Distribution vectors of a Lie-group representation

Definition

Let $G$ be a Lie group and let $(\pi,\mathcal H)$ be a strongly continuous unitary representation. Write $\mathcal H^\infty$ for its space of smooth vectors, with the usual Fréchet topology. The space of distribution vectors is

\mathcal H^{-\infty}=(\mathcal H^\infty)'_{\mathrm{anti}},

the continuous antilinear dual of $\mathcal H^\infty$ , equipped usually with its strong dual topology. The Hilbert-space pairing gives continuous dense inclusions

\mathcal H^\infty\subseteq\mathcal H\subseteq\mathcal H^{-\infty}.

Thus a distribution vector is a generalized vector acting continuously on smooth test vectors, not necessarily an element of $\mathcal H$ .

The smooth-vector topology

Choose a finite basis of the complexified Lie algebra and take all ordered monomials in that basis. Equivalently, let $D$ range over a vector-space basis of the universal enveloping algebra. Seminorms of the form

v\longmapsto\lVert d\pi(D)v\rVert,\qquad D\in U(\mathfrak g_{\mathbb C}),

define the Fréchet topology on $\mathcal H^\infty$ . Different finite bases or generating sets of $\mathfrak g_{\mathbb C}$ , together with all monomials in them, give equivalent countable seminorm families. No single finite collection of these seminorms generally suffices. Continuity with respect to the full family distinguishes distribution vectors from arbitrary algebraic functionals Wallach, §4.4.

Extended group and Lie-algebra actions

The contragredient action is defined by

(\pi^{-\infty}(g)\lambda)(v)=\lambda(\pi(g^{-1})v).

Differentiating on the test-vector side extends the derived representation to distribution vectors:

(d\pi^{-\infty}(X)\lambda)(v)=-\lambda(d\pi(X)v).

These formulas are well-defined because the original action preserves $\mathcal H^\infty$ continuously.

Use in representation theory

Distribution vectors allow equivariant maps to be encoded by generalized matrix coefficients and invariant functionals. Delta distributions in geometric realizations and automorphic distribution vectors are typical examples. In the representation theory of real reductive groups, the passage between smooth globalizations and their distribution duals is a basic tool; it is broader than the Hilbert-space representation and must retain the chosen locally convex topology.

References

Nolan R. Wallach, Real Reductive Groups I, Pure and Applied Mathematics 132, Academic Press, 1988. WorldCat record. Relevant: §4.4 on smooth and distribution vectors.
W. Casselman, “Canonical Extensions of Harish-Chandra Modules to Representations of $G$ ,” Canadian Journal of Mathematics 41 (1989), 385–438. DOI record. Relevant: smooth globalizations and distributional duality.