Английская Википедия:Harish-Chandra module

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Шаблон:Refimprove 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 <math>(\mathfrak{g},K)</math>-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 <math>(\pi,V)</math> is a representation of G, then the Harish-Chandra module of <math>\pi</math> 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 <math>\varphi_v : G \longrightarrow V</math> via

<math>\varphi_v(g) = \pi(g)v</math>

is smooth, and the subspace

<math>\text{span}\{\pi(k)v : k\in K\}</math>

is finite-dimensional.

Notes

In 1973, Lepowsky showed that any irreducible <math>(\mathfrak{g},K)</math>-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 <math>(\mathfrak{g},K)</math>-module with a positive definite Hermitian form satisfying

<math>

\langle k\cdot v, w \rangle = \langle v, k^{-1}\cdot w \rangle </math> and

<math>

\langle Y\cdot v, w \rangle = -\langle v, Y\cdot w \rangle </math> for all <math>Y\in \mathfrak{g}</math> and <math>k\in K</math>, then X is the Harish-Chandra module of a unique irreducible unitary representation of G.

References

• Шаблон:Citation

See also

• (g,K)-module
• Admissible representation
• Unitary representation