18.755 eighth problems, due Monday, November 16, 2015
As background for the first problem, suppose that V is an n-dimensional vector space
(can be over any field k of characteristic not two) endowed with a nondegenerate quadratic
form Q. In class I defined a complete orthogonal flag to be a chain of subspaces
0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn−1 ⊂ Vn = V,
dim Vj = j,
Vj⊥ = Vn−j .
I stated that such flags exist if k is algebraically closed, and that the orthogonal group O(V )
acts transitively on them.
To say more, write
n = 2m + ǫ
(ǫ = 0 or 1).
Because of the relations
dim(W ⊥ ) = n − dim W,
W is isotropic ⇐⇒ W ⊂ W ⊥ ,
we see that an isotropic subspace must have dimension at most m.
Definition. A Lagrangian subspace of V is an m-dimensional isotropic subspace. Define
L(V ) to be the set of all Lagrangian subspaces of V . This is (pretty obviously) closed in
the Grassmannian Grm (V ) of all m-dimensional subspaces of V . (To make sense of this as
a statement about smooth manifolds, we need to have k equal to C. But as a statement
about algebraic varieties, it makes sense for any k.)
Essentially I stated in class
Proposition. A complete orthogonal flag in V is the same thing as
(1) a Lagrangian subspace Wm ⊂ V , and
(2) a complete flag W0 ⊂ W1 ⊂ · · · ⊂ Wm in Wm .
General hint: problems 2, 3, and 5 are meant to be quite easy, and 6 is pretty easy.
Numbers 1 and 4 test whether you understand the Gram-Schmidt process; they may take a
little longer to write down. I include problem 7 just because I’m mean.
1. Suppose n = 2m is even, and that W is an m-dimensional Lagrangian subspace of a
2m-dimensional V as above. Choose any basis (w1 , . . . , wm ) for W , and let (λ1 , . . . , λm )
be the dual basis of W ∗ (see the solutions to Problem Set 7). Prove that there are unique
∗
) satisfying
vectors (w1∗ , . . . , wm
Q(wj , wk∗ ) = δjk ,
Q(wj∗ , wk∗ ) = 0
(1 ≤ j, k ≤ m).
For the next problems we’re back in the world of Lie groups. Recall that I stated in
class the existence of an exact sequence for computing fundamental groups of homogeneous
spaces:
π1 (H) → π1 (G) → π1 (G/H) → π0 (H) → π0 (G) → π0 (G/H) → 0.
Here H is any closed subgroup of a Lie group G. You can also use a generalization: if
H1 ⊂ H2 ⊂ G are closed subgroups, then
π1 (H2 /H1 ) → π1 (G/H1 ) → π1 (G/H2 ) → π0 (H2 /H1 ) → π0 (G/H1 ) → π0 (G/H2 ) → 0.
2
2. Prove that if n ≥ 3, then the natural map
π1 (SO(n − 1)) → π1 (SO(n))
induced by the inclusion SO(n − 1) ֒→ SO(n) is onto. (You can use things done in class
about spheres.)
3. Suppose H is any closed connected subgroup of SO(n) containing SO(2). Prove that
G/H is simply connected.
4. Suppose VR is an n-dimensional real vector space with a positive definite quadratic
form QR , and that VC is its complexification. I proved in class that any vector w ∈ VC of
length zero must be of the form
w = u + iv,
(u, v ∈ VR , QR (u, v) = 0, QR (u, u) = QR (v, v)).
Suppose that Wm is an m-dimensional Lagrangian subspace of VC , and that W0 ⊂ · · · ⊂ Wm
′
) of W such that
is a complete flag in Wm . Prove that there is a basis (w1′ , . . . , wm
(1) Wj = hw1′ , . . . , wj′ i,
√
(2) wj′ = (uj + ivj )/ 2, and
(3) (u1 , v1 , . . . , um , vm ) is an orthonormal basis of VR .
Prove furthermore that the m oriented planes
Pj = huj , vj i ⊂ VR
are uniquely determined by the flag.
Problem 4 essentially completes what I stated in class: that the space of orthogonal flags
in an even-dimensional VC is isomorphic to the union of two copies of SO(2m)/SO(2)m .
Then Problem 3 says that this space of flags is simply connected. The odd-dimensional case
is no more difficult; I left it out just to simplify the notation a bit.
5. Give an example of a Lie group G with a closed subgroup H so that G/H is compact,
but there is no compact subgroup K of G such that K/K ∩ H = G/H.
6. Same problem as above, but this time require that



 1 x z

G =  0 1 y  | x, y, z ∈ R .


0 0 1
(Yes, you can get full credit for problems 5 and 6 by solving only problem 6.)
7. Is there a connected Lie group G with a closed connected subgroup H such that
(1) G/H is compact, but
(2) there is no compact subgroup K of G such that K/K ∩ H = G/H?
I don’t know the answer to this question; should be no, but I can’t think of a proof.