Continuous and unitary representations

Let $G$ be a compact Lie group with identity $e$, and let $V$ be a finite-dimensional complex vector space. A finite-dimensional complex representation of $G$ on $V$ is a continuous homomorphism $\pi :G⟶\mathrm{GL}(V),$ where $\mathrm{GL}(V)$ denotes the group of invertible complex-linear endomorphisms of $V$ with its standard finite-dimensional smooth structure; a homomorphism here is a group homomorphism that is a smooth map, in the sense of Lie-group homomorphism, isomorphism, and automorphism. Thus $\pi (gh)=\pi (g)\pi (h)$ and $\pi (e)={\mathrm{i}\mathrm{d}}_V$, and the entries of $\pi$ in any basis of $V$ are continuous functions on $G$. The dimension $\dim \pi$ of the representation is ${\dim }_{\mathbb{C}}V$.

The representation is unitary relative to an inner product $⟨\cdot ,\cdot ⟩$ on $V$ in the sense of Real and complex inner-product spaces and their induced length when every operator $\pi (g)$ preserves that inner product, $⟨\pi (g)v,\pi (g)w⟩=⟨v,w⟩(v,w\in V,g\in G)\mathrm{.}$ This says exactly that each $\pi (g)$ is a unitary operator for that inner product. Unless explicitly stated otherwise, representations in this page are finite-dimensional, and bases of $V$ are chosen so that the matrix of a unitary representation is unitary in the usual sense.

Infinite-dimensional Hilbert-space representations are named explicitly at the few places where they occur, namely for the left and right regular representations on $L^2(G)$; a representation of $G$ on a complex Hilbert space $H$ is a group homomorphism $\pi :G\to U(H)$, where $U(H)$ is the group of unitary bounded operators on $H$, such that $g\mapsto \pi (g)\xi$ is continuous for every $\xi \in H$.

• The word continuous is part of the data: a finite-dimensional representation of a Lie group is required to be a continuous (equivalently, smooth) homomorphism, and the smooth structure on $\mathrm{GL}(V)$ is the one induced by $\det \mathrm{\ne }0$ in $\mathrm{End}(V)$.
• Equivalence. Two representations $\pi$ on $V$ and $\sigma$ on $W$ are equivalent (or isomorphic) when there is a linear isomorphism $T:V\to W$ intertwining them, $T\pi (g)=\sigma (g)T$ for all $g\in G$; a subrepresentation is a linear subspace $W\subseteq V$ with $\pi (g)W\subseteq W$ for all $g$. A representation is irreducible when it is nonzero and has no nonzero proper subrepresentation. These conventions are used throughout this page.