Lecture 9: Maximal Tori

Tori Maximal tori Conjugacy of maximal tori

Daniel Bump

May 5, 2020 Tori Maximal tori Conjugacy of maximal tori

Tori

In this section a torus will be a compact connected abelian Lie group T with Lie algebra t. The Lie algebra t is abelian, so if X, Y ∈ t then [X, Y] = 0. This implies that eXeY = eX+Y = eY eX, the simplest case of the Campbell-Hausdorff formula; see Proposition 15.2 for a proof. Tori Maximal tori Conjugacy of maximal tori

Structure of tori

Proposition R A torus of dimension r is isomorphic to (R/Z) . Indeed the exponential map t → T is a homomorphism that is a local homeomorphism. Its image is a subgroup that contains a neighborhood of the identity; since T is compact, it is generated by this neighborhood. Thus exp is surjective and T =∼ t/Λ.

The subgroup Λ is discrete (since exp is a local homeomorphism) and cocompact, so it is a lattice and we may r r identify t = R , Λ = Z . Tori Maximal tori Conjugacy of maximal tori

Kronecker’s Theorem

A generator of T is an element t0 such that the powers of t0 are dense in T. Theorem (Kronecker) r Let (t1,..., tr) ∈ R , and let t be the image of this point in r T = (R/Z) . Then t is a generator of T if and only if 1, t1,..., tr are linearly independent over Q. Tori Maximal tori Conjugacy of maximal tori

Proof of Kronecker’s theorem

Let H be the closure of the group hti generated by t in r T = (R/Z) . Then T/H is a compact Abelian group, and if it is not 1 it has a nonzero character χ. We may regard this as a character of T that is trivial on H. It has the form

P 2πi kjxj (x1,..., xr) 7→ e , P for some ki ∈ Z. Since t itself is in H, this means that kjtj ∈ Z, so 1, t1,..., tr are linearly dependent. The existence of nontrivial characters of T/H is thus equivalent to the linear dependence of 1, t1,..., tr and the Kronecker’s theorem follows. Tori Maximal tori Conjugacy of maximal tori

Existence of generators

Corollary Each compact torus T has a generator. Indeed, generators are dense in T. By Kronecker’s Theorem, what we must show is that r-tuples (t1,..., tr) such that 1, t1,..., tr are linearly independent over Q r are dense in R . If 1, t1,..., ti−1 are linearly independent, then linear independence of 1, t1,..., ti excludes only countably many ti, and the result follows from the uncountability of R. Tori Maximal tori Conjugacy of maximal tori

The automorphism group of a torus

Proposition The automorphism group of a compact torus of dimension r is isomorphic to GL(r, Z). Consider the commutative diagram

exp ker(exp) t T ∼ =∼ =∼ = r r r r Z R R /Z

The differential of an automorphism of T is a linear transformation of t taking ker(exp) to itself. It induces a linear r r transformation of R taking Z to itself, an element of GL(r, Z). Tori Maximal tori Conjugacy of maximal tori

The automorphism group of a torus is discrete

The group GL(r, Z) is discrete in its relative topology as a subgroup of GL(r, R).

So we expect that T has no continuous group of automorphisms. That is, H is a connected topological space and f : H → Aut(T) is a map such that (h, t) → f (h)t is continuous, then the map f is constant. Tori Maximal tori Conjugacy of maximal tori

A theorem of von Neumann

Von Neumann proved that if G is a Lie group and H a closed subgroup, then H is a Lie subgroup. The special case where H is abelian is proved in the book as Theorem 15.2. Proofs of the general case may be found in the text of Bröcker and Tom Dieck, or in Knapp’s book Lie groups beyond an introduction.

We will take this fact for granted when H is abelian. Thus a closed abelian subgroup of a compact Lie group is a torus. Tori Maximal tori Conjugacy of maximal tori

The Weyl group

Proposition Let G be a compact Lie group and T a maximal torus. Then T is the connected component in the identity of its normalizer: N(T)◦ = T. The quotient group N(T)/T is ﬁnite.

