K-finite vector - Imported Knowlpedia Content

Definition

Let $\pi$ be a representation of a group $G$ on a complex vector space $V$ , and let $K\subseteq G$ be compact. A vector $v\in V$ is $K$ -finite when

\operatorname{span}_{\mathbb C}\{\pi(k)v:k\in K\}

is finite-dimensional. The $K$ -finite vectors form a $K$ -stable linear subspace $V_K$ , called the $K$ -finite part. For a strongly continuous unitary representation, the finite-dimensional orbit span is a finite direct sum of irreducible $K$ -representations; equivalently, a $K$ -finite vector has nonzero components in only finitely many irreducible summands.

Density and algebraic role

If $K$ is compact and $\pi|_K$ is continuous and unitary, then $V_K$ is dense in the Hilbert space. This follows by applying the Peter–Weyl theorem to the restricted representation. When $G$ is a real reductive group and $K$ a maximal compact subgroup, the $K$ -finite part is the algebraic carrier on which the differentiated Lie-algebra action and the $K$ -action combine into a $(\mathfrak g,K)$ -module Wallach, §3.3. Its decomposition is organized by the $K$ -types occurring in the restricted representation.

Example

Let $K=S^1$ act on $L^2(S^1)$ by rotations. The characters $z\mapsto z^m$ are $K$ -finite, and every trigonometric polynomial is $K$ -finite because it involves finitely many characters. A function with infinitely many nonzero Fourier coefficients is not $K$ -finite, although such functions are limits of $K$ -finite vectors.

Conventions and scope

The letter $K$ usually denotes a maximal compact subgroup in real-reductive representation theory, but the definition makes sense for any compact subgroup. It concerns the algebraic span of an orbit, not whether the orbit is a finite set. Thus a one-dimensional character has $K$ -finite vectors even when its image is infinite.

References

Nolan R. Wallach, Real Reductive Groups I, Academic Press, 1988. Publisher record. Relevant: Chapter 3, §3.3 on $(\mathfrak g,K)$ -modules and $K$ -finite vectors.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. DOI record. Relevant: Chapter VIII on admissible representations and $K$ -finite vectors.