Let be a semisimple Lie group, and be an integral weight for that group. naturally defines a one-dimensional representation of the Borel subgroup of , by simply pulling back the representation on the maximal torus where is the unipotent radical of . Since we can think of the projection map as a principle -bundle (http://planetmath.org/PrincipleBundle), to each , we get an associated fiber bundle on , which is obviously a line bundle. Identifying with its sheaf of holomorphic sections, we consider the sheaf cohomology groups . Realizing , the Lie algebra of , as vector fields on , we see that acts on the sections of over any open set, and so we get an action on cohomology groups. This integrates to an action of , which on is simply the obvious action of the group.
If is dominant, than is generated by global sections, and thus determines a map
This map is an obvious one, which takes the coset of to the highest weight vector of . This can be extended by equivariance since fixes . This provides an alternate description of .
For example, consider . is , the Riemann sphere, and an integral weight is specified simply by an integer , and . The line bundle is simply , whose sections are the homogeneous polynomials of degree . This gives us in one stroke the representation theory of : is the standard representation, and is its th symmetric power. We even have a unified decription of the action of the Lie algebra, derived from its realization as vector fields on : if are the standard generators of , then
|Date of creation||2013-03-22 13:50:52|
|Last modified on||2013-03-22 13:50:52|
|Last modified by||mathcam (2727)|