Irreducible highest weight modules and equivariant quantization

Highest-weight Theory: Verma Modules

We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or, equivalently, of a compact Lie group). It turns out that such representations can be characterized by their “highest-weight”. The first method we’ll consider is purely Lie-algebraic, it begins by constructing a universal representation with a given high...

Laplace transform and unitary highest weight modules

The unitarizable modules in the analytic continuation of the holomorphic discrete series for tube type domains are realized as Hilbert spaces obtained through the Laplace transform.

Characterization of Simple Highest Weight Modules

We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.

Monomial Irreducible sln-Modules

In this article, we introduce monomial irreducible representations of the special linear Lie algebra $sln$. We will show that this kind of representations have bases for which the action of the Chevalley generators of the Lie algebra on the basis elements can be given by a simple formula.

Geometric Quantization and Equivariant Cohomology Geometric Quantization and Equivariant Cohomology