Topics in Representation Theory: Maximal Tori and the Weyl Group
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چکیده
where the Hilbert space direct sum is over all irreducible representations of G. Under the left action of G, the infinite dimensional representation on L(G) breaks up into finite dimensional representation spaces Vπ with multiplicity dim Vπ. What the Peter-Weyl theorem doesn’t do is tell us what the representations are and how to tell them apart. We would like to know how to project from L(G) onto each of the Vπ ⊗ V ∗ π . This requires classifying the irreducible representations and finding their characters.
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