Torus actions and maximal tori, part 1

Take a look at this:

Theorem 1. If a torus T acts on a compact smooth manifnew M with finitely many fixed points, then $\mid M^T\mid = \chi(M)$. Here $M^T$ is the fixed point set of the action, and $\chi(M)$ is the Euler characteristic of M.

This is neat. Think of a circle acting freely on itself by rotation (agreeing with $\chi(S^1) = 0$), or on $S^2$ by rotation about a fixed axis, with the north and south pole as fixed points (agreeing with $\chi(S^2) = 2$). This immediately imposes some restrictions on possible torus actions: for example, any torus action on a manifnew of negative Euler characteristic (e.g. a multiple-holed torus) cannot have finitely many fixed points; or, by Poincaré duality, any odd-dimensional manifnew has Euler characteristic 0, so a torus action with finitely many fixed points has to act freely.

One reason to care about torus actions comes from the representation theory of compact Lie groups. From now on, G will mean a compact connected Lie group. Just as for finite groups, any finite-dimensional complex representation V of G is completely reducible, i.e. is the direct sum of irreducibles. In particular, if G is abelian, Schur’s lemma says that any irreducible representation of G is one-dimensional, so the action of G on V is diagonalizable. Even if G is not abelian, we can restrict its action on V to an abelian subgroup T to get something nice and diagonalizable, and hope this tells us something about the action of G. To get the most mileage out of this, we should choose T as large as possible. (This turns out to work quite well: for example, applying it to the adjoint representation of G on its Lie algebra, one gets the root space decomposition of the Lie algebra, and is eventually led to the classification of compact Lie groups.)

After realizing that I used T in the previous paragraph and hence that any compact connected abelian Lie group is a torus, this leads us to think about maximal tori in G: closed subgroups isomorphic to a torus and not contained in a larger torus. Such things certainly exist: the trivial subgroup is a torus, and we can only go up so far, since a containment of (closed) connected Lie groups of the same dimension is an equality.

Example. Take G to be the group of $n \times n$ unitary matrices, U(n). The diagonal matrices in U(n) form a maximal torus: the entries along the diagonal have to be complex numbers of norm 1, and it’s easy to check that a matrix commuting with all diagonal matrices is diagonal.  But there’s more to be said here: the spectral theorem shows that any unitary matrix is unitarily diagonalizable. This is the U(n) case of a more general result:

Theorem 2. a) Every element of G lies in some maximal torus. b) Any two maximal tori are conjugate.

One nice, immediate application of this theorem: the exponential map of G is surjective, because it is for a torus, and every element of G lies in a torus.

Let’s see how to prove Theorem 2 from Theorem 1. First of all, we want to be talking about fixed points, so:

Theorem 2’. a) If $x$ is in G, there is a maximal torus T such that the action (left translation) of $x$ on G/T has a fixed point. b) If S, T are maximal tori, the action of S on G/T has a fixed point.

It’s easy to write down what it means to have a fixed point in each case and see that Theorem 2’ is (a,b)-wise equivalent to Theorem 2.

We’ll need the Lefschetz fixed point theorem. Say X is a compact smooth manifnew (more generally, a retract of a finite simplicial complex), and $f : X \to X$ is continuous. Write $f_{*,n}$ for the induced map $H_n(X, \mathbb{Q}) \to H_n(X, \mathbb{Q})$; this is a map from a finite-dimensional $Q$-vector space to itself. Now define the Lefschetz number of $f$ by $L(f) = \sum_{n\geq 0} (-1)^n \text{trace}(f_{*,n})$. The Lefschetz fixed point theorem says: if L($f$) is nonzero, then $f$ has a fixed point. (A simple fact to think of at this point: if one has a permutation $p$ of a finite set S and extends it to a linear transformation $P$ of the vector space with basis S, then the trace of $P$ is the number of fixed points of $p$). In particular, if $f$ is the identity, or just homotopic to the identity, then L($f$) = χ(X).

To prove Theorem 2’, we pull a bait and switch. The fixed points of the action of a maximal torus T on G/T are easy to understand: they’re exactly the members of the Weyl group N(T)/T, where N(T) is the normalizer of T in G. In particular, there are more than 0 of them, so χ(G/T) is nonzero by Theorem 1. But now by a second appeal to Theorem 1, the action of another (maximal) torus S on G/T must also have a fixed point, which is part (b) of Theorem 2’. (One detail has been glossed over, that N(T)/T is actually finite. But this comes easily enough from thinking about how there’s a continuous map from N(T) to the discrete group $\text{Aut}(T) \simeq \text{GL}(n,\mathbb{Z})$.)

For part (a) of Theorem 2’, notice that the action on G/T of any $x$ in G is homotopic to the identity, simply because G is path-connected. Therefore L($x$) = χ(G/T) is nonzero, so $x$ has a fixed point by the Lefschetz fixed point theorem. (Incidentally, we’ve shown that dim(G/T) is even, a fact which is also apparent from the root space decomposition of the Lie algebra of G.)

In part 2 I’ll sketch how to prove Theorem 1 (Lefschetz again! plus some geometry), and give a generalization to arbitrary torus actions.