mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, more specifically in the

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

of reductive Lie groups, a

(\mathfrak,K)

-module is an algebraic object, first introduced by Harish-Chandra, 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 In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

, ''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 operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

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

maximal compact subgroup In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. T ...

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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

''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 i ...

\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 ...

of ''K'' (without regard to the

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

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 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 n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...

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