  As sharpened in terms of Alesker's theory of valuations on manifolds, a
classic theorem of Weyl asserts that the coefficients of the tube polynomial of
an isometrically embedded riemannian manifold $M \hookrightarrow \mathbb R^n$
constitute a canonical finite dimensional subalgebra $\mathcal {L K}(M)$ of the
algebra $\mathcal{V} (M)$ of all smooth valuations on $M$, isomorphic to the
algebra of valuations on Euclidean space that are invariant under rigid
motions. We construct an analogous, larger, canonical subalgebra
$\mathcal{KLK}(M)\subset \mathcal{V}(M)$ for K\"ahler manifolds $M$: i) if
$\dim M = n $, then $\mathcal{KLK}(M)\simeq \mathrm{Val}^{\mathrm{U}(n)}$, the
algebra of valuations on $\mathbb{C}^n$ invariant under the holomorphic
isometry group, and ii) if $M\hookrightarrow \tilde M$ is a K\"ahler embedding,
then the restriction map $\mathcal{V}(\tilde M) \to \mathcal{V}(M)$ induces a
surjection $\mathcal{KLK}(\tilde M)\to \mathcal{KLK}(M)$. This answers a
question posed by Alesker in 2010 and gives a structural explanation for some
previously known, but mysterious phenomena in hermitian integral geometry.
