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Free homogeneous spaces

Abstract:
Let $G$ be a compact Lie group, $K\subset G$. By $I_0(G/K)$ we denote the spectrum of the representation of the group $G$ in the functions on $G/K$. We say that a pair $(G,K)$ is free, if $I_0(G/K)$ is a free generated abelian semigroup.

Let $G/K$ be a non-free irreducible simply connected spherical space. Then $\dim Z_K=1$, $\dim Z_G=0$. Construct $G^\otimes = G \otimes U_1$, and let $Z_K \subset G^\otimes$ diagonally.

I assert that {\it $G^\otimes/K$ is a free irreducible spherical space.}

Denote by $\zeta$ some weight of $Z_{G^\otimes}\simeq Z_K$. Let $\{\varphi_1, ..., \varphi_p\}$ be a set of generators of the free semigroup $I_0 (G^\otimes/K)$, $r_i = \deg _\zeta \varphi_i$. Obviously,
$I_0 (G/K) \simeq \left\{ a_i \in {\mathbb Z}: \ a_i \geqm 0, \ \sum _{i=1}^p r_i a_i = 0 \right\} .$

{\bf Question.} Let $G/K$ be an arbitrary non-free homogeneous space. In what cases we can represent $I_0 (G/K)$ as a plane in a free semigroup $I_0 (G^\otimes/K)$?