In mathematics, the Harish-Chandra character, named after
Harish-Chandra, of a representation of a
semisimple Lie group ''G'' on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''H'' is a
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
on the group ''G'' that is analogous to the character of a finite-dimensional representation of a
compact group.
Definition
Suppose that π is an irreducible
unitary representation of ''G'' on a Hilbert space ''H''.
If ''f'' is a
compactly supported smooth function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
on the group ''G'', then the operator on ''H''
:
is of
trace class, and the distribution
:
is called the character (or global character or Harish-Chandra character) of the representation.
The character Θ
π is a distribution on ''G'' that is invariant under conjugation, and is an eigendistribution of the center of
the
universal enveloping algebra of ''G'', in other words an invariant eigendistribution, with eigenvalue the
infinitesimal character of the representation π.
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.
References
*A. W. Knapp, ''Representation Theory of Semisimple Groups: An Overview Based on Examples.'' {{isbn, 0-691-09089-0
Representation theory of Lie groups