Peter-Weyl theorem

A theorem on the approximation of functions on a compact topological group by means of representation functions (cf. Representation function). Let $ \pi $ run through a family $ \Sigma $ of representatives of all equivalence classes for the irreducible continuous unitary representations of a compact group $ G $( cf. Representation of a topological group). Let $ \mathop{\rm dim} \pi $ be the dimension of a representation $ \pi $ and let $ u _ {ij} ^ {( \pi ) } $ be its matrix elements in some orthonormal basis. The Peter–Weyl theorem asserts that the functions

$$ \sqrt { \mathop{\rm dim} \pi } u _ {ij} ^ {( \pi ) } \ ( \pi \in \Sigma ) $$

form an orthonormal basis in the space $ L _ {2} (G) $ of square-summable functions with respect to the Haar measure on $ G $( the measure of the entire group is taken to be 1). The algebra of all complex-valued representation functions on $ G $, which coincides with the set of finite linear combinations of the functions $ u _ {ij} ^ {( \pi ) } $, $ \pi \in \Sigma $, is uniformly dense in the space of all continuous complex-valued functions in $ G $.

If $ G=T $ 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 $ G $ is dense in the algebra of all continuous functions on $ G $, constant on classes of conjugate elements. Another consequence is that for any element $ a \in G $, $ a \neq e $, there is an irreducible continuous representation $ \phi $ of $ G $ such that $ \phi (a) \neq e $; if, on the other hand, $ G $ is a compact Lie group, then $ G $ 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 $ \phi $ of a compact group $ G $ in a Fréchet space $ E $. Then the subspace of representation elements of $ E $ is dense in $ E $. Here an element $ v \in E $ is called a representation, or spherical or almost-invariant, element if the orbit $ \phi (G)v $ generates a finite-dimensional subspace in $ E $. This is applicable in particular to the case where $ E $ is the space of sections of a certain smoothness class of smooth vector $ G $- 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 $ G $.

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 $ G $- finite element.

The statement that the algebra of complex-valued representation functions is uniformly dense in the algebra of continuous functions on $ G $ 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 $ L _ {2} (G) $( 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