On Normalizers of Maximal Tori in Classical Lie Groups

Home , Classical group, Group extension, Maximal torus, Tits group, Weyl group

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The fact that for Cℓ-series of classical Lie groups the considered construction provides an embedding of the extension of the Weyl group by 2-torsion elements of a maximal torus, called the Tits group but not an embedding of the Weyl group itself looks rather surprising in this context. We propose an explanation for this phenomenon based on non-commutativity of quaternions. The main argument is based on the known fact that while general linear and orthogonal Lie groups are naturally associated with matrix groups over complex and real numbers symplectic groups allow description in terms of matrix groups over quaternions. This leads to a modification of the standard notion of maximal torus by taking into account the non-commutative nature of quaternions. As a consequence the notions of normalizer of maximal torus and the Weyl group are also modified. The standard Weyl groups of symplectic Lie groups appear as subgroups of thus defined Weyl groups. The Tits extension of the symplectic Weyl group then arises via construction of universal cover of Aut(H)= SO3 by the three-dimensional spinor group Spin3. Concretely this might be traced back to the fact that that additional quaternionic unit  squares to minus one. It is worth mentioning that the standard algorithm of the Gelfand-Zetlin construction of bases in finite-dimensional irreducible representations fails in the case of symplectic Lie groups. This fact obviously reverberates with the absence of the splitting of (1.1) in symplectic case. We believe that this is not accidental coincidence and the issue may be clarified using quaternionic geometry. Let us note in this respect that description of classical Lie group in terms of matrix algebras over division algebras might be generalized to other, non-classical Lie groups (see e.g. [B] and references therein). We expect that the (im)possibility of embedding of the Weyl groups into the corresponding Lie groups may be elucidated in all these cases generalizing our considerations for symplectic Lie groups and quaternionic matrix algebras. In this note we also consider the problem of construction of a section of p in (1.1) for unimodular linear groups SLℓ+1, special orthogonal groups SOℓ+1, pinor/spinor groups Pinℓ+1 and Spinℓ+1. Although these groups are not classical Lie groups in strict sense they are related to classical Lie groups either via central extensions or via taking unimodular subgroups. In the case of central extension the resulting section is apparently given by a central extension of the corresponding Weyl group. Moreover the case of unimodular subgroups preliminary draft 2 December 5, 2019 1:46 is also covered by central extension of the Weyl group. We provide explicit description of maximal torus normalizers in terms of generators and relations in all these cases. The plan of the paper is as follows. In Section 2 we recall required facts on semisimple Lie algebras and groups including normalizers of maximal tori (see also Appendix 10 for detailed discussion of classical Lie groups). In Section 3 we recall constructions of classical Lie groups as fixed point subgroups of appropriate involutions of general Lie groups after E.Cartan. In Section 4 the structure of normalizers of maximal tori of general linear Lie groups is considered in details including explicit lifts of the Weyl groups. In Section 5 we present construction of the lifts of the Weyl groups of the classical Lie groups in terms of those for general Lie groups (see Theorem 5.1). In Section 6 we further clarify the explicit formulas of the previous Section 5 by establishing connections between maximal tori normalizers, the Weyl groups and their lifts for general linear groups and its classical subgroups. In Section 7 the underlying reasons for the special properties of symplectic groups are considered. In

Section 8 we extend our analysis to the groups SLℓ+1, SOℓ+1, Pinℓ+1 and Spinℓ+1. In all these cases we construct explicit sections of p in (1.1) realized by central extensions of the corresponding Weyl groups. Finally in Section 9 we compute the adjoint action of the lifts of the Weyl groups on Lie algebra g = Lie(G) for all classical Lie groups. Various technical details of the construction are provided in Appendix. Acknowledgments: The research of the second author was supported by RSF grant 16- 11-10075. The work of the third author was partially supported by the EPSRC grant EP/L000865/1.

