoscopic since their dual groups are GL_{2n} and Sp_{2n} and we have Sp_{2n} \hookrightarrow GL_{2n} .

As part of a more general conjecture called Langlands functoriality, we would like to relate the automorphic representations of G to the automorphic representations of all possible endoscopic groups G' .  Ngo’s proof of the Fundamental Lemma completes the proof of this relationship.

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Algebraic geometry without prime ideals

The first definition in “Grothendieck-style” algebraic geometry is the affine scheme Spec R for any ring R. This is a topological space whose set of points in the set of prime ideals in R. Then one defines a scheme to be a locally ringed space locally isomorphic to an affine scheme.

The definition of Spec R goes against intuition since it involves prime ideals, not just maximal ideals. Maximal ideals are more natural, since if R = k[x_1, \dots, x_n]/I for some alg closed field k , then the set of maximal ideals of R is in bijection with the vanishing set in the affine space k^n of the ideal I . (Of course one can give a geometric meaning to the prime ideals in terms of subvarieties, but it is less natural.)

However, in Daniel Perrin’s text Algebraic geometry, an introduction, he states/implies that one can define affine schemes just using maximal ideals (at least for finitely-generated k algebras) and still get a good theory of schemes and varieties. Is this true?  If so why don’t we all learn it this way? (One answer to the this latter question could be that some people are interested in non-algebraically closed fields.)

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Three geometric constructions of the irreducible representations of GL_n

The past few weeks there has been a summer school and conference on geometric representation theory and extended affine Lie algebras at University of Ottawa. As part of this event, I gave a week long lecture series entitled “three geometric constructions of the irreducible representations of GL_n “. Specifically I discussed the Borel-Weil theorem, Ginzburg’s construction using Springer fibres, and the geometric Satake correspondence. I focused on GL_n to keep the root system combinatorics and the geometry as elementary as possible.

The typed lecture notes from my talk are now available. If you do read them, please let me know if you have any comments/corrections. (You can also find videos of the talks.)

The other lectures at the summer school were given by Neher, Kang, Wang, Savage, and Chari. I recommend reading their notes/watching their videos if you want to learn more about geometric representation theory, crystals, and affine Lie algebras.

Hall algebras and Donaldson-Thomas invariants I

I would like to tell you about recent work of Dominic Joyce and others (Bridgeland, Kontsevich-Soibelman, Behrend, Pandaripande-Thomas, etc) on Hall algebras and Donaldson-Thomas invariants.  I don’t completely understand this work, but it seems very exciting to me. This post will largely be based on talks by Bridgeland and Joyce that I heard last month at MSRI.

In this post, I will concentrate on different versions of Hall algebras. Let us start with the most elementary one. Suppose I have an abelian category \mathcal{A} which has the following strong finiteness properties: namely Hom(A,B) and Ext^1(A,B) are finite for any objects A, B . Then one can define an algebra, called the Hall algebra of \mathcal{A}, which has a basis given by isomorphism classes of objects of \mathcal{A} and whose structure constants c_{[M], [N]}^{[P]} are the number of subobjects of P which are isomorphic to N and whose quotient is isomorphic to M .

The main source of interest of Hall algebras for me is the Ringel-Green theorem which states that if you start with a quiver Q, then the Hall algebra of the category of representation of Q over a finite field \mathbb{F}_q is isomorphic to the upper half of the quantum group corresponding to Q at the parameter q^{1/2}.

The obvious question concerning Hall algebras is to come up with a framework for understanding them when the Hom and Ext sets are not finite. This is what Joyce has done and he has applied it where A is the category of coherent sheaves on a Calabi-Yau 3-fold.

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The Newlander-Nirenberg theorem

As you may have noticed, I haven’t written a post in a long time.  Fortunately my fellow bloggers have been doing a great job in my absence.  Though I haven’t been posting, I have been reading and enjoying the blog regularly.

If you want to know what I have been up to in the meantime, the short answer is “Bella”. For a longer, more mathematical answer, you could check out this post or look at the recent papers that I’ve posted on the arxiv with Sabin Cautis and Tony Licata.

In this post, however, I would like to discuss something completely different, namely the Newlander-Nirenberg theorem. Let me begin by recalling the setting. An almost complex structure on a smooth manifold M is an endomorphism J of the tangent bundle such that J^2 = -1 .

One way to get an almost complex structure is to start with a complex manifold. The definition of a complex manifold is just like that of a smooth manifold — ie in terms of an atlas — except that we require that the transition functions be holomorphic.

The underlying real manifold of a complex manifold has an almost complex structure. An almost complex structure which arises in this way is called integrable. The N-N theorem gives you a criterion for testing whether an almost complex structure is integrable.

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Tannakian construction of the fundamental group and Kapranov’s fundamental Lie algebra

This post is a report on a talk that Mikhail Kapranov gave in Berkeley a few weeks ago on the “fundamental Lie algebra” — it is a Lie algebra associated to a manifold and a point on the manifold much like the fundamental group. Before telling about what Kapranov said, I’d like to start with some background.

Let M be a manifold and x \in M. Suppose we consider pairs \mathcal{E} = (E, \nabla) where E is a vector bundle and \nabla is a flat connection on E. Two such pairs \mathcal{E}, \mathcal{E'} may be tensored together to produce a new vector bundle with flat connection \mathcal{E}\otimes \mathcal{E'}.

Define a group G by the following procedure. An element g \in G is a collection (g_\mathcal{E}) of linear isomorphisms g_{\mathcal{E}} : E_x \rightarrow E_x for each \mathcal{E} which are natural with respect to maps \mathcal{E} \rightarrow \mathcal{E'} and which obey the rule g_{\mathcal{E} \otimes \mathcal{E'}} = g_{\mathcal{E}} \otimes g_{\mathcal{E'}}.

In other words, an element of G is a linear automorphism of the fibre at x of any vector bundle with flat connection. So, what is G?

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Geometric Langlands from a TQFT perspective

In my continuing attempts to clear the backlog in my brain, I would like to tell you about the talks of Nadler and Gukov from Miami workshop which I was at a month ago. (Actually I really want to tell you about Kontsevich’s talks but I don’t think that I understand them well enough to do that.)

Ever since the work of Kapustin and Witten a couple of years ago, a TQFT interpretation of geometric Langlands has been available. However, I had never “internalized” it until these talks. It gives a nice conceptual picture which makes some constructions in geometric Langlands less mysterious and hopefully makes the whole subject a bit more accessible.

In this view of things, geometric Langlands concerns the equality of two 4D TQFTs, which will denote by A and B. A and B both depend on the choice of a semisimple algebraic group G. Or more precisely, if we want A = B, then we should have A depending on G and B depending on its Langlands dual group G^\vee. They are 4D TQFTs, so they assign a number to a (closed) 4-manifold, a vector space to a 3-manifold, a category to a 2-manifold etc and related morphisms to bordisms of such objects.

I will start with a 2-manifold C. The first surprise is that A(C) and B(C) depend on more than just a topological structure for C — in particular we assume that C is actually endowed with the structure of smooth projective algebraic curve. Then we define A(C) = D-mod(Bun_G(C)) and B(C) = QCoh(Conn_{G^\vee}(C)). Here Bun_G(C) is the moduli space of algebraic principal G bundles on C and Conn_{G^\vee}(C)  is the moduli space of algebraic principal G^\vee bundles with connection on C. To continue the explanation, D-mod means the category of modules for the sheaf of differential operators (equivalently the category of perverse sheaves) and QCoh means the category of quasi-coherent sheaves.

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