rt.representation theory - Examples of applications of the Borel-Weil-Bott theorem?

I'll just elaborate for you the example mentioned by Scott:
In the stereographic projection coordinates of S², the symplectic 2-form is given by:

ω = dz^dz̄/(1+zz̄)²

Classically, one can construct three hamiltonian functions representing the generators of the Lie algebra su(2) which constitute of a subalgebra of the Poisson algebra corresponding to ω

T_X = (z+z̄)/(1+zz̄)

T_Y = -i(z-z̄)/(1+zz̄)

T_Z = (1-zz̄)/(1+zz̄)

Quantum mechanically, the representation of spin j of SU(2) is realized on a (reproducing kernel) Hilbert space generated by holomorphic sections of a line bundle whose expressions in the stereographic coordinates are 1, z, . . . , z^2j. The su(2) Lie algebra can be realized on this space by means of the differential operators:

s_X = -(1-z²)∂/∂z + 2jz

s_Y = -i(1+z²)∂/∂z - 2ijz

s_Z = -2z∂/∂z -2j

Theories of geometric quantization offer systematic methods to make these constructions for a general compact Lie group for a concrete realization of the Bore-Weil-Bott theorem.

I would like to mention that many representation theoretical computations can be made using this realization of the representation theory of compact Lie groups. Also, this realization is connected to Perelomov's generalized coherent states.

There are some generalizations to representations on non-compact Lie groups. Also, the Borel-Weil-Bott theorem can be connected in many ways to supersymmetry.

The "linearization' of the classical mechanics is achievd through the realization of the quantum Hilbert space by sections of a "line" bundle. These sections also relates this realization to projective geometry via Kodaira's embedding thorem.