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?



Is there a Morse theory proof of the Bruhat decomposition?
Here's a partial answer. What I'm about to say is taken from Section 2.4 of Chriss and Ginzburg's *Representation Theory and Complex Geometry*. The references I use will be from this book. There aren't complete proofs there, but there are references to complete proofs.


The Bruhat decomposition on $G$ (we'll assume $G$ is reductive, the general case follows from this) can be seen as the preimage of the Bruhat decomposition of its flag variety $G/B$. It's easier to work with $G/B$ since it's a projective variety.


Here $G/B$ has a torus (= ${\bf C}^\*$) action with finitely many fixed points $\{w\_1, \dots, w\_n\}$, so one has a Bialynicki-Birula cell (= affine space) decomposition given by the attracting sets


$X\_i = \{ x \in G/B \mid \lim\_{t \to 0} t \* x = w\_i \}$.


[Normally, one thinks of $G/B$ as having finitely many fixed points with respect to $({\bf C}^\* )^r$, where $r$ is the rank of $G$, but we can take ${\bf C}^\* $ to be a subgroup in general position]


In our case, the fixed points are indexed by the Weyl group, and the attracting sets will be the Bruhat cells.


From this, one can construct a Morse function $H$ (Lemma 2.4.17) whose Morse decomposition coincides with the B-B cell decomposition (Corollary 2.4.24).


So even though it's a very special case, one case where cell decompositions come from Morse functions is when the manifold is a projective algebraic variety with finitely many torus fixed points.


