February 12, 2010

Action of the Casimir on highest weight modules; characters

Posted by Akhil Mathew under algebra, representation theory | Tags: , , , , |
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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…)

 

July 25, 2009

The enveloping algebra

Posted by Akhil Mathew under algebra | Tags: , , |
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As we saw in the first post, a representation of a finite group {G} can be thought of simply as a module over a certain ring: the group ring. The analog for Lie algebras is the enveloping algebra. That’s the topic of this post.

Definition

The basic idea is as follows. Just as a representation of a finite group {G} was a group-homomorphism {G \rightarrow Aut(V)} for a vector space, a representation of a Lie algebra {\mathfrak{g}} is a Lie-algebra homomorphism {\mathfrak{g} \rightarrow \mathfrak{g}l(V)}. Now, {\mathfrak{g}l(V)} is the Lie algebra constructed from an associative algebra, {End(V)}—just as {Aut(V)} is the group constructed from {End(V)} taking invertible elements.

(more…)