Character of a representation of a group

In the case of a finite-dimensional representation $ \pi $ this is the function $ \chi _ \pi $ on the group $ G $ defined by the formula

$$ \chi _ \pi ( g) = \ \mathop{\rm tr} \pi ( g),\ \ g \in G. $$

For arbitrary continuous representations of a topological group $ G $ over $ \mathbf C $ this definition is generalized as follows:

$$ \chi _ \pi ( g) = \ \chi ( \pi ( g)) \ \ \textrm{ for } \ g \in G, $$

where $ \chi $ is a linear functional defined on some ideal $ I $ of the algebra $ A $ generated by the family of operators $ \pi ( g) $, $ g \in G $, that is invariant under inner automorphisms of $ A $. In certain cases the character of a representation $ \pi $ is defined as that of the representation of a certain group algebra of $ G $ determined by $ \pi $ (see Character of a representation of an associative algebra).

The character of a direct sum (of a tensor product) of finite-dimensional representations is equal to the sum (the product) of the characters of these representations. The character of a finite-dimensional representation of a group is a function that is constant on classes of conjugate elements; the character of a continuous finite-dimensional unitary representation of a group is a continuous positive-definite function on the group.

In many cases the character of a representation of a group determines the representation uniquely up to equivalence; for example, the character of an irreducible finite-dimensional representation over a field of characteristic 0 determines the representation uniquely up to spatial equivalence; the character of a finite-dimensional continuous unitary representation of a compact group is determining up to unitary equivalence.

The character of a representation of a locally compact group $ G $ admitting an extension to a representation of the algebra of continuous functions of compact support on $ G $ can be defined by a measure on $ G $; in particular, the character of the regular representation of a unimodular group is given by a probability point measure concentrated at the unit element of $ G $. The character of a representation $ \pi $ of a Lie group $ G $ admitting an extension to a representation of the algebra $ C _ {0} ^ \infty ( G) $ of infinitely-differentiable functions of compact support on $ G $ can be defined as a generalized function on $ G $. If $ G $ is a nilpotent or a linear semi-simple Lie group, then the characters of irreducible unitary representations $ \pi $ of $ G $ are defined by locally integrable functions $ \psi _ \pi $ according to the formula

$$ \chi _ \pi ( f ) = \ \int\limits _ { G } f ( g) \psi _ \pi ( g) dg,\ \ f \in C _ {0} ^ \infty ( G). $$

These characters determine the representation $ \pi $ uniquely up to unitary equivalence.

If the group $ G $ is compact, every continuous positive-definite function on $ G $ that is constant on classes of conjugate elements can be expanded into a series with respect to the characters of the irreducible representations $ \pi _ \alpha $ of $ G $. The series converges uniformly on $ G $ and the characters $ \chi _ {\pi _ \alpha } $ form an orthonormal system in the space $ L _ {2} ( G) $ that is complete in the subspace of functions in $ L _ {2} ( G) $ that are constant on classes of conjugate elements in $ G $. If $ \chi _ \rho = \sum _ \alpha m _ \alpha \chi _ {\pi _ \alpha } $ is the expansion of the character of a continuous finite-dimensional representation $ \rho $ of the group $ G $ with respect to the characters $ \chi _ {\pi _ \alpha } $, then the $ m _ \alpha $ are integers, namely, the multiplicities with which the $ \pi _ \alpha $ occur in $ \rho $. If $ \rho $ is a continuous representation of $ G $ in a quasi-complete, barrelled, locally convex topological space $ E $, then there exists a maximal subspace $ E _ \alpha $ of $ E $ such that the restriction of $ \rho $ to $ E _ \alpha $ is a multiple of $ \pi _ \alpha $, and there is a continuous projection $ P _ \alpha $ of $ E $ onto $ E _ \alpha $, defined by

$$ P _ \alpha = \ \chi _ {\pi _ \alpha } ( e) \int\limits _ { G } {\chi _ {\pi _ \alpha } ( g) } bar \rho ( g) dg, $$

where $ dg $ is the Haar measure on $ G $ for which $ \int _ {G} dg = 1 $.

References

[1]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[2]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[3]	J. Dixmier, "$C^\star$ algebras" , North-Holland (1977) (Translated from French)
[4]	G.F. Frobenius, J.-P. Serre (ed.) , Gesammelte Abhandlungen , Springer (1968)
[5]	M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian)
[6]	D.E. Littlewood, "The theory of group characters and matrix representations of groups" , Clarendon Press (1950)