Is there a Morse theory proof of the Bruhat decomposition?

Question

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?

Answer 1

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.

Answer 2

It may be useful to expand Steven Sam's partial answer. This isn't really an answer to the original question, but is too long for a comment. The discussion in Chriss-Ginzburg 2.4, which relies on many other background sources, brings out nicely the interaction between Morse theory and the Bialynicki-Birula work on torus actions with finitely many fixed points. Of course, whether this leads to a full proof of the Bruhat decomposition in $G/B$ depends a lot on how you define "Bruhat decomposition". In Chevalley's development of structure theory over arbitrary algebraically closed fields, on the way to classifying semisimple algebraic groups, the Borel fixed point theorem is refined to show that a maximal torus has finitely many fixed points on the flag variety. This leads to a simultaneous study of the Weyl group $W=N_G(T)/T$, the root system, and the detailed form of the Bruhat decomposition needed to pin down the exact structure of the group. In particular, one sees that the number of fixed points is $|W|$, while the Bruhat cell corresponding to $w \in W$ has dimension equal to the length of $w$, etc.

The approach over $\mathbb{C}$ via Morse theory gives a nice perspective on the cell decomposition as such, but is limited to characteristic 0 and probably can't reveal all the finer points needed in this special case(?)

In any case, the work of Bialynicki-Birula (following earlier work by Iversen) led people to appreciate better the way in which Bruhat decomposition fits into the general pattern of torus actions on projective varieties having only finitely many fixed points (at least in characteristic 0). This theme was developed quite a bit in papers by Jim Carrell and by his 1977 Ph.D. student at UBC, Ersan Akyildiz. One hard-to-find paper is easy to overlook:

MR628640 (83a:14035) 14L30 (14M17 20G05) Akyıldız, Ersan, Bruhat decomposition via $G_m$-action. (Russian summary) Bull. Acad. Polon. Sci. S´er. Sci. Math. 28 (1980), no. 11-12, 541–547 (1981).

Here as in Chriss-Ginzburg the multiplicative group corresponds to a regular character of a maximal torus. To quote the abstract: "In this note we will obtain the Bruhat decomposition of $G/B$ from a $G_m$-action. Moreover we will show that a theorem of A. Bialynicki-Birula is the generalization of the Bruhat decomposition, which was a question of B. Iversen."

Answer 3

Yes, there is!

The best place I know where this is done are the initial sections of

J. J. Duistermaat, J. A. C. Kolk, and V. S. Varadarajan. Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups. Compositio Math., 49(3):309–398, 1983

I don't know much about complex groups (I work in the real context), but the above article also considers the complex case. Below I give you a picture of what happens for a real Lie semisimple noncompact Lie group $G$, hope this can help.

The key is the action of a regular split-element of the Lie algebra on the maximal flag manifold $\mathbb{F} = G/P$ of $G$: this action is Morse, and the Morse function is beautifully simple: it is the height function of a natural embeeding of $\mathbb{F}$ in the Lie algebra of $G$ under an appropriate metric on $\mathbb{F}$. Its critical points are computed to be $wP$ and their stable manifolds are computed to be $PwP$.

In case you are interested, they even prove the double-coset Bruhat decomposition $$G = \coprod_w P_\Theta w P_\Delta,$$ which is disjoint when $w$ runs trough the double coset $ P_\Theta \backslash W / P_\Delta $. Here $P_\Theta, P_\Delta$ are the standart parabolic subgroups of type $\Theta$ and $\Delta$: they contain the minimal parabolic subgroup, which plays the role of the Borel subgroup in the real theory. The key here is the action of a (possibly not regular) split-elemet on $\mathbb{F}$: this action is Morse-Bott with the same beatifull Morse function! Its critical manifolds are computed to be the orbits of $wP_\Delta$ and their stable manifolds are computed to be the orbits of $P_\Theta w P_\Delta$ on the flag manifold. The hard part is to show that the above critical manifolds are indeed disjoint: to do this the above article appeals to algebraic constructions involving Tits buildings and other things I don't understand... This was disappointing to me since I expected a self-contained purely dynamical solution!

After some stubborn tries I was able to do this step by purely dynamical arguments. It turns out that these critical manifolds are again flag manifolds: actually, flag manifolds of semisimple subgroups of $G$! This came as a nice surprise to me and my PhD advisor. Using this fact and the previous regular Bruhat decomposition, one can show by some simple dynamical arguments that these critical manifolds are disjoint. So the question can be settled by purely dynamical arguments and in a nice inductive manner. This is the content of my article

Seco, L. . "A Note on the Bruhat Decomposition of Semisimple Lie Groups". Journal of Lie Theory, v. 18, p. 725-731, 2008.

I would be very interested to know if the same method applies in the complex case to obtain an analogous double-coset Bruhat decomposition.

Answer 4

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 $A$.

Answer 5

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.