Harish-Chandra module explained
In mathematics, specifically in the representation theory of Lie groups, a Harish-Chandra module, named after the Indian mathematician and physicist Harish-Chandra, is a representation of a real Lie group, associated to a general representation, with regularity and finiteness conditions. When the associated representation is a
-module, then its Harish-Chandra module is a representation with desirable factorization properties.
Definition
Let G be a Lie group and K a compact subgroup of G. If
is a representation of
G, then the
Harish-Chandra module of
is the subspace
X of
V consisting of the
K-finite smooth vectors in
V. This means that
X includes exactly those vectors
v such that the map
\varphiv:G\longrightarrowV
via
is smooth, and the subspace
is finite-dimensional.
Notes
In 1973, Lepowsky showed that any irreducible
-module
X is isomorphic to the Harish-Chandra module of an irreducible representation of
G on a
Hilbert space. Such representations are
admissible, meaning that they decompose in a manner analogous to the prime factorization of integers. (Of course, the decomposition may have infinitely many distinct factors!) Further, a result of Harish-Chandra indicates that if
G is a
reductive Lie group with maximal compact subgroup
K, and
X is an irreducible
-module with a positive definite Hermitian form satisfying
\langlek ⋅ v,w\rangle=\langlev,k-1 ⋅ w\rangle
and
\langleY ⋅ v,w\rangle=-\langlev,Y ⋅ w\rangle
for all
and
, then
X is the Harish-Chandra module of a unique irreducible unitary representation of
G.
See also
