A THEOREM OF TITS, NORMALIZERS OF MAXIMAL TORI AND FIBREWISE BOUSFIELD-KAN COMPLETIONS

FRANK NEUMANN

Abstract. We use a theorem of Tits on the algebraic presentation with generators and relations of the normalizer of a maximal torus of a compact connected semisimple Lie group to derive several equivalent conditions for the splitting of the associated normalizer group extension in terms of p-adic fibrewise homotopy theory.

Introduction

It was shown by Curtis, Wiederhold and Williams [C-W-W] that the isomorphism type of a compact connected semisimple Lie group G is completely determined by the isomorphism type of the normalizer of the maximal torus T (G). This was generalized much later by Notbohm for any compact connected Lie group [N]. In their paper Curtis, Wiederhold and Williams also studied the question when the group extension

0 ⟶ T (G) ⟶ N(G) ⟶^{π} W (G) ⟶ 1

is a split extension and the normalizer N(G) be completely determined by the action of the Weyl group W (G) on the maximal torus T (G). Using a theorem of Tits [T2] giving an explicit description of the normalizer in terms of generators and relations, they could decide case-by-case for which simple Lie groups the above normalizer sequence is split exact.

In this note we use the theorem of Tits to derive the following more conceptional criterion equivalent to the splitting of the above group extension in terms of fibrewise p-adic Bousfield-Kan completion of the associated fibration of classifying spaces, namely

Theorem. Let G be a compact connected Lie group. The group extension

0 ⟶ T (G) ⟶ N(G) ⟶^{π} W (G) ⟶ 1

is a split extension if and only if the fibration

BT (G)_{2}^{∧} ⟶ BN(G)_{2}^{∘} ⟶^{Bπ} BW (G)

has a section, i.e. is fibre homotopy equivalent to the fibration

BT (G)_{2}^{∧} ⟶ EW (G) ×_{W (G)} BT (G)_{2}^{∧} ⟶ BW (G).

1991 Mathematics Subject Classification. 20 F 55, 20 G 20, 20 J 06, 22 E 15, 57 T 10 Key words and phrases. compact Lie groups, normalizer of maximal tori, group extensions, Coxeter groups, classifying spaces, fibrewise homototopy theory

Moreover, it turns out that for odd primes p the fibrewise p-adic completed fibration of classifying spaces

BT (G)_{p}^{∧} ⟶ BN(G)_{p}^{∘} ⟶^{Bπ} BW (G)

has always a section. Here we just consider the Lie group case, but it is interesting to ask if one can derive similiar results for the more general case of a connected finite loop space with maximal torus. The results of Andersen [A] show that similiar statements can be obtained for connected p-compact groups in the sense of Dwyer and Wilkerson [D-W], which are adequate homotopy theoretic replacements of compact Lie groups. Instead of using the result of Tits Andersen calculated the lowdimensional cohomology groups of the Weyl groups in all cases. Dwyer-Wilkerson and Notbohm also announced proofs for the splitting of the normalizer of a maximal torus in the case of connected p-compact groups. A detailed discussion of the splitting problem can be found in the overview article of Lannes [L]. But it is still an open problem if there is an analogous statement of the theorem of Tits, which might allow to decide the splitting question in the same manner as described in this note. In the case of p-compact groups the Weyl groups are p-adic pseudoreflection groups, so not real reflection groups anymore as in the Lie group case. The work of Broué, Malle and Rouquier [B-M-R] on complex reflection groups and their associated braid groups seems to be closely related with this question. It is also interesting to ask what would happen in the case of a split semisimple reductive algebraic group G, where the theorem of Tits is also known [T2].

Acknowledgements. We are very grateful to J. Møller, D. Notbohm L. Smith and A. Viruel for many useful discussions and suggestions. Special thanks also to K. S. Andersen for telling me about his work.

1. A Theorem of Tits and the Splitting of the Normalizer of a maximal torus of a compact connected Lie group

Let G be a compact connected semisimple Lie group. Fix a maximal torus T (G) of G and let N(G) be the normalizer of T (G) in G. The Weyl group of G is given by W (G) = N(G)/T (G). We recall some basic facts from Lie theory [B-tD]. The normalizer N(G) acts on T (G) by conjugation

N(G) × T (G) → T (G), n · t = ntn^{−1}

The action restricted to T (G) is trivial, so factors through the quotient W (G) = N(G)/T (G) inducing an action of W (G) on T (G)

W (G) × T (G) → T (G), n · T (G) = ntn^{−1}.

