Английская Википедия:(g,K)-module
In mathematics, more specifically in the representation theory of reductive Lie groups, a <math>(\mathfrak{g},K)</math>-module is an algebraic object, first introduced by Harish-Chandra,[1] 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 <math>(\mathfrak{g},K)</math>-modules, where <math>\mathfrak{g}</math> is the Lie algebra of G and K is a maximal compact subgroup of G.[2]
Definition
Let G be a real Lie group. Let <math>\mathfrak{g}</math> be its Lie algebra, and K a maximal compact subgroup with Lie algebra <math>\mathfrak{k}</math>. A <math>(\mathfrak{g},K)</math>-module is defined as follows:[3] it is a vector space V that is both a Lie algebra representation of <math>\mathfrak{g}</math> and a group representation of K (without regard to the topology of K) satisfying the following three conditions
- 1. for any v ∈ V, k ∈ K, and X ∈ <math>\mathfrak{g}</math>
- <math>k\cdot (X\cdot v)=(\operatorname{Ad}(k)X)\cdot (k\cdot v)</math>
- 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 ∈ <math>\mathfrak{k}</math>
- <math>\left.\left(\frac{d}{dt}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v.</math>
In the above, the dot, <math>\cdot</math>, denotes both the action of <math>\mathfrak{g}</math> on V and that of K. The notation Ad(k) denotes the adjoint action of G on <math>\mathfrak{g}</math>, and Kv is the set of vectors <math>k\cdot v</math> as k varies over all of K.
The first condition can be understood as follows: if G is the general linear group GL(n, R), then <math>\mathfrak{g}</math> 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
- <math>kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv.</math>
In other words, it is a compatibility requirement among the actions of K on V, <math>\mathfrak{g}</math> on V, and K on <math>\mathfrak{g}</math>. The third condition is also a compatibility condition, this time between the action of <math>\mathfrak{k}</math> on V viewed as a sub-Lie algebra of <math>\mathfrak{g}</math> and its action viewed as the differential of the action of K on V.
Notes
References
- ↑ Page 73 of Шаблон:Harvnb
- ↑ Page 12 of Шаблон:Harvnb
- ↑ This is James Lepowsky's more general definition, as given in section 3.3.1 of Шаблон:Harvnb
