Proposition 1

— For any compact Lie group H, the $\mathbf Z$-*module* $X(H)$ is of finite type. It is free if H is connected.

Assume first that H is connected; every element of $X(H)$ vanishes on the derived group $D(H)$ of H, hence we have a homomorphism $X(H/D(H))\longrightarrow X(H)$. But $H/D(H)$ is connected and commutative, hence is a torus, and $X(H/D(H))$ is a free $\mathbf Z$-module of finite type (Spectral Theories, Chap. II, §2, no. 1, Cor. 2 of Prop. 1). In the general case, it follows from the exactness of the sequence

where $X(H_0)$ is free of finite type and $X(H/H_0)$ is finite, that $X(H)$ is of finite type.