= Equivariant Fredholm index
{title2=$\operatorname{ind}_G T=[\ker T]-[\operatorname{coker}T]$}
An equivariant <Fredholm operator> between <unitary representations> of a <compact group> has finite-dimensional invariant kernel and cokernel. Their formal difference is its index in the <representation ring of a compact group>. If domain and target decompose into irreducibles with finite multiplicities $m_\pi,m'_\pi$, Schur's lemma makes each intertwining block a matrix on the multiplicity spaces. Its rank cancels between kernel and cokernel, giving $\operatorname{ind}_G T=\sum_\pi(m_\pi-m'_\pi)[\pi]$, with finitely many nonzero terms.
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