In mathematics, the Harish-Chandra character, named after
Harish-Chandra
Harish-Chandra (né Harishchandra) FRS (11 October 1923 – 16 October 1983) was an Indian-American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.
Early ...
, of a representation of a
semisimple Lie group
In mathematics, a simple Lie group is a connected space, connected nonabelian group, non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple ...
''G'' on a
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
''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 varia ...
on the group ''G'' that is analogous to the character of a finite-dimensional representation of a
compact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
.
Definition
Suppose that π is an irreducible
unitary representation
In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in the ca ...
of ''G'' on a Hilbert space ''H''.
If ''f'' is a
compactly supported
In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set ...
smooth function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
on the group ''G'', then the operator on ''H''
:
is of
trace class
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of tra ...
, 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
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representa ...
of ''G'', in other words an invariant eigendistribution, with eigenvalue the
infinitesimal character
In mathematics, the infinitesimal character of an irreducible representation \rho of a semisimple Lie group ''G'' on a vector space ''V'' is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagon ...
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
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function (mathematics), function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importanc ...
.
References
*A. W. Knapp, ''Representation Theory of Semisimple Groups: An Overview Based on Examples.'' {{isbn, 0-691-09089-0
Representation theory of Lie groups