Normalizers of maximal tori and real forms of Lie groups

Given a complex connected reductive Lie group $G$ with a maximal torus $H\subset G$, Tits defined an extension $W_G^T$ of the corresponding Weyl group $W_G$. The extended group is supplied with an embedding into the normalizer $N_G(H)$, such that $W_G^T$ together with $H$ generate $N_G(H)$. In this paper we propose an interpretation of the Tits classical construction in terms of the maximal split real form $G(\mathbb{R})\subset G$, which leads to the simple topological description of $W^T_G$. We also consider a variation of the Tits construction associated with compact real form $U$ of $G$. In this case we define an extension $W_G^U$ of the Weyl group $W_G$, naturally embedded into the group extension $\widetilde{U}:=U\rtimes\Gamma$ of the compact real form $U$ by the Galois group $\Gamma={\rm Gal}(\mathbb{C}/\mathbb{R})$. Generators of $W^U_G$ are squared to identity as in the Weyl group $W_G$. However, the non-trivial action of $\Gamma$ by outer automorphisms requires $W^U_G$ to be a non-trivial extension of $W_G$. This gives a specific presentation of the maximal torus normalizer of the group extension $\widetilde{U}$. Finally, we describe explicitly the adjoint action of $W_G^T$ and $W^U_G$ on the Lie algebra of $G$.


Introduction
In the standard approach to classification of complex semisimple Lie groups the problem is reduced to an equivalent problem of classification of root data. In other words, the root data (the roots/co-roots and the lattices of characters/co-characters of a maximal torus H ⊂ G) define the corresponding semisimple Lie group up to an isomorphism. Curtis, Wiederhold and Williams [CWW] demonstrate that for classification of compact connected semisimple Lie groups G it is enough to classify the normalizers N G (H) of maximal tori H ⊂ G. The normalizer provides an information about the action of the Weyl group W G := N G (H)/H on H, but this is not enough for classification, as one also needs the precise structure of the group extension of W G by H. Thus, for the classification problem one might replace the original object, the semisimple Lie group G, by the group extension of the finite group W G by the abelian Lie group H. One perspective to grasp this equivalence is to look at N G (H) as a kind of degeneration of G [CWW]. An apparently related but more conceptual approach is based on attempts to look at N G (H) as the Lie group G defined over some non-standard base field (akin to the mysterious field F 1 "with one element" introduced by Tits [T3], probably with regard to this subject). In this way an equivalence of the two classification problems for compact semisimple Lie groups and normalizers looks like a manifestation of a general principle (due to C. Chevalley [C]), saying that a classification of semisimple algebraic groups should not essentially depend on the nature of the base algebraically closed field.
The reasoning above demands a more detailed study of the group extension structure on N G (H). The key fact is that this extension does not split in general [D], [T1], [T2], [CWW], [AH]. To get a universal description of N G (H) one should look for a set-theoretic section of the projection N G (H) → W G generating an extension of W G . Such construction was proposed by Demazure [D] and Tits [T1], [T2]. It may be naturally formulated in terms of the Tits extension W T G of Weyl group W G by the subgroup of order 2 elements in H. This construction allows an explicit presentation of N G (H) by generators and relations.
Although the Tits construction is known for a long time, its forthright explanation involves scheme-theoretic arguments. Precisely, for a Chevalley group scheme G over Z the Tits extension is the group of Z-points of the normalizer of the Z-split torus of G. In this paper we use set-theoretic arguments to explain the Tits construction in the case of complex reductive Lie groups (for recent discussion on Tits groups see e.g. [N], [DW], [AH]). After reminding general results on normalizers of maximal tori in Section 2 we revisit the Tits construction in Section 3. We stress that the Tits group construction is defined for the split real form G(R) ⊂ G of a complex semisimple group G. This enables us to present in Proposition 3.1 a simple purely topological description of the Tits extension of the Weyl group W G (our considerations appear to be pretty close to the final section of [BT]).
