Casselman–Wallach globalization theorem - Imported Knowlpedia Content

Statement

Let $G$ be a real reductive group with maximal compact subgroup $K$ . The Casselman–Wallach globalization theorem states that the functor

E\longmapsto E_K

from Casselman–Wallach representations with continuous $G$ -maps to Harish–Chandra modules with $(\mathfrak g,K)$ -maps is an equivalence of categories. Thus every Harish–Chandra module $V$ has a smooth admissible moderate-growth Fréchet globalization $V^\infty$ , unique up to a unique continuous $G$ -isomorphism compatible with the identification $(V^\infty)_K\cong V$ . Moreover, every $(\mathfrak g,K)$ -map between Harish–Chandra modules extends uniquely to a continuous $G$ -map between their globalizations.

What uniqueness means

A Harish–Chandra module can be completed in many inequivalent Banach or Hilbert norms. The theorem does not identify all such completions. It says that after passing to the smooth Fréchet, moderate-growth category, the globalization is canonical up to the categorical uniqueness stated in the core. In Casselman’s formulation, a finitely generated Harish–Chandra module has exactly one smooth representation of moderate growth with the prescribed underlying $(\mathfrak g,K)$ -module, up to canonical topological isomorphism Casselman, introduction.

Consequences

Submodules, quotients, extensions, and morphisms may be studied algebraically and then globalized. In particular, irreducibility and finite length correspond across the equivalence. If $V^\infty$ denotes the globalization, Bernstein and Krötz identify it as

V^\infty=\pi(\mathcal S(G))V,

the span obtained by acting on $V$ with the Harish–Chandra Schwartz algebra. Their proof also shows that minimal and maximal smooth Fréchet globalizations coincide Bernstein–Krötz, Introduction and §8.3.

Boundary of the theorem

References

W. Casselman, “Canonical Extensions of Harish-Chandra Modules to Representations of $G$ ,” Canadian Journal of Mathematics 41 (1989), 385–438. DOI record. Relevant: introduction and the canonical smooth extension.
Joseph Bernstein and Bernhard Krötz, “Smooth Fréchet Globalizations of Harish-Chandra Modules,” Israel Journal of Mathematics 199 (2014), 45–111. DOI record. Relevant: Introduction, §§5–8, especially §8.3.