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…)

The next basic tool we’re going to need is the theory of distributions.

Distributions

Distributions are extremely useful because they are both fairly general (including both all integrable functions and things like the Dirac delta function) but also allow for operations such as differentiation. So oftentimes we can obtain distribution solutions to differential equations we are interested in.

Actually, we’ll only discuss here tempered distributions. A tempered distribution is a linear functional ${\phi: \mathcal{S} \rightarrow \mathbb{C}}$. Clearly the tempered distributions form a vector space ${\mathcal{S}'}$; it is a locally convex space if we endow it with the weak* topology. It must now be seen how distributions generalize functions. So, if ${f \in L^p(\mathbb{R}^n)}$ for any ${p, 1 \leq p \leq \infty}$, then ${f}$ can be made into a distribution

$\displaystyle w \rightarrow \int_{\mathbb{R}^n} wf dx, \ w \in \mathcal{S}.$

In fact, we could just assume that ${f}$ grows polynomially. In particular, we have an imbedding ${\mathcal{S} \rightarrow S'}$.

An example of a distribution that is not a function is the Dirac distribution ${\delta}$ mapping ${f \rightarrow f(0)}$. (more…)

So, as I’ve already indicated, I’m planning to talk about PDEs for the next month or so, both the general theory and specific posts on the equations of mathematical physics.  There are some preliminaries I’ll have to do first such as Fourier transforms.  Today, I’ll get up to the inversion formula for Schwarz functions.

The Schwarz Class

The Schwarz class ${\mathcal{S}}$ consists of smooth functions ${f: \mathbb{R}^n \rightarrow \mathbb{R}}$ such that for all multi-indices ${\alpha=(\alpha_1, \dots, \alpha_n), \beta=(\beta_1, \dots, \beta_n)}$,

$\displaystyle x^{\alpha} D^{\beta} f := x_1^{\alpha_1} \dots x_n^{\alpha_n} \left( \frac{\partial}{\partial x_1}\right)^{\beta_1} \dots \left( \frac{\partial}{\partial x_n}\right)^{\beta_n}f$

is bounded. For instance, the function ${e^{-|x|^2}}$ is in ${\mathcal{S}}$, as is any ${C^{\infty}}$ function with compact support. Elements in ${\mathcal{S}}$ are loosely speaking, functions that decrease rapidly at ${\infty}$ with all their partial derivatives.

There is a way to make the space ${\mathcal{S}}$ into a Frechet space, by the countable family of seminorms

$\displaystyle ||f||_{a,b} := \sup |x^{a} D^{b} f(x) |.$ (more…)