We study the group extension

0 ⟶ T (G) ⟶ N(G) ⟶^{π} W (G) ⟶ 1

and derive several equivalent conditions under which this extension is a split extension with

N(G) ≅ T (G)×W (G).

Denote by L(G) the Lie algebra of G which we identify with the tangent space to G at the identity element e of G. Formally we may split L(G) ≅ L(T (G))⊕L(G/T (G)), where L(G/T (G)) is the orthogonal complement of L(T (G)) with respect to the W (G)-invariant inner product on L(T (G)).

Let Ψ(G) be the set of roots and Ψ^{0}(G) be a simple root system of G. For any root α ∈ Ψ^{0}(G) let s_{α} be the corresponding reflection of L(T (G)). The Weyl group W (G) has a presentation as a finite real reflection group with generators the set of reflections S = {s_{α} : α ∈ Ψ^{0}(G)} subject to the relations

(1) s_{α}^{2} = 1

(2) (s_{α}s_{β})^{m_{αβ}} = 1 for all α ≠ β and m_{αβ} ∈ {2, 3, 4, 6} The W (G)-invariant inner product allows us to identify the W (G)-representation

L(T (G)) with the dual representation L(T (G))^{∗} = Hom_{ℝ}(L(T (G)), ℝ). Denote this isomorphism by κ : L(T (G)) → L(T (G))^{∗}.

For α ∈ Ψ(G) and x ∈ L(T (G)) we have

s_{α}(x) = x− < α, x > α^{∗}

where α^{∗} = 2α/ < α, α > is the coroot corresponding to α. For the set of coroots of G let us write Ψ(G)^{∗} = {α^{∗} : α ∈ ψ(G)}. If it becomes necessary to distinguish between L(T (G)) and L(T (G))^{∗} we will denote by α^{∧} = κ(α^{∗}) ∈ L(T (G))^{∗}, the inverse root of α. Let exp : LT (G) → T (G) be the exponential map and set I = ker(exp) and I^{∗} = {α ∈ L(T (G)^{∗} : αI ⊂ ℤ}. I is called the integral lattice and I^{∗} the lattice of integral forms. We get the following commutative diagram

[figure: 0; ℤ^{n}; ℝ^{n}; exp; T (G); 1; 0; I; L(T (G)); α; exp; T (G); θ_{α}; 1; 0; ℤ; ℝ; e; U(1); 1]

where e : ℝ → U(1) is given by t ↦ exp(2πit) and θ_{α} : T (G) → U(1) is defined by exp(H) ↦ exp(2πiα(H)) as the global root corresponding to α ∈ L(T (G))^{∗}. It is easy to see that Ψ(G) ⊂ I^{∗} and Ψ(G)^{∗} ⊂ I [B-tD].

For every root α ∈ Ψ(G) we now define an element h_{α} ∈ T (G) by h_{α} = exp(\frac{1}{2}α^{∧}). We get immediately from the above considerations that h_{α}^{2} = 1 and h_{α} = h_{−α}.

The normalizer N(G) of the maximal torus T (G) also has a presentation with generators and relations as a braid group mixed with the toral part as given in the following theorem of Tits [T1]. See also [C-W-W]. In [T2] Tits proved a similiar theorem in the case of a split semisimple reductive algebraic group G.

Theorem 1.1 (Tits). Let G be a compact connected semisimple Lie group, T (G) a maximal torus and N(G) its normalizer in G. For every simple root α ∈ Ψ^{0}(G) there exists an element q_{α} ∈ N(G) such that

π^{−1}(s_{α}) = T (G) ∪ q_{α}T (G)

with the following properties

(1) q_{α}^{2} = h_{α}

(2) (Braid condition)

q_{α}q_{β}q_{α} … = q_{β}q_{α}q_{β} … for all α ≠ β where the factors are repeated m_{αβ} times, where m_{αβ} ∈ {2, 3, 4, 6}.

(3) q_{α}tq_{α}^{−1} = s_{α}(t) for t ∈ T (G).

Moreover, the normalizer N(G) is isomorphic to the group generated by the set T (G) ∪ {q_{α} : α ∈ Ψ^{0}(G)} subject to the relations (1), (2), (3) and those coming from T (G). □

Let T_{∞}(G) be the subgroup of T (G) of elements having finite order, called the discrete approximation of T (G). We have T_{∞}(G) ≅ (ℚ/ℤ)^{n} where n is the rank of T (G). T_{∞}(G) is mapped to itself under the action of the Weyl group W (G). For all α ∈ Ψ^{0}(G) the element h_{α} lies in T_{∞}(G). Let N_{∞}(G) be the subgroup of N(G) generated by the set T_{∞}(G) ∪ {q_{α} : α ∈ Ψ^{0}(G)} subject to the relations (1), (2), (3) in the theorem of Tits and those from T_{∞}(G). Then N_{∞}(G) is the discrete approximation of N(G).