As a variation of the Tits construction we consider its analog for the maximal compact group U ⊂ G. Precisely, we define an extension W U G of W G embedded into a semi-direct product U = U ⋊ Γ of the maximal compact subgroup U ⊂ G and the Galois group Γ = Gal(C/R) acting in the standard (non-twisted) way on U. In this case the natural generators of the corresponding extension W U G of the Weyl group are squared to identity (in contrast with the case of the Tits group), while the non-trivial extension of W G arises from the action of Γ via on W U G by outer automorphisms. The main result of this paper (Theorem 4.1 in Section 4) describes the structure of maximal tori normalizers in the Galois group extension U of the compact connected semisimple Lie group U. Note that our construction is dealing with the extended group U and thus, it differs from the constructions given in [DW], [AH]. In Section 5 we calculate explicitly the adjoint action of the Tits group and of its unitary analog on the Lie algebra g = Lie(G). This action, in contrast with the adjoint action on h ⊂ g, depends on the lift of W G into G. Finally in Section 6 we provide details of the proof of Theorem 4.1.

Normalizers of maximal tori and Weyl groups
We start with recalling standard facts on normalizers of maximal tori and the associated Weyl groups. Let G be a complex connected semisimple Lie group, H ⊂ G be a maximal torus and N G (H) be its normalizer in G. Then there is the following exact sequence where p is the projection on the finite group W G := N G (H)/H, the Weyl group of G. The Weyl group W G does not actually depend on the choice of H ⊂ G, and thus appears to be an invariant of G. Let g := Lie(G) and let I be the set of vertices of the Dynkin diagram associated to G, where |I| = rank(g). Let (∆, ∆ ∨ ) be the root-coroot system corresponding to G, {α i , i ∈ I} be a set of positive simple roots, and {α ∨ i , i ∈ I} be the corresponding set of positive simple co-roots. Let a ij , a ij = α ∨ i , α j , be the Cartan matrix of (∆, ∆ ∨ ). The Weyl group W G has a simple presentation in terms of generators and relations. Precisely, W G is generated by simple root reflections {s i , i ∈ I} subjected to where m ij = 2, 3, 4, 6 for a ij a ji = 0, 1, 2, 3, respectively. Equivalently these relations may be written in the Coxeter form: The exact sequence (2.1) defines the canonical action of W G on H. Let h i ∈ h = Lie(H) be the generators corresponding to the simple co-roots α ∨ i , then the W G -action on h ⊂ g and on its dual is as follows (2.5) The exact sequence (2.1) does not split in general, i.e. N G (H) is not necessarily isomorphic to a semi-direct product of W G and H. A delicate situation in this regard is described by the following result due to [CWW], [AH].
Theorem 2.1 Assume G is a simple complex Lie group and let Z(G) be the center of G. Then the exact sequence (2.1) splits in the following and only the following cases: • Type A ℓ , ℓ ≥ 1, ℓ = 3, such that |Z(G)| is odd; • Type B ℓ , ℓ > 1, for the adjoint form; • Type D ℓ , ℓ > 2, for all forms except Spin(2ℓ); • Type G 2 .
Thus to have an explicit description of the normalizer N G (H) one should look for appropriate set-theoretic section of the projection map p in (2.1) generating some extension of W G . In the following section we provide the construction of the resulting extension of the Weyl group by a finite group. Let us note that for a normal finite subgroup Z ⊂ G one has: if (2.1) splits for G then it splits for G/Z as well. In the following, for simplicity, we consider only the case of simply-connected complex groups.

The Tits extension of Weyl group
To describe the extension (2.1) in terms of generators and relations Tits proposed the following extension W T G of the Weyl group W G by a discrete group [T1], [T2] (closely related results were obtained by Demazure [D]).
Definition 3.1 Let A = a ij be the Cartan matrix corresponding to a semisimple Lie algebra g = Lie(G) and let m ij = 2, 3, 4, 6 for a ij a ji = 0, 1, 2, 3, respectively. The Tits group W T G is an extension of the Weyl group W G by an abelian group Z |I| 2 generated by {τ i , θ i , i ∈ I} subjected to the following relations: where the abelian subgroup is generated by Let {h i , e i , f i : i ∈ I} be the Chevalley-Serre generators of the Lie algebra g = Lie(G), satisfying the standard relations where A = a ij is the Cartan matrix i.e. a ij = α ∨ i , α j . According to [BT] (see also [T1]) there exists a subset {ζ i , i ∈ I} ⊂ H of canonical elements of order two satisfying the following relations where s i , i ∈ I are the generators of the Weyl group W G (2.2), (2.3).
