Harish Chandra module - Alchetron, The Free Social Encyclopedia

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 ) -module, then its Harish-Chandra module is a representation with desirable factorization properties.

Definition

Let G be a Lie group and K a compact subgroup of G. If ( π , V ) is a representation of G, then the Harish-Chandra module of π 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 φ v : G ⟶ V via

φ v ( g ) = π ( g ) v

is smooth, and the subspace

span { π ( k ) v : k ∈ K }

is finite-dimensional.

References

Harish-Chandra module Wikipedia

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