Sphere correlation functions and Verma modules

Semi–holonomic Verma Modules

Verma modules arise geometrically through the jets of homogeneous vector bundles. We consider in this article, the modules that arise from the semi-holonomic jets of a homogeneous vector bundle. We are particularly concerned with the case of a sphere under MMbius transformations. In this case there are immediate applications in the theory of conformally invariant diierential operators.

Generalized Verma Modules

Twisted Verma modules

Using principal series Harish-Chandra modules, local cohomology with support in Schubert cells and twisting functors we construct certain modules parametrized by the Weyl group and a highest weight in the subcategory O of the category of representations of a complex semisimple Lie algebra. These are in a sense modules between a Verma module and its dual. We prove that the three different approa...

Harish-Chandra’s homomorphism, Verma modules

The Harish-Chandra homomorphism is due to [Harish-Chandra 1951]. Attention to universal modules with highest weights is in ]Harish-Chandra 1951], [Cartier 1955], as well as [Verma 1968], [BernsteinGelfand-Gelfand 1971a], [Bernstein-Gelfand-Gelfand 1971b], [Bernstein-Gelfand-Gelfand 1975]. See also [Jantzen 1979]. [1] We treat sl(2) in as simple a style as possible, to highlight ideas. Then sl(3...

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...