So.  First off, surely the five remaining readers of this increasingly erratic blog have noticed the change of theme.

I want to next discuss the second inequality in class field theory, which is an upper bound on the norm index of the idele group.  There are two ways I know of to prove this: one analytic, one algebraic.  I will first sketch the analytic one. I say sketch because to do a full proof would get into the details of Dirichlet series, lattice points in homogeneously expanding domains, the construction of a certain fundamental domain for the action of the units, etc., etc., and I’d rather outline those ideas rather than do all the details because this is a series on class field theory. What I do plan on doing properly, however, is the algebraic (due to Chevalley in 1940) proof of the second inequality, which heavily uses results of field theory (e.g. Kummer theory) and local fields (e.g. power index computations).  I still thought it worthwhile to sketch the analytic approach, though. Rather than jumping right into it (I have to first say something about how the ideal and idele groups are connected), I decided to give an expository post on L-functions and Dirichlet’s theorem—in the case of the rational numbers.

1. Ramblings on the Riemann-zeta function

Recall that the Riemann-zeta function is defined by ${\zeta(s) = \sum n^{-s}}$, and that it is intimately connected with the distribution of the prime numbers because of the product formula

$\displaystyle \zeta(s) = \prod_p (1 - p^{-s})^{-1}$

valid for ${Re(s)>1}$, and which is a simple example of unique factorization. In particular, we have

$\displaystyle \log \zeta(s) = \sum_p p^{-s} + O(1) , \ s \rightarrow 1^+.$

It is known that ${\zeta(s)}$ has an analytic continuation to the whole plane with a simple pole with residue one at ${1}$. The easiest way to see this is to construct the analytic continuation for ${Re (s)>0}$. For instance, ${\zeta(s) - \frac{1}{s-1}}$ can be represented as a certain integral for ${Re(s)>1}$ that actually converges for ${Re(s)>0}$ though. (The functional equation is then used for the rest of the analytic continuation.) The details are here for instance. As a corollary, it follows that

$\displaystyle \sum_p p^{-s} = \log \frac{1}{s-1} + O(1) , \ s \rightarrow 1^+.$

This fact can be used in deducing properties about the prime numbers. (Maybe sometime I’ll discuss the proof of the prime number theorem on this blog.) Much simpler than that, however, is the proof of Dirichlet’s theorem on the infinitude of primes in arithmetic progressions. I will briefly outline the proof of this theorem, since it will motivate the idea of L-functions.

Theorem 1 (Dirichlet) Let ${\{an+b\}_{n \in \mathbb{Z}}}$ be an arithmetic progression with ${a,b}$ relatively prime. Then it contains infinitely many primes.

The idea of this proof is to note that the elements of the arithmetic progression ${\{an+b\}}$ can be characterized by so-called “Dirichlet characters.” This is actually a general and very useful (though technically trivial) fact about abelian groups, which I will describe now.

So, the blog stats show that semisimple Lie algebras haven’t exactly been popular.  Traffic has actually been unusually high, but people have been reading about the heat equation or Ricci curvature rather than Verma modules.  Which is interesting, since I thought there was a dearth of analysts in the mathosphere.  At MathOverflow, for instance, there have been a few complaints that everyone there is an algebraic geometer.     Anyway, there wasn’t going to be that much more I would say about semisimple Lie algebras in the near future, so for the next few weeks I plan random and totally disconnected posts at varying levels (but loosely related to algebra or algebraic geometry, in general).

I learned a while back that there is a classification of the simple modules over the semidirect product between a group and an commutative algebra which works the same way as the (more specific case) between a group and an abelian group.  The result for abelian groups rather than commutative rings appears in a lot of places, e.g. Serre’s Linear Representations of Finite Groups or Pavel Etingof’s notes.  I couldn’t find a source for the more general result though.  I wanted to work that out here, though I got a bit confused near the end, at which point I’ll toss out a bleg.

Let ${G}$ be a finite group acting on a finite-dimensional commutative algebra ${A}$over an algebraically closed field ${k}$ of characteristic prime to the order of ${G}$. Then, the irreducible representations of ${A}$ correspond to maximal ideals in ${A}$, or equivalently (by Hilbert’s Nullstellensatz!) homomorphisms ${\chi: A \rightarrow k}$, called characters. In other words, ${A}$ acts on a 1-dimensional space via the character ${\chi}$. (more…)

Now choose a dominant integral weight ${\lambda}$. By yesterday, we have:

$\displaystyle \mathrm{ch} L(\lambda) = \sum_{\mu < \lambda} b(\lambda, \mu) \mathrm{ch} V(\mu).$

Our first aim is to prove

Proposition 1 ${b(\lambda, w \cdot \lambda) = (-1)^w}$ for ${w \in W}$, the Weyl group, and ${\cdot}$ the dot action. For ${\mu \notin W\lambda}$, we have ${b(\lambda, \mu)=0}$.

After this, it will be relatively easy to obtain WCF using a few formal manipulations. To prove it, though, we use a few such formal manipulations already.

