View source for Character of a representation of a group

In the case of a finite-dimensional representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c0215701.png" /> this is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c0215702.png" /> on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c0215703.png" /> defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c0215704.png" /></td> </tr></table>

For arbitrary continuous representations of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c0215705.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c0215706.png" /> this definition is generalized as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c0215707.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c0215708.png" /> is a linear functional defined on some ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c0215709.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157010.png" /> generated by the family of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157012.png" />, that is invariant under inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157013.png" />. In certain cases the character of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157014.png" /> is defined as that of the representation of a certain [[Group algebra|group algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157015.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157016.png" /> (see [[Character of a representation of an associative algebra|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157017.png" /> admitting an extension to a representation of the algebra of continuous functions of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157018.png" /> can be defined by a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157019.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157020.png" />. The character of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157021.png" /> of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157022.png" /> admitting an extension to a representation of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157023.png" /> of infinitely-differentiable functions of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157024.png" /> can be defined as a generalized function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157026.png" /> is a nilpotent or a linear semi-simple Lie group, then the characters of irreducible unitary representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157028.png" /> are defined by locally integrable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157029.png" /> according to the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157030.png" /></td> </tr></table>

These characters determine the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157031.png" /> uniquely up to unitary equivalence.

If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157032.png" /> is compact, every continuous positive-definite function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157033.png" /> that is constant on classes of conjugate elements can be expanded into a series with respect to the characters of the irreducible representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157035.png" />. The series converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157036.png" /> and the characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157037.png" /> form an orthonormal system in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157038.png" /> that is complete in the subspace of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157039.png" /> that are constant on classes of conjugate elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157040.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157041.png" /> is the expansion of the character of a continuous finite-dimensional representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157042.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157043.png" /> with respect to the characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157044.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157045.png" /> are integers, namely, the multiplicities with which the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157046.png" /> occur in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157047.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157048.png" /> is a continuous representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157049.png" /> in a quasi-complete, barrelled, locally convex topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157050.png" />, then there exists a maximal subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157051.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157052.png" /> such that the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157053.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157054.png" /> is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157055.png" />, and there is a continuous projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157057.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157058.png" />, defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157059.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157060.png" /> is the Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157061.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157062.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov,   "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Dixmier,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021570/c02157063.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.F. Frobenius,   J.-P. Serre (ed.) , ''Gesammelte Abhandlungen'' , Springer  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Naimark,   "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.E. Littlewood,   "The theory of group characters and matrix representations of groups" , Clarendon Press  (1950)</TD></TR></table>