Let me give a very elemntary descriptiopn of the Morse-theoretical interpretation of the Bruhat decomposition in a simple case, of thecomplex Grasmannians. $\DeclareMathOperator{\Gr}{Gr}$ Denote by $\Gr_{k,n}$ the Grassmannian of complex $k$-dimensional subspaces of $\mathbb{C}^n$.
We can identify $\Gr_{k,n}$ with a submanifold of the vector space $\mathscr{S}_n$ of hermitian $n\times n$ matrices by associating to a subspace $L$ the orthogonal projection $P_L$. Let $A\in\mathscr{S}_n$ a hermitian matrix with distinct eigenvalues. Define $\DeclareMathOperator{\tr}{tr}$
$$f_A:\Gr_{k,n}\to\mathbb{R},\;\;L\mapsto $f_A:{\Gr}_{k,n}\to\mathbb{R},\;\;L\mapsto \tr(AP_L).$$
Denote by $X_A$ the (negative) gradient of $f_AS$ f_A$ with respect to the metric on $\Gr_{k,n}$ induced by the Euclidean metric on $\mathscr{S}_n$. Then the unstable manifolds of the flow generated by $X_A$ are the Schubert cells on $\Gr_{k,n}$ with respect to a falg flag determined by the eigenvectors of $A$.
