= Complete reducibility of rational GL and SL representations
For finite-dimensional <rational representations> of complex $GL_m$ or $SL_m$, average a Hermitian <inner product> over the <compact group> $U(m)$ or $SU(m)$ using normalized <Haar measure>. <Orthogonal complements> become invariant under the compact group and its complexified <Lie algebra>. The latter generates the complex group, giving invariant complements and hence complete reducibility. This conclusion does not hold for arbitrary <affine algebraic groups>, such as the additive group.
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