Manipulations in the group ring

I will now define something that is close to an “inverse” of the Verma module character ${p \ast e(\lambda)}$ for ${p(\lambda)}$ the Kostant partition function evaluated at ${-\lambda}$ (inverse meaning in the group ring ${\mathbb{Z}[L]}$, where ${L}$ is the weight lattice of ${\beta}$ with ${<\beta, \delta> \in \mathbb{Z} \ \forall \delta \in \Delta}$). Define ${q}$ by

$\displaystyle q = \prod_{\alpha \in \Phi^+} \left( e(\alpha/2) - e(-\alpha/2) \right).$

I claim that

$\displaystyle q = e(\rho) \prod_{\alpha \in \Phi^+} (1 - e(-\alpha)), \ \ wp = (-1)^w p, \quad \forall w \in W.$

(Note that since ${w}$ acts on the weight lattice ${L}$, it clearly acts on the group ring. Here, as usual, ${\rho = \frac{1}{2} \sum_{\gamma \in \Phi^+} \gamma}$.)

The first claim is obvious. The second follows because the minimal expression of ${w}$ as a product of reflections has precisely as many terms as the number of positive roots that get sent into negative roots by ${w}$, and a reflection has determinant ${-1}$. (more…)

I’m now aiming to get to the major character formulas for the (finite-dimensional) simple quotients ${L(\lambda)}$ for ${\lambda}$ dominant integral. They will follow from formal manipulations with character symbols and a bit of reasoning with the Weyl group. First, however, it is necessary to express ${\mathrm{ch} L(\lambda)}$ as a sum of characters of Verma modules. We will do this by considering any highest weight module of weight ${\lambda}$ for ${\lambda}$ integral but not necessarily dominant, and considering a filtration on it whose quotients are simple modules ${L(\mu)}$ where there are only finitely many possibilities for ${\mu}$. Applying this to the Verma module, we will then get an expression for ${\mathrm{ch} V(\lambda)}$ in terms of ${\mathrm{ch} L(\lambda)}$, which we can then invert.

First, it is necessary to study the action of the Casimir element (w.r.t. the Killing form). Recall that this is defined as follows: consider a basis ${B}$ for the semisimple Lie algebra ${\mathfrak{g}}$ and its dual basis ${B'}$ under the Killing form isomorphism ${\mathfrak{g} \rightarrow \mathfrak{g}^{\vee}}$. Then the Casimir element is

$\displaystyle \sum_{b \in B} b b^{\vee} \in U \mathfrak{g}$

for ${b^{\vee} \in B'}$ dual to ${b}$. As we saw, this is a central element. I claim now that the Casimir acts by a constant factor on any highest weight module, and that the constant factor is determined by the weight in such a sense as to give information about the preceding filtration (this will become clear shortly).

Central characters

Let ${D \in Z(\mathfrak{g}) := \mathrm{cent} \ U \mathfrak{g}}$ and let ${v_+ \in V(\lambda)}$ be the Verma module. Then ${Dv_+}$ is also a vector with weight ${v_+}$, so it is a constant multiple of ${v_+}$. Since ${v_+}$ generates ${V(\lambda)}$ and ${D}$ is central, it follows that ${D}$ acts on ${V(\lambda)}$ by a scalar ${\mathrm{ch}i_{\lambda}(D)}$. Then ${\mathrm{ch}i_{\lambda}}$ becomes a character ${ Z(\mathfrak{g}) \rightarrow \mathbb{C}}$, i.e. an algebra-homomorphism. There is in fact a theorem of Harish-Chandra that states that ${\mathrm{ch}i_{\lambda}}$ determines the weight ${\lambda}$ up to “linkage” (i.e. up to orbits of the dot action of the Weyl group: ${w \dot \lambda := w(\lambda + \rho) - \rho}$), though I shall not prove this here. (more…)

Let ${X}$ be a representation of a semisimple Lie algebra ${\mathfrak{g}}$, a Cartan subalgebra ${\mathfrak{h}}$, and some choice of splitting ${\Phi = \Phi^+ \cup \Phi^-}$ on the roots.

Recall from the representation theory of finite groups that to each representation of a finite group ${G}$ one can associate a character function ${\chi}$, and that the ring generated by the characters is the Grothendieck ring of the semisimple tensor category ${Rep(G)}$. There is something similar to be said for semisimple Lie algebras. So, assume ${\mathfrak{h}}$ acts semisimply on ${X}$ and that the weight spaces are finite-dimensional, and set formally

$\displaystyle \mathrm{ch}(X) := \sum_{\lambda} \dim X_{\lambda} e(\lambda).$

In other words, we define the character so as to include all the information on the size of the weight spaces at once.

It is necessary, however, to define what ${e(\lambda)}$ for ${\lambda \in \mathfrak{h}^{\vee}}$. Basically, it is just a formal symbol; ${\mathrm{ch}(X)}$ can more rigorously be thought of as a function ${\mathfrak{h}^{\vee} \rightarrow \mathbb{Z}_{\geq 0}}$. Nevertheless, we want to think of ${e(\lambda)}$ as a formal exponential in a sense; we want to have ${e(\lambda) e(\lambda') = e(\lambda + \lambda')}$. The reason is that we can tensor two representations, and we want to talk about multiplying to characters.

I now claim that the above condition on ${X}$ makes sense for ${X \in \mathcal{O}}$, the BGG category. This will follow because it is true for highest weight modules and we have:

Proposition 1 If ${M \in \mathcal{O}}$, then there is a finite filtration on ${M}$ whose quotients are highest weight modules. (more…)