# 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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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