TITLE: Is there a Morse theory proof of the Bruhat decomposition?
QUESTION [27 upvotes]: Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup_{w \in W} BwB$ of cells given by double cosets.
One way I have seen to obtain cell decompositions of manifolds is using Morse theory. Is there a way to prove the Bruhat decomposition using Morse theory for a certain well-chosen function $f$? More generally, are there heuristics when obtaining a certain cell decomposition through Morse theory is likely to work?

REPLY [8 votes]: Let me give  a very elementary description 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 $\DeclareMathOperator{\tr}{tr}$
$$f_A:{\Gr}_{k,n}\to\mathbb{R},\;\;L\mapsto  \tr(AP_L).$$
Denote by $X_A$ the (negative) gradient of $f_A$ 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 flag determined by the eigenvectors of $A$.

REPLY [8 votes]: My answer has some overlap with Lucas', but a different point of view.
In the appendix to our paper 


Cycles of Bott-Samelson type for taut representations 


G. Thorbergsson and I considered generalized real flag manifolds $G/P$ where $G$ is a noncompact real semisimple Lie group and $P$ is a parabolic subgroup. It is well known that those arise as orbits of representations of isotropy representations of symmetric spaces. In a celebrated paper in the 1950's, Bott and Samelson had constructed concrete cycles for those spaces which represent a basis for the $\mathbb Z_2$-homology. We wrote 
in the appendix an explicit (Lie-theoretic) proof that 


the images of the cycles of Bott and Samelson equal the closures of the Bruhat cells


(such a proof had been previously given by Hansen in the case $G$ is a complex Lie group
in "On cycles in flag manifolds", Math. Scand. 33 (1973), 269-274).
As an application of the cycles of Bott and Samelson, (squared) distance functions to orbits of isotropy representations of symetric spaces are perfect in the sense that the Morse inequalities are in fact equalities; submanifolds with such property are usually called taut. Because of their tautness, the Bruhat decomposition of he spaces $G/P$ is minimal in the sense that the number of cells in dimension $k$ is equal to the 
$k$th $\mathbb Z_2$-Betti number of $G/P$. Notice that in the case $G$ is a complex Lie group, the cells are all even-dimensional which makes the minimality of the Bruhat decompostion trivial, but such easy arguments do not apply otherwise. Another 
Morse-theoretic interpretation of the Bruhat cells is their appearence and the unstable manifolds of suitable height functions on $G/P$ seen as orbits of isotropy representations of symmetric spaces, see M. Atiyah, "Convexity and commuting Hamiltonians", Bull. London. Math. Soc. 14 (1982), 1-15 and  R. R. Kocherlakota, "Integral homology of real flag manifolds and loop spaces of symmetric spaces", Adv. Math. 110 (1995), 1-46.