2 Preliminaries on semisimple Lie algebras and groups

Let g be a complex semisimple Lie algebra and let h g be the Cartan subalgebra of g, ⊂ ∗ so that dim(h) = rank(g) = ℓ. LetΠ= αi, i I h be the set of simple roots of g, indexed by the set I of vertexes of the Dynkin{ ∈ diagram} ⊂ Γ. Let Π∨ = α∨, i I h be { i ∈ } ⊂ the set of corresponding coroots of g. Let Φ be the set of roots and let Φ∨ be the set of co-roots of g, then let (Π, Φ; Π∨, Φ∨) be the root system associated with g, supplied with a non-degenerate Z-valued pairing

, : Φ Φ∨ Z . (2.1) h i × −→ Introduce the weight lattice,

Λ = γ h∗ : γ, α∨ Z , α∨ Φ∨ h∗ , (2.2) W ∈ h i ∈ ∀ ∈ ⊂ and deﬁne the fundamental weights ̟ , i I by i ∈ ̟ , α∨ = δ . (2.3) h i j i ij

Fundamental weights provide a basis of the weight lattice ΛW and we have

ℓ α = a ̟ , a = α , α∨ , (2.4) CartanMatrix j ij i ij h j i i i=1 X preliminary draft 3 December 5, 2019 1:46 where A = a is the Cartan matrix of g. The root lattice Λ h∗ is generated by k ijk R ⊂ simple roots αi Π, i I and appears to be a sublattice of the weight lattice ΛW , so that Λ Λ . The∈ quotient∈ group Λ /Λ is a ﬁnite group of order R ⊂ W W R Λ /Λ = det(A) . (2.5) FundamentalGroup | W R| ∨ ∨ Cℓ The co-weight and co-root lattices ΛW , ΛR h in the Euclidean space h = ; , are generated by the fundamental co-weights and⊂ simple co-roots, given by h· ·i ℓ ℓ α , ̟∨ = δ , ̟∨ = c α∨ , α∨ = a ̟∨ , (2.6) coweights h i j i ij j ji i j jk k i=1 k X X=1 where C = c =(AT )−1 is the inverse transposed Cartan matrix A. k ijk The corresponding Weyl group W (Φ) is generated by simple roots reﬂections,

s (α )= α α ,α∨ α = α a α , (2.7) WeylRoots i j j −h j i i i j − ij i and its action in h∗ preserves the root system Φ h∗. Each symmetry of the Dynkin diagram Γ = Γ(Φ) induces an automorphism of Φ, and the⊂ group Aut(Φ) of all automorphisms of the root system Φ h∗ contains W (Φ) as a normal subgroup. Moreover, the following holds: ⊂ Aut(Φ) = W (Φ) ⋊ Out(Φ) , (2.8) AutRoots with Out(Φ) being a group of symmetries of Γ(Φ). Let h = h , e = e , f = e , i I be the standard set of generators of g: { i αi i αi i −αi ∈ }

[hi, hj]=0 , h , e = a e , h , f = a f , (2.9) ChSe i j ij j i j − ij j e , f = h , e , f = 0 , i = j ; i i i i j 6 a 1−aij 1− ij (2.10) Serre2 adei (ej) = adfi (fj)=0. In the following a slightly modiﬁed presentation of (2.9) will be useful. Namely, the Lie algebra g can be generated by ̟∨ , e , f : i I subjected to the following relations: { i i i ∈ } ℓ [̟∨, e ] = e δ , [̟∨, f ] = f δ , [e , f ] = δ a ̟∨ . (2.11) LieAlgebra i j j ij i j − j ij i j ij jk k k X=1

Now let Gc be a connected compact semisimple Lie group of rank ℓ, such that gc = Lie(Gc) is the tangent Lie algebra at 1 G . Let G = G C be the complexiﬁcation of G and let ∈ c c ⊗ c g = Lie(G)= gc C be its Lie algebra. Then the Lie algebra gc is generated by the following elements: ⊗

