Peter-Weyl theorem

A theorem on the approximation of functions on a compact topological group by means of representation functions (cf. Representation function). Let run through a family of representatives of all equivalence classes for the irreducible continuous unitary representations of a compact group (cf. Representation of a topological group). Let be the dimension of a representation and let be its matrix elements in some orthonormal basis. The Peter–Weyl theorem asserts that the functions

form an orthonormal basis in the space of square-summable functions with respect to the Haar measure on (the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on , which coincides with the set of finite linear combinations of the functions , , is uniformly dense in the space of all continuous complex-valued functions in .

If is the rotation group for the plane, this assertion coincides with an elementary theorem on approximating periodic continuous functions by trigonometric polynomials.

A consequence of the Peter–Weyl theorem is that the set of linear combinations of characters of the irreducible representations of is dense in the algebra of all continuous functions on , constant on classes of conjugate elements. Another consequence is that for any element , , there is an irreducible continuous representation of such that ; if, on the other hand, is a compact Lie group, then has a faithful linear representation.

The Peter–Weyl theorem implies also the following more general assertion [5], [6]. Suppose one is given a continuous linear representation of a compact group in a Fréchet space . Then the subspace of representation elements of is dense in . Here an element is called a representation, or spherical or almost-invariant, element if the orbit generates a finite-dimensional subspace in . This is applicable in particular to the case where is the space of sections of a certain smoothness class of smooth vector -fibrations, for example, the space of tensor fields of a certain type or given smoothness class on a smooth manifold with a smooth action of a compact Lie group .

The Peter–Weyl theorem was proved in 1927 by F. Peter and H. Weyl .

References

[1a]	F. Peter, H. Weyl, "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe" Math. Ann. , 97 (1927) pp. 737–755
[1b]	F. Peter, H. Weyl, "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe" , Gesammelte Abhandlungen H. Weyl , III : 73 , Springer (1968) pp. 58–75
[2]	L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[3]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1979)
[4]	C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946)
[5]	R.S. Palais, T.E. Stewart, "The cohomology of differentiable transformation groups" Amer. J. Math. , 83 : 4 (1961) pp. 623–644
[6]	G.D. Mostow, "Cohomology of topological groups and solvmanifolds" Ann. of Math. , 73 : 1 (1961) pp. 20–48

Comments

A representation element is now usually called a -finite element.

The statement that the algebra of complex-valued representation functions is uniformly dense in the algebra of continuous functions on is known as the Weyl approximation theorem. The Peter–Weyl theorem gives a complete description of the (left or right) regular representation in terms of its irreducible components. In particular, each irreducible component occurs with a multiplicity equal to its dimension, cf. [a1], Chapt. 7, §2. There exists a generalized Peter–Weyl theorem for unimodular Lie groups, cf. [a1], Chapt. 14, §2. The description of (and the other unitary representations) in terms of the irreducible representations, including the fact that the irreducible unitary representations are finite dimensional, is known as Peter–Weyl theory, cf. e.g. [a2].

References

[a1]	A.O. Barut, R. Raçzka, "Theory of group representations and applications" , 1–2 , PWN (1977)
[a2]	A. Wawrzyńczyk, "Group representations and special functions" , Reidel (1984) pp. Sect. 4.4
[a3]	A.W. Knapp, "Representation theory of semisimple groups" , Princeton Univ. Press (1988) pp. 17