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 be a compact group, let be a Banach space and let be a representation. If is a Hilbert space and is a unitary operator for every , then is called a unitary representation. There always is an equivalent norm in for which is unitary.

Every irreducible unitary representation (cf. Irreducible representation) of a compact group is finite-dimensional. Let be the family of all possible pairwise inequivalent irreducible unitary representations of the group . Every unitary representation of is an orthogonal sum of unique representations , , such that is an orthogonal sum, possibly zero, of a set of representations equivalent to .

If is finite, then the family is also finite and contains as many elements as there are distinct conjugacy classes in (moreover, ). 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 is a connected, simply-connected, compact Lie group and is its complexification (cf. Complexification of a Lie group), then the description of the family for amounts (by restricting the representations to ) to the description of the family of all irreducible pairwise inequivalent finite-dimensional rational representations of the reductive algebraic group . 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 -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, "" , 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)
[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

References

[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