The inclusion i : T_{∞}(G) → T (G) induces a homomorphism of group extensions

[figure: 0; T_{∞}(G); i; N_{∞}(G); π; W (G); 1; 0; T (G); N(G); π; W (G); 1]

For a fixed prime p let T_{p^{∞}}(G) denote the subgroup of T_{∞}(G) of elements having order a power of p. We have T_{p^{∞}}(G) ≅ (ℤ/p^{∞})^{n} where n is again the rank of T (G) and ℤ/p^{∞} denotes the group ℤ[\frac{1}{p}]/ℤ = colim_{s}ℤ/p^{s}. The Weyl group W (G) acts on T_{p^{∞}}(G) and the inclusion

j : ∏_{p prime} T_{p^{∞}}(G) ⟶^{≅} T_{∞}(G)

is a W (G)-equivariant isomorphism.

From the universal property of the product we get therefore a unique homomorphism π_{p^{∞}} : T_{∞}(G) → T_{p^{∞}}(G) making the diagram

[figure: ∏_{p prime}T_{p^{∞}}(G); pr; j; T_{∞}(G); π_{p^{∞}}; T_{p^{∞}}(G)]

commutative. The map π_{p^{∞}} is also a W (G)-equvariant map.

Now let N_{p^{∞}}(G) be the subgroup of N_{∞}(G) generated by the set T_{p^{∞}}(G) ∪ {q_{α} : α ∈ Ψ^{0}(G)} subject to the relations (1), (2), (3) in the theorem of Tits and those coming from T_{p^{∞}}(G). The homomorphism π_{p^{∞}} induces a homomorphism of group extensions

[figure: 0; T_{∞}(G); π_{p^{∞}}; N_{∞}(G); π; W (G); 1; 0; T_{p^{∞}}(G); N_{p^{∞}}(G); π; W (G); 1]

Since h_{α}^{2} = 1 for all α ∈ Ψ^{0}(G) it follows that h_{α} ∈ T_{2^{∞}}(G). Therefore for all odd primes p we have π_{p^{∞}}(h_{α}) = 1.

From this it follows immediately that for odd primes p the group N_{p^{∞}}(G) is generated by the set T_{p^{∞}}(G) ∪ {q_{α} : α ∈ Ψ^{0}(G)} subject only to the relations (1’) q_{α}^{2} = 1 (2’) (q_{α}q_{β})^{m_{αβ}} = 1 for all α ≠ β (3’) q_{α}tq_{α}^{−1} = s_{α}(t) for t ∈ T (G). Let B_{p^{∞}}(G) be the subgroup of N_{p^{∞}}(G) generated only by the set {q_{α} : α ∈ Ψ^{0}(G)}

subject to the relations (1’), (2’), (3’). The homomorphism π : N_{p^{∞}}(G) → W (G) maps B_{p^{∞}}(G) isomorphically onto W (G) since the elements π(q_{α}) = s_{α} for α ∈ Ψ^{0}(G) generate the Weyl group W (G) and B_{p^{∞}}(G)∩T_{p^{∞}}(G) = ∅. Therefore we have shown the following theorem:

Theorem 1.2. Let G be a compact connected semisimple Lie group. For each prime p ≠ 2 the group extension

0 ⟶ T_{p^{∞}}(G) ⟶ N_{p^{∞}}(G) ⟶^{π} W (G) ⟶ 1

is a split extension with N_{p^{∞}}(G) ≅ T_{p^{∞}}(G)×W (G). □

In the special case that h_{α} = 1 for all α ∈ Ψ^{0}(G) we can derive a stronger result. Then the group N(G) is generated by the set T (G) ∪ {q_{α} : α ∈ Ψ^{0}(G)} subject to the relations

(1) q_{α}^{2} = 1

(2) (q_{α}q_{β})^{m_{αβ}} = 1 for all α ≠ β

(3) q_{α}tq_{α}^{−1} = s_{α}(t) for t ∈ T (G). and in the same way as above we get immediately N(G) ≅ T (G)×W (G). From this we can conclude the following equivalent algebraic conditions for the splitting of the normalizer exact sequence:

Theorem 1.3. Let G be a compact connected semisimple Lie group. The following statements are equivalent:

(1) For every α ∈ Ψ^{0}(G) it is h_{α} = 1.

