Irreducible representation of a Lie group - Imported Knowlpedia Content

Let $G$ be a Lie group and let $\pi:G\to \mathrm{GL}(V)$ be a (finite-dimensional) representation.

Definition (Irreducible). The representation $(\pi,V)$ is irreducible if the only $G$ -invariant subspaces of $V$ are $\{0\}$ and $V$ , i.e. there is no proper nonzero subspace $W\subset V$ with $\pi(g)W\subseteq W$ for all $g\in G$ .

Remarks

Link with the Lie algebra (connected case). Assume $G$ is connected and let $d\pi:\mathfrak g\to \mathfrak{gl}(V)$ be the differential representation (compare differentiation is a Lie algebra homomorphism). Then a subspace $W\subset V$ is $G$ -invariant if and only if it is invariant under $d\pi(\mathfrak g)$ . Consequently, for connected $G$ , irreducibility of $\pi$ is equivalent to irreducibility of the induced Lie algebra representation $(d\pi,V)$ .

Context. For compact connected groups, irreducible unitary representations are classified by highest weights (see highest-weight theorem and Peter–Weyl).