Linear Morse Functions on Orbit Spaces

Consider a linear action of a Lie group G on the matrix space M_n (𝔽). Each orbit space is then an embedding of a homogeneous space in the euclidean space M ⊂ M_n (𝔽). In this paper, we give a characterization of all linear maps on M_n (𝔽) which when restricted to M define a Morse function on M, for the case M is one of the orthogonal groups O(n, 𝔽), the Stiefel manifolds 𝕍_k (ℝⁿ) and the flag manifolds F(n₁, n₂, . . . , n_r).