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ID: past-exam-of-the-lie-groups-and-differentiable-manifolds-course-of-the-university-of-oxford/2008/a/4/a/solution

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Past exam of the lie Groups and Differentiable Manifolds course of the University of Oxford / 2008 / A / 4 / a / Solution by

Codex  0 2026-10-31

As usual here, representations are finite-dimensional and continuous. Start with any positive definite Hermitian form

h_{0}

and normalized Haar measure

μ

on the compact group. Define

h (v, w) = \int_{G} h_{0} (gv, g w) d μ (g) .

This is Hermitian, sesquilinear, and positive definite: for

v \neq = 0

its continuous integrand

h_{0} (gv, gv)

is strictly positive. Bi-invariance of Haar measure shows

h (a v, a w) = h (v, w)

for every

a \in G

.

If

U \subset V

is invariant, then so is its orthogonal complement: for

v ⊥ U

and

u \in U

,

h (a v, u) = h (v, a^{- 1} u) = 0

. Hence

V = U \oplus U^{⊥}

as a representation. Induct on dimension, splitting whenever a nontrivial proper invariant subspace exists. This proves every such representation is a direct sum of irreducible representations.

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