On modality and complexity of affine embeddings

Let $G$ be a reductive algebraic group and let $H$ be a reductive subgroup of $G$. The modality of a $G$-variety $X$ is the largest number of the parameters in a continuous family of $G$-orbits in $X$. A precise formula for the maximum value of the modality over all affine embeddings of the homogeneous space $G/H$ is obtained.