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

[1]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[2]	M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[3]	D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013
[4]	S. Lang, "${\rm SL}_2({\bf R})$", Addison-Wesley (1975)
[5]	I.M. Gel'fand, M.I. Graev, I.I. Pyatetskii-Shapiro, "Generalized functions" , 6. Representation theory and automorphic functions , Saunders (1969) (Translated from Russian) Zbl 0801.33020 Zbl 0699.33012 Zbl 0159.18301 Zbl 0355.46017 Zbl 0144.17202 Zbl 0115.33101 Zbl 0108.29601
[6]	J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0181643 Zbl 0143.05901 Zbl 0128.26303
[7]	C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) MR0082628 MR0015396 Zbl 0063.00842
[a1]	N. Bourbaki, "Groupes et algèbres de Lie" , Eléments de mathématiques , Masson (1982) pp. Chapt. 9. Groupes de Lie réels compacts MR0682756 Zbl 0505.22006
[a2]	Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) MR0781344 Zbl 0581.22009
[a3]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , II , Springer (1970) MR0262773 Zbl 0213.40103
[a4]	A. Wawrzyńczyk, "Group representations and special functions" , Reidel & PWN (1984) MR0750113 Zbl 0545.43001