Theorem 3.1 (Demazure-Tits) Let W T G be the Tits group associated with the complex semisimple simply connected Lie group G, then the map defines an embedding of the Tits group W T G into N G (H), such that p(W T G ) = W G for the projection p in (2.1). In particular, the normalizer group N G (H) is generated by H and by the image of the Tits group, so that the following relations hold: The appearance of the Tits extension W T G via a specific choice of a set-theoretic section looks a bit ad hoc. As it is mentioned in Introduction one may use a scheme-theoretic argument to support this particular choice of extension of W G . In the following we propose a set-theoretic argument based on consideration of the split real form G(R) of G to elucidate the construction of W T G . For the split real form G(R) ⊂ G there is an analog of (2.1): with the real split maximal torus given by the intersection of the complex maximal torus with the split real subgroup. Note that the set-theoretic section of (3.10) defining W T G provides a set-theoretic section of (2.1) thus embedding W T G into N G(R) (H(R)). The group H(R) allows the product decomposition is the connected exponential group A = exp(h(R)) without torsion. Therefore, H(R) is not connected and consists of 2 |I| components, and the group M may be identified with the discrete group of connected components of H(R): (3.13) Considering the groups of connected components of the topological groups entering (3.10) we obtain the induced exact sequence This provides a canonical extension of W G by the abelian group M of order 2 |I| .
Explicitly, the group of connected components may be identified with the quotients by the connected normal subgroup A and we have the exact sequence Lemma 3.1 The exact sequence (3.16) splits and thus π 0 (N G(R) (H(R))) allows an embed- Proof. The extension (3.16) is an instance of extensions of π 0 (N G(R) (H(R))) by A. Such extensions are classified by the group H 2 (π 0 (N G(R) (H(R))), A). The triviality of this group follows from the fact that π 0 (N G(R) (H(R))) is a finite group and A is an instance of abelian torsion-free group. By the standard cohomology argument (see e.g. [B], Chapter VI), the second cohomology of any finite group with coefficients in a torsion-free abelian group is trivial. Thus the extension (3.16) is necessarily trivial and therefore the required embedding exists. ✷ Up to now we have constructed a canonical extension of W G given by (3.14). It is easy to see that this extension is isomorphic to the Tits group.
Proposition 3.1 The following isomorphism holds Proof. Recall that the imagesṡ i , ζ i , i ∈ I of the Tits generators belong to the maximally split real subgroup G(R) ⊂ G. Thus we have the embedding W T G into N G(R) (H(R)) providing section of the exact sequence Note that splitting of this exact sequence may be independently verified by using an argument similar to the one used in our proof of Lemma 3.1. Therefore the group N G(R) (H(R)) allows a representation as semidirect product N G(R) (H(R)) = A ⋊ W T G . By considering the connected components we deduce the assertion (3.17). ✷ Example 3.2 For maximal split form SL 2 (R) ⊂ SL 2 (C) we have (3.20) Elements g ∈ N SL 2 (R) (H(R)) are defined by the condition that for each λ ∈ R * there exists λ ∈ R * such that One might check that the normalizer group N SL 2 (R) (H(R)) is a union of two components:

22)
where N 1 is a set of diagonal elements with c = b = 0, ad = 1 = 0, and N s is the set of anti-diagonal elements with a = d = 0, cb = −1. Each of the co-sets N 1 , N s splits further into two connected components depending on the sign of the entries c, d in the last row of g.
The group π 0 (N SL 2 (R) (H(R))) is isomorphic to the quotient of N SL 2 (R) (H(R)) by A ≃ R * + and consists of four elements corresponding to the connected components N ± 1 , N ± s of the group N SL 2 (H(R)), allowing the following parameterization: Clearly, the group π 0 (N SL 2 (R) (H(R))) is generated by representatives of the connected components N ± 1,s , so indeed it is isomorphic to the order four cyclic group generated byṡ, in concordance with (3.17).

Weyl group and Galois extension of the compact real form
As we have demonstrated in the previous Section the Tits group extension W T G appears quite naturally if we consider the split real subgroup G(R) ⊂ G. This motivates to look for analogs of the Tits construction associated with other real forms of G.
Let {h i , e i , f i ; i ∈ I} be the Chevalley-Serre generators of the Lie algebra g = Lie(G), satisfying (3.4), (3.5). We fix the split real structure by assuming the generators {h i , e i , f i ; i ∈ I} to be real. Let ⊤ : g → g be the Cartan anti-involution, associated with the split real structure: Let U ⊂ G be the connected compact real form of the Lie group G: where g → g † is the composition of the Cartan anti-involution (4.1) with the complex conjugation associated with the split real structure). The Galois group Γ := Gal(C/R) ≃ Z 2 of the extension R ⊂ C is generated by γ, γ 2 = 1. The group Γ acts both on G and on U by the complex conjugation, so let us introduce the following semidirect products: Since the generators {e i , f i , h i ; i ∈ I} are real, they are fixed by γ ∈ Γ. Let us note that the Γ-fixed subgroup of U is a maximal compact subgroup K ⊂ G(R) of the split real form G(R).

