Knowlpedia

Definition

Gårding subspace

The subspace generated by applying compactly supported smooth convolution operators to vectors in a continuous Lie-group representation.

Definition

Let GG be a finite-dimensional with left , and let (π,H)(\pi,\mathcal H) be a . For f∈Cc∞(G)f\in C_c^\infty(G), define

π(f)v=∫Gf(g)π(g)v dg.\pi(f)v=\int_G f(g)\pi(g)v\,dg.

The Gårding subspace is

HG=span⁡{π(f)v:f∈Cc∞(G), v∈H}.\mathcal H_G=\operatorname{span}\{\pi(f)v:f\in C_c^\infty(G),\ v\in\mathcal H\}.

Thus it is the image span of the smooth compactly supported convolution algebra under the . Its elements are called Gårding vectors. The subspace is GG-invariant, dense in H\mathcal H, and contained in the smooth-vector space.

Smoothing mechanism

Differentiation may be moved from the to the test function:

dπ(X)π(f)v=−π(Xrightf)vd\pi(X)\pi(f)v=-\pi(X_{\mathrm{right}}f)v

with the sign and invariant determined by the convolution convention. Since every iterated derivative of ff is again smooth and compactly supported, π(f)v\pi(f)v is a . This is the representation-theoretic analogue of regularization by convolution.

Density and factorization

A smooth approximate identity fjf_j supported near the identity satisfies π(fj)v→v\pi(f_j)v\to v, which proves density; this is the . The deeper strengthens the inclusion HG⊆H∞\mathcal H_G\subseteq\mathcal H^\infty to equality.

Conventions and scope

Some authors call HG\mathcal H_G the Gårding domain and reserve “Gårding space” for a completed locally convex version. The algebraic span is essential: the set of individual vectors π(f)v\pi(f)v need not itself be closed under addition in an obvious single-factor form. The same construction works for continuous representations on suitable complete , with the integral interpreted there.

References
  1. Lars Gårding, Note on Continuous Representations of Lie Groups, Proceedings of the National Academy of Sciences 33 (1947), 331–332. DOI record. Relevant: smoothing by compactly supported functions and density.
  2. Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Springer, 1972. DOI record. Relevant: §4.4 on Gårding vectors and differentiable vectors.