Tuesday, 25 September 2012

rt.representation theory - weight space for a Lie group representation

I understand how weights are defined for a Lie algebra representation.



How are weight spaces defined for a Lie group action (with respect to a fixed torus)?



I know this is a very embarrassing basic question, but i've looked through Harris+Fulton with no satisfactory explanation, and the only thing I can think of is using the exponential map somehow to reduce it to a Lie algebra, which seems unefficient computationally. Surely there must be a better way.

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