# Representation of a compact group(2)

A homomorphism of a compact group into the group of continuous linear automorphisms of a (complex) Banach space that is continuous with respect to the strong operator topology.

Let $G$ be a compact group, let $V$ be a Banach space and let $\phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V)$ be a representation. If $V = H$ is a Hilbert space and $\phi (g)$ is a unitary operator for every $g \in G$ , then $\phi$ is called a unitary representation. There always is an equivalent norm in $H$ for which $\phi$ is unitary.

Every irreducible unitary representation (cf. Irreducible representation) of a compact group $G$ is finite-dimensional. Let $\{ {\rho ^ \alpha } : {\alpha \in I} \}$ be the family of all possible pairwise inequivalent irreducible unitary representations of the group $G$ . Every unitary representation $\phi$ of $G$ is an orthogonal sum of unique representations $\phi ^ \alpha$ , $\alpha \in I$ , such that $\phi ^ \alpha$ is an orthogonal sum, possibly zero, of a set of representations equivalent to $\rho ^ \alpha$ .

If $G$ is finite, then the family $\{ \rho ^ \alpha \}$ is also finite and contains as many elements as there are distinct conjugacy classes in $G$ ( moreover, $\sum _ {\alpha \in I} ( \mathop{\rm dim}\nolimits \ \rho ^ \alpha ) ^{2} = | G |$ ). The problem of studying these representations (computing their characters, finding explicit realizations, etc.) is the subject of an extensive theory (cf. Finite group, representation of a).

If $G$ is a connected, simply-connected, compact Lie group and $G _ {\mathbf C}$ is its complexification (cf. Complexification of a Lie group), then the description of the family $\{ {\rho ^ \alpha } : {\alpha \in I} \}$ for $G$ amounts (by restricting the representations to $G$ ) to the description of the family of all irreducible pairwise inequivalent finite-dimensional rational representations of the reductive algebraic group $G _ {\mathbf C}$ . The latter family, in turn, allows of a complete description by considering highest weights (cf. Representation with a highest weight vector).

In modern number theory and algebraic geometry one considers $\ell$-adic representations of compact totally-disconnected groups (cf. [5], [6]).

#### References

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