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| In mathematics, the '''Harish-Chandra character''', named after [[Harish-Chandra]], of a representation of a [[semisimple Lie group]] ''G'' on a [[Hilbert space]] ''H'' is a [[distribution (mathematics)|distribution]] on the group ''G'' that is analogous to the character of a finite dimensional representation of a [[compact group]].
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| ==Definition==
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| Suppose that π is an irreducible [[unitary representation]] of ''G'' on a Hilbert space ''H''.
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| If ''f'' is a [[compactly supported]] [[smooth function]] on the group ''G'', then the operator on ''H''
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| :<math>\pi(f) = \int_Gf(x)\pi(x)\,dx</math>
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| is of [[trace class]], and the distribution
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| :<math>\Theta_\pi:f\mapsto \operatorname{Tr}(\pi(f))</math> | |
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| is called the '''character''' (or '''global character''' or '''Harish-Chandra character''') of the representation.
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| The character Θ<sub>π</sub> is a distribution on ''G'' that is invariant under conjugation, and is an eigendistribution of the center of
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| the [[universal enveloping algebra]] of ''G'', in other words an invariant eigendistribution, with eigenvalue the [[infinitesimal character]] of the representation π.
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| [[Harish-Chandra's regularity theorem]] states that any invariant eigendistribution, and in particular any character of an irreducible unitary representation on a Hilbert space, is given by a [[locally integrable function]].
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| ==References==
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| *A. W. Knapp, ''Representation Theory of Semisimple Groups: An Overview Based on Examples.'' ISBN 0-691-09089-0
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| [[Category:Representation theory of Lie groups]]
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