Lemma 4.2 The map Π acting on the generators by
extends to an involutive automorphism of the group W U G .
For a ji even we have to prove (4.11), which reads This follows from (4.5), (σ jσj ) σ i (σ jσj ) −1 = σ i , by multiplying both sides by σ iσi and using (4.7). ✷ Proposition 4.1 The group W U G is given by the following extension (4.14) of the Weyl group W G by the abelian group Z |I| 2 ⊂ W U G .
Proof. Introduce the elements η i := σ iσi , i ∈ I. They have order two, and they pairwise commute by (4.7): (4.15) and are invariant under involution Π. Consider a Π-stable subgroup H η ⊂ W U G generated by {η i , i ∈ I}. The relation (4.5) may be equivalently written in the following form: (4.16) Indeed, from (4.5) we have σ i η j = η j Π 1−a ji (σ i ), which for a ji even reads σ i η j = η j σ i and for a ji odd (4.5) is equivalent to σ i η j = η jσi = η j η i σ i . This implies the first equation in (4.16). The second relation in (4.16) is obtained by applying the automorphism Π to the first one. The identities (4.16) yield that the subgroup H η ⊂ W U G generated by η i , i ∈ I is normal. A proof of the fact that |H η | = 2 |I| is given in Lemma 6.4 below.
Next, we consider the quotient group W U G /Z |I| 2 . It is generated by s i := π(σ i ), i ∈ I and satisfying the standard relations (2.2), (2.3) of the group W G . Indeed, π(σ i ) = π(σ i ) implies that the relations (4.4) are mapped to the relations (2.2), relations (4.5) equivalent to (4.16) become identities, and the braid relations (4.6) of W U G are mapped to the braid relations (2.3) of W G . Thus, we have a surjective homomorphism π This gives the exact sequence (4.14). ✷ The following analog of Theorem 3.1 holds.
Theorem 4.1 Let U ⊂ G be a maximal compact subgroup of the complex semisimple simply connected Lie group G. Let (∆, W G ) be the root system of g = Lie(G) with the Cartan matrix a ij . Let γ be the generator of the Galois group Γ = Gal(C/R) of the field extension R ⊂ C and let ı ∈ C be the imaginary unit. Then the following map defines an injective homomorphism W U G −→ U , with U given by (4.3). The elements ς i , i ∈ I and γ together with the maximal torus H generate the group N G (H) ⋊ Γ.
We give a proof of Theorem 4.1 in Section 6 below.
Let us stress a clear analogy between the constructions of W T G and W U G . On the one hand, in the Tits setting the finite group is embedded into the maximal compact subgroup K ⊂ G(R) of the maximally split real form G(R) ⊂ G. On the other hand the extension W U G constructed above is embedded into Galois group extensionŨ of maximal compact subgroup U ⊂ G (note that the action of complex conjugation on U may be equivalently represented by the action of the Cartan anti-involution (4.1)). In this wayŨ plays the role analogous to K ⊂ G(R) in the Tits construction, while W U G looks like a "complex" analog of the finite group W T G , where the relations τ 2 i = θ i are replaced by σ iσi = η i . Let us note that for each i ∈ I we have the cyclic subgroup ṡ i : (ṡ i ) 4 = 1 = Z 4 ⊂ W T G in the Tits construction. Meanwhile in the case of W U G for each i ∈ I the group σ i ,σ i : σ 2 i =σ 2 i = (σ iσi ) 2 = 1 = (Z 2 ) 2 ⊂ W U G appears. Note that these two instances exhaust possible extensions of Z 2 by Z 2 . It is natural to expect that with other real forms of the complex group G one can associate appropriate extensions of the Weyl group W G . These extensions presumably would be combinations of both constructions considered above.

Adjoint action of the extended Weyl groups
While the action of W G on the maximal commutative subalgebra h = Lie(H) is defined canonically (2.5) and does not depend on a lift of W G into N G (H) its action on the whole Lie algebra g = Lie(G) does depend on the lift. Above we have considered two extensions of the Weyl group W G together with their homomorphisms into the corresponding Lie group. Here we describe their induced adjoint actions on g.
