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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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up vote 3 down vote accepted

In the case of a finite dimensional representation of a compact Lie group, one picks a basis in which the action of a maximal torus T is diagonal. The weight associated to a vector in this basis is the homomorphism

lambda: T-->T^1 : t_1^lambda_1 . t_2^lambda_2 ... . t_n^lambda_n

by which the maximal torus acts on the vector. The weight space of the group representation is the set of weights of the representation in the charcter lattice of the maximal torus.

A clear exposition of this material can be found in Pressley and Segal : Loop groups chapter 2.

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It's actually easier than in the Lie algebra case, because (a) the torus action is automatically semisimple, and (b) the lattice support of the weight diagram is self-evident. –  Greg Kuperberg Dec 19 '09 at 6:39
    
There's also a goofy thing you can do in the Lie group case: pick a general enough element $t$ of the torus to have $\langle t\rangle$ dense in $T$. Then when you diagonalize the action of $t$, you've automatically found a weight basis for $T$. –  Allen Knutson Apr 13 '11 at 13:00
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