Computation of the Cartan spaces of affine homogeneous spaces

Let $G$ be a reductive algebraic group and $H$ its reductive subgroup. Fix a Borel subgroup $B\subset G$ and a maximal torus $T\subset B$. The Cartan space ${\text{a}}_{G,G\mathrm{/}H}$ is, by definition, the subspace of $\text{Lie}(T{)}^{\ast }$ generated by the weights of $B$-semiinvariant rational functions on $G\mathrm{/}H$. We compute the spaces ${\text{a}}_{G,G\mathrm{/}H}$.