mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, more specifically in the

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

reductive Lie group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

s, a

(\mathfrak,K)

-module is an algebraic object, first introduced by

Harish-Chandra Harish-Chandra Fellow of the Royal Society, 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. ...

, used to deal with continuous infinite-dimensional representations using algebraic techniques. Harish-Chandra showed that the study of irreducible unitary representations of a real reductive Lie group, ''G'', could be reduced to the study of irreducible

(\mathfrak,K)

-modules, where

\mathfrak

is the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

of ''G'' and ''K'' is a

maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classi ...

of ''G''.

Definition

Let ''G'' be a real Lie group. Let

\mathfrak

be its Lie algebra, and ''K'' a maximal compact subgroup with Lie algebra

\mathfrak

. A

(\mathfrak,K)

-module is defined as follows:This is James Lepowsky's more general definition, as given in section 3.3.1 of it is a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

''V'' that is both a

Lie algebra representation In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is g ...

\mathfrak

and a

group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to re ...

of ''K'' (without regard to the

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

of ''K'') satisfying the following three conditions :1. for any ''v'' ∈ ''V'', ''k'' ∈ ''K'', and ''X'' ∈

\mathfrak

k\cdot (X\cdot v)=(\operatorname(k)X)\cdot (k\cdot v)

:2. for any ''v'' ∈ ''V'', ''Kv'' spans a ''finite-dimensional'' subspace of ''V'' on which the action of ''K'' is continuous :3. for any ''v'' ∈ ''V'' and ''Y'' ∈

\mathfrak

\left.\left(\frac\exp(tY)\cdot v\right)\_=Y\cdot v.

In the above, the dot,

\cdot

, denotes both the action of

\mathfrak

on ''V'' and that of ''K''. The notation Ad(''k'') denotes the

adjoint action In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL( ...

of ''G'' on

\mathfrak

, and ''Kv'' is the set of vectors

k\cdot v

as ''k'' varies over all of ''K''. The first condition can be understood as follows: if ''G'' is the

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

GL(''n'', R), then

\mathfrak

is the algebra of all ''n'' by ''n'' matrices, and the adjoint action of ''k'' on ''X'' is ''kXk''⁻¹; condition 1 can then be read as :

kXv=kXk^kv=\left(kXk^\right)kv.

In other words, it is a compatibility requirement among the actions of ''K'' on ''V'',

\mathfrak

on ''V'', and ''K'' on

\mathfrak

. The third condition is also a compatibility condition, this time between the action of

\mathfrak

on ''V'' viewed as a sub-Lie algebra of

\mathfrak

and its action viewed as the differential of the action of ''K'' on ''V''.

Notes

References

* * {{DEFAULTSORT:(G,K)-Module Representation theory of Lie groups