Proposition 5.1 The adjoint action of the Tits group W T G on the Lie algebra g = Lie(G) via homomorphism (3.7) is given by , Proof. Relations (5.1) are actually relations for sl 2 Lie subalgebras generated by {e i , h i , f i , i ∈ I} and may easily be checked using for example the standard faithful representation (6.3). Relations (5.2) trivially follow from the Lie algebra relations (3.4). Thus we need to prove (5.3). Let us introduce the following notation:ṡ i (a) := Adṡ i (a). Then for the conjugated generators we have These relations fix the r.h.s. of (5.3) up to coefficients. Let us calculate the coefficients by taking into account only the terms of the right weights. We havė where we have used the Serre relations (3.5) and denote by · · · the terms of the "wrong" weight. Taking into account Ad e f i (e i ) = e i + · · · , (5.7) we obtain the first relation in (5.3). The second relation is obtained quite similarly using the following equality (for a proof see Lemma 6.1) (5.8) In this case we havė , (5.13) Now we describe the action on g of the Weyl group extension W U G introduced in Section 4. It is convenient to express it in terms of purely imaginary generators ıe i , ıf i , i ∈ I.
Proposition 5.2 The elements of the group W U G act on the Lie algebra g = Lie(G) via homomorphism (4.18) as follows (5.14) and Next, we use the following representation for ς i (see Lemma 6.1 for details): and Proposition 5.1 to obtain (5.16) and (5.14). ✷ 6 Proof of Theorem 4.1 We start with establishing an explicit relation between the generators ς i ,ς i (4.18) and the Tits generatorsṡ i .
Lemma 6.1 For each i ∈ I the following identities holḋ (6.1) Thus the generators {ς i ,ς i , i ∈ I} defined by (4.18) may be represented as follows Proof. The identities (6.1) follow from the corresponding relations in SL 2 ⊂ G, using the standard faithful two-dimensional representation φ : (6.8) The identityς 2 i = 1 follows fromς i = γς i γ, i ∈ I. ✷ Now let us verify that the generators ς i ,ς i and ξ i = ς iςi , i ∈ I satisfy the remaining defining relations (4.16), (4.6) for the group W U G : and where m ij = 2, 3, 4, 6 for a ij a ji = 0, 1, 2, 3, respectively.
For the relation (6.10), on the left side we have (6.13) Sinceṡ iṡj · · · m ij =ṡ jṡi · · · m ij holds due to (3.3), the identity (6.10) reduces to the following (6.14) In turn the identity (6.14) may be proved by invoking the following fact.
Lemma 6.3 For each pair i, j ∈ I, i = j, the following holds: (1 − s j + s i s j − . . . Proof. Consider the order 2m ij Coxeter subgroup of W G generated by a pair of the simple root reflections s i , s j , i = j : s i , s j : (s i s j ) m ij = s 2 i = s 2 j = 1 ⊂ W G . (6.16) This group is isomorphic to the dihedral group D m ij ⊂ O 2 (R) of symmetries of m ij -gone in the real plane V ij = Rh i ⊕ Rh j . The dihedral group may be equivalently written in the following form: D m ij = t, r : t m ij = r 2 = 1, rtr −1 = t −1 , t = s i s j , r = s i . (6.17) We have two projectors in the plane V ij = Rh i ⊕ Rh j : P ± = 1 ± s i 2 : V ij −→ V ij , P 2 ± = P ± , P ± P ∓ = 0, (6.18) such that Therefore, the identity (6.15) is equivalent to the following: (1 − s j + s i s j − . . . (6.20) where (−1) g := det(g) is the sign character of D m ij ⊂ O 2 (R). The kernel of the sign character is a normal subgroup, Z m ij = t : t m ij = 1 ⊂ D m ij , (6.21) which consists of the rotations by 2πk m ij , 0 ≤ k < m ij of the plane V ij . The non-trivial co-set in D m ij / t consists of the reflections {r k = t k r : 0 ≤ k < m ij } with r 0 = r being a reflection sending h i to −h i . Thus we have (−1) r k = det(r k ) = −1 and D m ij = {t k : 0 ≤ k < m ij } ⊔ {r k = rt k : 0 ≤ k < m ij }, (6.22) hence the identity (6.20) reads