K-type - Imported Knowlpedia Content

Definition

Let $K$ be a compact group and $V$ a complex $K$ -representation that is a direct sum of finite-dimensional irreducibles, as happens for the $K$ -finite part of a continuous unitary representation. A $K$ -type of $V$ is an equivalence class $\tau$ of irreducible finite-dimensional $K$ -representations for which

\operatorname{Hom}_K(E_\tau,V)\neq 0,

where $E_\tau$ is a representative. Its multiplicity is $\dim\operatorname{Hom}_K(E_\tau,V)$ . The $\tau$ -isotypic component $V(\tau)$ is the sum of all $K$ -submodules isomorphic to $E_\tau$ ; it is naturally isomorphic to $E_\tau\otimes\operatorname{Hom}_K(E_\tau,V)$ .

Decomposition

For a unitary representation of $K$ , the Peter–Weyl theorem gives an orthogonal Hilbert direct sum of isotypic components. Its $K$ -finite part is the algebraic direct sum

V_K=\bigoplus_{\tau\in\widehat K}V(\tau).

Thus each $K$ -finite vector involves only finitely many $K$ -types, although the representation may contain infinitely many types overall. These decompositions underlie the treatment of admissible representations in Knapp, Chapter VIII.

Example

For $K=S^1$ , every irreducible representation is a character $z\mapsto z^m$ , $m\in\mathbb Z$ . Under rotations on $L^2(S^1)$ , the $m$ -th $K$ -type is the one-dimensional span of the Fourier mode $z^m$ , and its multiplicity is one. A general compact group can have higher-dimensional irreducible $K$ -types and multiplicities larger than one.

Conventions and scope

The phrase “a $K$ -type” may mean either the equivalence class $\tau$ , a chosen model $E_\tau$ , or the associated isotypic component; context should distinguish them. A $K$ -type is not an individual vector. In real-reductive theory, $K$ is normally a maximal compact subgroup.

References

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