Casselman–Wallach representation - Imported Knowlpedia Content

Definition

Let $G$ be a real reductive group and $K$ a maximal compact subgroup. A Casselman–Wallach representation is a continuous representation $(\pi,E)$ on a complex Fréchet space such that the action is smooth, its $K$ -finite part $E_K$ is a Harish–Chandra module, and the action has moderate growth. The latter means that, for every continuous seminorm $p$ on $E$ , there are a continuous seminorm $q$ and $N\geq0$ with

p(\pi(g)v)\leq \lVert g\rVert^Nq(v)

for all $g\in G$ and $v\in E$ , where $\lVert\cdot\rVert$ is any fixed algebraic scale on $G$ .

The role of the three conditions

Smoothness requires every orbit map $g\mapsto\pi(g)v$ to be smooth. Admissibility is algebraic: $E_K$ has finite $K$ -multiplicities and is finitely generated over $U(\mathfrak g)$ . Moderate growth controls every defining seminorm uniformly by a polynomial scale on $G$ . Replacing the algebraic scale by an equivalent one does not change the condition. Bernstein and Krötz formulate these objects as smooth admissible moderate-growth Fréchet representations, the objects of their category $\mathcal{SAF}$ Bernstein–Krötz, Introduction and §1.3.

Structure and consequences

The underlying Fréchet space of a Casselman–Wallach representation is nuclear, and continuous $G$ -maps between such representations are controlled by their restrictions to $K$ -finite vectors. The category is closed under kernels and cokernels in the form dictated by the corresponding exact operations on Harish–Chandra modules. Most importantly, the Casselman–Wallach globalization theorem says that taking $K$ -finite vectors loses no categorical information.

Examples and near-misses

If $V$ is a Harish–Chandra module, its canonical smooth globalization $V^\infty$ is the basic example. Smooth vectors in a Hilbert globalization also give a Casselman–Wallach representation once their $K$ -finite part is $V$ and moderate growth is verified Bernstein–Krötz, §4.1. A smooth Fréchet representation of moderate growth whose $K$ -finite vectors have an infinite $K$ -type multiplicity is a near-miss: it fails admissibility.

Conventions and scope

The synonymous abbreviations “SAF” and “SF” are not completely uniform across sources. In Bernstein–Krötz, an SF-representation means a smooth Fréchet representation satisfying the relevant growth condition, while $\mathcal{SAF}$ additionally imposes admissibility. Here “Casselman–Wallach representation” includes admissibility, so its $K$ -finite part is a Harish–Chandra module. Nuclearity is a consequence in this setting, not an extra axiom in the core definition.

References

Joseph Bernstein and Bernhard Krötz, “Smooth Fréchet Globalizations of Harish-Chandra Modules,” Israel Journal of Mathematics 199 (2014), 45–111. DOI record. Relevant: Introduction, §1.3 on moderate growth, and §4.1 on globalizations.
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 smooth moderate-growth category.