“(g,K)-module”的意思、由来-开放百科全书

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(g,K)-module

释义

Definition
Notes
References

In mathematics, more specifically in the representation theory of reductive Lie groups, a -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 -modules, where 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 be its Lie algebra, and K a maximal compact subgroup with Lie algebra . A -module is defined as follows:^[3] it is a vector space V that is both a Lie algebra representation of 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 ∈

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 ∈

In the above, the dot, , denotes both the action of on V and that of K. The notation Ad(k) denotes the adjoint action of G on , and Kv is the set of vectors 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 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

In other words, it is a compatibility requirement among the actions of K on V, on V, and K on . The third condition is also a compatibility condition, this time between the action of on V viewed as a sub-Lie algebra of and its action viewed as the differential of the action of K on V.

Notes

1. ^Page 73 of {{harvnb|Wallach|1988}}
2. ^Page 12 of {{harvnb|Doran|Varadarajan|2000}}
3. ^This is James Lepowsky's more general definition, as given in section 3.3.1 of {{harvnb|Wallach|1988}}

References

{{Citation

| editor1-last=Doran
| editor1-first=Robert S.
| editor2-last=Varadarajan
| editor2-first=V. S.
| title=The mathematical legacy of Harish-Chandra
| publisher=AMS
| series=Proceedings of Symposia in Pure Mathematics
| volume=68
| year=2000
| mr=1767886
| isbn=978-0-8218-1197-9
}}

{{Citation

| last=Wallach
| first=Nolan R.
| title=Real reductive groups I
| year=1988
| publisher=Academic Press
| series=Pure and Applied Mathematics
| volume=132
| mr=0929683
| isbn=978-0-12-732960-4
}}{{DEFAULTSORT:(G,K)-Module}}

1 : Representation theory of Lie groups

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