Calculating Cartan spaces for affine homogeneous spaces

Let $G$ be a connected reductive algebraic group and $H$ a reductive subgroup of it. Fix a Borel subgroup $B$ of $G$ and a maximal torus $T\subset B$. The Cartan subspace $\mathfrak a_{G,G/H}$ is by definition a subspace of $\mathfrak t^*$ (where $\mathfrak t$ is the Lie algebra of the group $T$) spanned by the weights of all the $B$-semi-invariant rational functions on $G/H$. The spaces $\mathfrak a_{G,G/H}$ are calculated in this paper.
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