(2) The group extension

0 ⟶ T_{2^{∞}}(G) ⟶ N_{2^{∞}}(G) ⟶^{π} W (G) ⟶ 1

splits, i.e. N_{2^{∞}}(G) ≅ T_{2^{∞}}(G)×W (G).

(3) The group extension

0 ⟶ T (G) ⟶ N(G) ⟶^{π} W (G) ⟶ 1

splits, i.e. N(G) ≅ T (G)×W (G). □

In [C-W-W] Curtis, Wiederhold and Williams used special elements in the Lie algebra L(T (G)) to check case-by-case for which simple Lie groups the normalizer sequence splits. This was also investigated earlier by Tits. He considered also groups obtained by quotienting out proper subgroups of centers. It turns out that for SU(2n + 1), SU(2n)/Z, SO(n), G_{2} the normalizers always splits, while the normalizers of SU(2n), Sp(n)/Z, Spin(n), F_{4}, E_{6}, E_{7}, E_{8} and their quotients modulo centers Z do not split.

2. The splitting of the normalizer sequence via Fibrewise Bousfield-Kan Completions

We would like to derive topological conditions for the splitting of the normalizer of a maximal torus of a compact connected Lie group, which are equivalent to the algebraic conditions derived in the previous section. Let us therefore consider the following diagram of group extensions and homomorphisms of group extensions for a compact connected Lie group G.

[figure: 0; T (G); N(G); π; W (G); 1; 0; T_{∞}(G); π_{p^{∞}}; i; N_{∞}(G); π; W (G); 1; 0; T_{p^{∞}}(G); N_{p^{∞}}(G); π; W (G); 1]

Applying the classifying space functor B(?) and fibrewise p-adic completions of the rows for a fixed prime p in the sense of Bousfield-Kan [B-K] yields the following diagram of fibrations and maps of fibrations:

[figure: BT (G)_{p}^{∧}; BN(G)_{p}^{∘}; Bπ; BW (G); BT_{∞}(G)_{p}^{∧}; ≃; ≃; BN_{∞}(G)_{p}^{∘}; ≃; ≃; Bπ; BW (G); BT_{p^{∞}}(G)_{p}^{∧}; BN_{p^{∞}}(G)_{p}^{∘}; Bπ; BW (G)]

with fibrewise homotopy equivalences. We obtain therefore the following interpretation of theorem 1.2 in terms of fibrewise homotopy theory:

Theorem 2.1. Let G be a compact connected Lie group. For each prime p ≠ 2 the fibration

BT (G)_{p}^{∧} ⟶ BN(G)_{p}^{∘} ⟶^{Bπ} BW (G)

has a section, i.e is fibre homotopy equivalent to the fibration

BT (G)_{p}^{∧} ⟶ EW (G) ×_{W (G)} BT (G)_{p}^{∧} ⟶ BW (G)

In other words

BN(G)_{p}^{∘} ≃ B(T (G)×W (G))_{p}^{∘} ≃ EW (G) ×_{W (G)} BT (G)_{p}^{∧}

where EW (G) is a free acyclic W (G)-space. □

Proof. First of all let Z(G) be the center of G. The compact Lie group G/Z(G) has maximal torus T (G)/Z(G) and Weyl group W (G) and the normalizer of the maximal torus is given by N(G)/Z(G). We have a commutative diagram

[figure: 0; T (G); i; N(G); π; W (G); 1; 0; T (G)/Z(G); N(G)/Z(G); W (G); 1] .

So the top extension splits if and only if the bottom one is a split extension. Therefore (see [B-tD], Theorem 7.1) we can always assume that G is simply connected. But if G is simply connected it is certainly semisimple (see [B-tD], Remark 7.13). So the theorem follows from the above considerations and theorem 1.2.

As an immediate consequence we get the following result in terms of group cohomology, which was also announced in [C-W-W], Appendix 2:

Corollary 2.2. Let G be a compact connected Lie group G. Then

H^{∗}(BN(G), ℤ[\frac{1}{2}]) ≅ H^{∗}(B(T (G)×W (G)), ℤ[\frac{1}{2}])

or equivalently

H^{∗}(BN(G), ℤ_{2}^{∧}) ≅ H^{∗}(B(T (G)×W (G)), ℤ_{2}^{∧}). □

For a compact Lie group G and a fixed prime p let S_{p}W (G) denote a p-Sylow subgroup of W (G) and S_{p}N(G) the inverse image of S_{p}W (G) in N(G). S_{p}N(G) is called a p-normalizer of N(G). Now we can state the main theorem of this section:

Theorem 2.3. Let G be a compact connected Lie group and p be a fixed prime. The following statements are equivalent:

(1) The group extension

0 ⟶ T_{p^{∞}}(G) ⟶ N_{p^{∞}}(G) ⟶ W (G) ⟶ 1

splits.

