Spaces of smooth and ultradifferentiable vectors associated with Lie group representations

The central objects of study in this thesis are spaces of regular vectors associated with Lie group representations and their role in various problems from functional analysis. Given a representation π of a Lie group G on a locally convex topological vector space E, one may look at the vectors v ∈ E for which the map G → E : x 7→ π(x)v is regular, that is to say, smooth, real analytic, or ultradifferentiable. They form the spaces of regular vectors. Notonly fill these spaces an important theoretical need in the abstract theory of infinite-dimensional representations, but, as it turns out, many classical spaces from functional analysis can be viewed as spaces of regular vectors associated with some Lie group representation. This useful point of view allows us to simultaneously treat some algebraic and functional analytic problems for a variety of function spaces via the broader perspective of regular vectors of suitable Lie group representations.

A first goal of this thesis is to rigorously introduce the underlying foundations of these spaces of regular vectors. In particular, many classical definitions and results are generalized; see for example Propositions 5.2.2, 5.4.7 and 5.5.3. We then use this framework to study two problems in analysis.

The first one concerns factorization problems, which are an important topic in harmonic analysis with a long tradition. If G is a compact Lie group, then, associated with any representation, there is a natural action Π of the convolution algebra of real analytic functions A(G) on the space of real analytic vectors E ω. We show that E ω factorizes over this algebra, that is, for any v ∈ Eω there exist w ∈ E ω and χ ∈ A( G) such that v = Π(χ)w. In fact, an even stronger result holds, cf. Theorem 7.1.1. This solves a particular, but important case of a recent conjecture about such factorization properties; see [60, Conjecture 6.4].

The second problem we consider is about two linear topological invariants in this context – namely, quasinormability and the property (Ω) – and more precisely, whether spaces of regular vectors inherit these invariants from the original representation space. Under very mild hypotheses, we show that the answer is positive; see Theorems 9.0.1 and 9.0.2.

These lifting properties are then made concrete for weighted Fréchet function spaces that are invariant under the right regular action of their
underlying Lie group; see Theorems 10.0.9 and 10.0.11.

The results of this thesis provide therefore a new powerful and systematic method to establish factorization properties, quasinormability, or the property (Ω), for large families of function spaces on the corresponding Lie groups.