weight space for a Lie group representation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T20:59:20Z http://mathoverflow.net/feeds/question/9347 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9347/weight-space-for-a-lie-group-representation weight space for a Lie group representation Vinoth 2009-12-19T05:18:32Z 2009-12-19T06:40:26Z <p>I understand how weights are defined for a Lie algebra representation. </p> <p>How are weight spaces defined for a Lie group action (with respect to a fixed torus)? </p> <p>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. </p> http://mathoverflow.net/questions/9347/weight-space-for-a-lie-group-representation/9353#9353 Answer by David Bar Moshe for weight space for a Lie group representation David Bar Moshe 2009-12-19T06:22:39Z 2009-12-19T06:40:26Z <p>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 </p> <p>lambda: T-->T^1 : t_1^lambda_1 . t_2^lambda_2 ... . t_n^lambda_n</p> <p>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. </p> <p>A clear exposition of this material can be found in Pressley and Segal : Loop groups chapter 2.</p>