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 (g,K)displaystyle (mathfrak g,K)-module, then its Harish-Chandra module is a representation with desirable factorization properties.

Definition[edit]

Let G be a Lie group and K a compact subgroup of G. If (π,V)displaystyle (pi ,V) is a representation of G, then the Harish-Chandra module of πdisplaystyle pi 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 φv:G⟶Vdisplaystyle varphi _v:Glongrightarrow V via

φv(g)=π(g)vdisplaystyle varphi _v(g)=pi (g)v

is smooth, and the subspace

spanπ(k)v:k∈Kdisplaystyle textspanpi (k)v:kin K

is finite-dimensional.

Notes[edit]

In 1973, Lepowsky showed that any irreducible (g,K)displaystyle (mathfrak g,K)-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
(g,K)displaystyle (mathfrak g,K)-module with a positive definite Hermitian form satisfying

⟨k⋅v,w⟩=⟨v,k−1⋅w⟩displaystyle langle kcdot v,wrangle =langle v,k^-1cdot wrangle

and

⟨Y⋅v,w⟩=−⟨v,Y⋅w⟩displaystyle langle Ycdot v,wrangle =-langle v,Ycdot wrangle

for all Y∈gdisplaystyle Yin mathfrak g and k∈Kdisplaystyle kin K, then X is the Harish-Chandra module of a unique irreducible unitary representation of G.

References[edit]

• 
  Vogan, Jr., David A. (1987), Unitary Representations of Reductive Lie Groups, Annals of Mathematics Studies, 118, Princeton University Press, ISBN 978-0-691-08482-4.mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
  

See also[edit]

• (g,K)-module


• Admissible representation


• Unitary representation