We have a homomorphism N(T)◦ → Aut(T) through the action by conjugation. Since N(T)◦ is connected and Aut(T) is discrete, this homomorphism is trivial. Therefore N(T)◦ centralizes T. Tori Maximal tori Conjugacy of maximal tori

The Weyl group (proof, continued)

We claim that N(T)◦ = T.

If N(T)◦ is strictly larger than T, then it contains a one-parameter subgroup R 3 t → n(t) and the closure of the group generated by T and n(t) is a closed connected abelian subgroup strictly larger than T. Hence T is not a maximal torus. This is a contradiction.

Now N(T) is a closed subgroup of the compact group G, so it is compact. Since N(T)◦ = T, the quotient N(T)/T is both compact and discrete, hence ﬁnite. This completes the proof. Tori Maximal tori Conjugacy of maximal tori

The Weyl group, continued

The Weyl group W = N(T)/T is called the Weyl group of G. It depends on the choice of maximal torus T, but we will see soon that all maximal tori are conjugate.

Example

Suppose that G = U(n). A maximal torus is    t1    ..  T =  .  |t1| = ··· = |tn| = 1 .    tn 

Its normalizer N(T) consists of all monomial matrices (matrices with a single nonzero entry in each row and column) so the ∼ quotient N(T)/T = Sn. Tori Maximal tori Conjugacy of maximal tori

Surjectivity of the exponential map

If G is a noncompact group, the exponential map may not be surjective. For example, if G = SL(2, R) then

a −1 1 , a−1 −1

where a < 0 are not in the image of the exponential map unless a = −1.

Theorem If G is a compact connected Lie group then exp : g → G is surjective.

We will assume this now, and discuss the proof later. Tori Maximal tori Conjugacy of maximal tori

Two major theorems

There are now two main theorems about maximal tori for a compact connected Lie group G. Theorem (I) Let G be a compact connected Lie group, T a maximal torus. Then [ G = gTg−1. g∈G

Theorem (II) Let G be a compact connected Lie group, T a maximal torus. Then any maximal torus is conjugate to T.

These are fundamental in representation theory. Both follow from the surjectivity of exp. Tori Maximal tori Conjugacy of maximal tori

Proof of Theorem (I)

We will prove that if g ∈ G there exists k ∈ G such that g ∈ kTk−1.

Let g and t be the Lie algebras of G and T, respectively. Let t0 be a generator of T. Using the surjectivity of exp, ﬁnd X ∈ g and X H H0 ∈ t such that e = g and e 0 = t0.

Since G is a compact group acting by Ad on the real vector space g, there exists on g an Ad(G)-invariant inner product for which we will denote the corresponding symmetric bilinear form as h , i. Choose k ∈ G so that the real value hX, Ad(k)H0i is −1 maximal, and let H = Ad(k)H0. Thus, exp(H) = kt0k generates kTk−1. Tori Maximal tori Conjugacy of maximal tori

Proof (continued)

Let Y ∈ g. Since X, Ad(etY )H has a maximum when t = 0

d 0 = X, Ad(etY )H = hX, ad(Y)Hi = − hX, [H, Y]i . dt t=0 Since the inner product h , i is invariant, this means h[H, X], Yi = 0 for all Y. And since the inner product is positive deﬁnite, it is nondegenerate, so [H, X] = 0.

H tX H Therefore e commutes with e for all t ∈ R. Since e generates the maximal torus kTk−1, it follows that the one-parameter subgroup {etX} is contained in the centralizer of kTk−1, and since kTk−1 is a maximal torus, it follows that {etX} ⊂ kTk−1. In particular, g = eX ∈ kTk−1. Tori Maximal tori Conjugacy of maximal tori

Conjugacy of maximal tori

We may now prove: Theorem (II) Let G be a compact connected Lie group, T a maximal torus. Then any maximal torus is conjugate to T.

Indeed, let T0 be another maximal torus, and t0 a generator of T0. Then t0 ∈ kTk−1 for some k. This implies that T0 ⊆ kTk−1. Both tori are maximal, so T0 = kTk−1.