(2) The group extension

0 ⟶ T (G) ⟶ S_{p}N(G) ⟶ S_{p}W (G) ⟶ 1

splits.

(3) The fibration

BT (G) ⟶ BS_{p}N(G) ⟶ BS_{p}W (G)

has a section.

(4) The fibration

BT (G)_{p}^{∧} ⟶ BN(G)_{p}^{∘} ⟶ BW (G)

has a section.

(5) The fibration

BT_{p^{∞}}(G) ⟶ BN_{p^{∞}}(G) ⟶ BW (G)

has a section.

Proof. That assertion (2) follows from (1) is an immediate consequence of the following commutative diagram combining a pullback and a pushout diagram:

[figure: 0; T_{p^{∞}}(G); N_{p^{∞}}(G); W (G); 1; 0; T_{p^{∞}}(G); S_{p}N_{p^{∞}}(G); S_{p}W (G); 1; 0; T (G); S_{p}N(G); S_{p}W (G); 1]

The assertion (3) follows at once from (2) by applying the classifying space functor B(?) which sends a splitting homomorphism to a section.

To show that (4) follows from (3) suppose now that the fibration

BT (G) ⟶ BS_{p}N(G) ⟶ BS_{p}W (G)

has a section σ. After fibrewise p-adic Bousfield-Kan completion we still have a section in the completed fibration

BT (G)_{p}^{∧} ⟶ BS_{p}N(G)_{p}^{∘} ⟶ BS_{p}W (G).

Since the homomorphism in cohomology

i^{∗} : H^{3}(BW (G), π_{2}(BT (G)_{p}^{∧})) ⟶ H^{3}(BS_{p}W (G), π_{2}(BT (G)_{p}^{∧}))

is a monomorphism (see also [N]) it follows that the fibration

BT (G)_{p}^{∧} ⟶ BN(G)_{p}^{∘} ⟶ BW (G)

has a section as required.

Now we show that (5) follows from (4). Let F be the homotopy fibre of the completion map

ϕ : BT_{p^{∞}}(G) ⟶ BT_{p^{∞}}(G)_{p}^{∧} ≃ BT (G)_{p}^{∧}

Then we have

F = hofib(ϕ) ≃ K((ℚ_{p}^{∧})^{n}, 1)

where n is the rank of T (G) and we get the following diagram of fibrations

[figure: F; F; ∗; BT_{p^{∞}}(G); BN_{p^{∞}}(G); BW (G); BT (G)_{p}^{∧}; BN(G)_{p}^{∘}; BW (G)]

where the bottom fibration has a section. Obstruction theory shows that this section can be lifted to a section of the middle fibration, because

H^{∗+1}(BW (G), π_{∗}(F)) = 0

since W (G) is a finite group and char(ℚ_{p}^{∧}) = 0 (see [B]).

Finally we have to prove that (1) follows from (5). So suppose the fibration

BT_{p^{∞}}(G) ⟶ BN_{p^{∞}}(G) ⟶ BW (G)

has a section σ. The groups T_{p^{∞}}(G), N_{p^{∞}}(G) and W (G) are discrete groups and therefore the long exact sequence of homotopy groups degenerate to the following short exact sequence of groups

0 ⟶ π_{1}(BT_{p^{∞}}(G)) ⟶ π_{1}(BN_{p^{∞}}(G)) ⟶ π_{1}(BW (G)) ⟶ 1

which is nothing else than the group extension

0 ⟶ T_{p^{∞}}(G) ⟶ N_{p^{∞}}(G) ⟶ W (G) ⟶ 1.

The section σ induces a splitting homomorphism in the short exact sequence of fundamental groups and hence in the group extension.

As an immediate corollary we get finally from the algebraic considerations of the first section the theorem as mentioned in the introduction:

Corollary 2.4. Let G be a compact connected Lie group. The group extension

0 ⟶ T (G) ⟶ N(G) ⟶^{π} W (G) ⟶ 1

is a split extension with N(G) ≅ T (G)×W (G) if and only if one and hence all of the statements in the preceeding theorem hold for the prime 2. □

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Mathematisches Institut der Georg-August-Universität, Bunsenstr. 3-5, D-37073 Göttingen, Fed. Rep. of Germany