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- Jun 20, 2014

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Let $G$ be a compact Lie group, and let $V$ be a finite-dimensional representation of $G$. Prove that if $\chi$ is the character associated with $V$, then $\int_G \chi(g)\, dg = \operatorname{dim}(V^G)$ where $V^G\subset V$ is the subspace of $G$-invariants of $V$.

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