2 mathifying

For actual applications, there are some; for example, the moment map image of G/B for the torus action is the convex hull of the weight diagram, and the weight multiplicities are approximated by the volume of fibers.

There's a similar business with tensor product multiplicities, where you look at the moment map image for G acting on the product of two coadjoint orbits. This one way of thinking about convexity of the support of tensor product multiplicities.

Also, if you look at the dimensions of the representations V_{n\lambda}$V_{n\lambda}$ for some fixed lambda, you get a polynomial of order the dimension of the corresponding coadjoint orbit, and leading term its symplectic volume.

 
 
 
 
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For actual applications, there are some; for example, the moment map image of G/B for the torus action is the convex hull of the weight diagram, and the weight multiplicities are approximated by the volume of fibers.

There's a similar business with tensor product multiplicities, where you look at the moment map image for G acting on the product of two coadjoint orbits. This one way of thinking about convexity of the support of tensor product multiplicities.

Also, if you look at the dimensions of the representations V_{n\lambda} for some fixed lambda, you get a polynomial of order the dimension of the corresponding coadjoint orbit, and leading term its symplectic volume.