H = ıh , J = f e , P = ı(e + f ) , k I. (2.12) HJP k k k k − k k k k ∈ preliminary draft 4 December 5, 2019 1:46 In particular, the following relations hold:

ℓ ∨ ∨ ∨ Jk, ı̟j = δkjPk , ı̟j , Pk = δkjJk , [Pk,Jk]=2 aki(ı̟i ) . (2.13) HJPrel i=1 X ∗ Let HG G be its maximal torus, such that h = Lie(HG), and let X (HG) be the group ⊂ 1 of (rational) characters χ : HG S . Then the diﬀerential dχ at 1 G of a character ∗ → ∈ ∗ ∗ χ X (HG) is a linear form on h; hence it provides an embedding X (HG) h as a discrete∈ subgroup (lattice), supplied with a scalar product , . ⊂ h i ∗ The dual X∗(HG) h of the lattice X (H) is isomorphic to hZ. Moreover, using the above notations of the⊂ (co)root and (co)weight lattices we have

∗ ∨ ∨ Λ X (H ) Λ , Λ X (H ) hZ Λ . (2.14) R ⊆ G ⊆ W R ⊆ ∗ G ≃ ⊆ W

The adjoint action of maximal torus HG on the complex Lie algebra g provides the Cartan decomposition:

g = hC g , g = X g : ad (X) = α(h)X, h h ; ⊕ α α ∈ h ∀ ∈ α∈Φ (2.15) M g = Ce , g = Cf , α Φ . α α −α α ∈ + ∗ The root system Φ and the lattice X (HG) (or its dual lattice X∗(HG) hZ) determine a unique (up to isomorphism) connected semisimple Lie group G. In particular,≃ G is simply ∗ connected if and only if X (H) ∼= ΛW . The center (G) of a complex connected Lie group G allows for the following presenta- Z tion:

(G) Λ∨ /X (H ) X∗(H )/Λ , (2.16) Z ≃ W ∗ G ≃ G R and the group (G) ⋊ Out(Φ) is isomorphic to the group of symmetries of the extended Z Dynkin diagram Γ(Φ). More speciﬁcally, in case of simply-connected G its center is isomorphic to the quotient group (2.5): e (G) Λ /Λ . (2.17) Z ≃ W R

In particular, for simply-connected complex Lie groups of types Aℓ, Bℓ, Cℓ and Dℓ we have (A ) Z/(ℓ + 1)Z , (B ) Z/2Z , (C ) Z/2Z . Z ℓ ≃ Z ℓ ≃ Z ℓ ≃ Z/2Z Z/2Z , ℓ 2Z (2.18) (Dℓ) ⊕ ∈ . Z ≃ ( Z/4Z , ℓ 1+2Z ∈ Let Aut(G) be the group of all automorphisms of a connected complex semisimple Lie group G. Its connected component can be identiﬁed with the group Int(G) of inner automorphisms, which is isomorphic to the adjoint group:

Int(G) G/ (G) . (2.19) ≃ Z preliminary draft 5 December 5, 2019 1:46 The quotient group over the connected component is the group of outer automorphisms:

Out(G) = Aut(G)/Int(G) . (2.20) OutG

In case of simply-connected Lie group G, the following holds (see e.g. [L], Chapter V Theorem 4.5.B):

Out(G) Out(Φ) . (2.21) OUT ≃

2.1 Normalizers of maximal tori

Given a connected semisimple complex Lie group G with a (ﬁxed) maximal torus H G G ⊂ of ﬁnite rank ℓ, let NG = NG(HG) be the normalizer of the maximal torus and let us write

p 1 H N W 1 . (2.22) NGH −→ G −→ G −→ G −→

The quotient group WG := NG/HG is finite and is isomorphic to the reflection group W (ΦG) associated with the corresponding root system ΦG. The group WG = W (ΦG) allows presentation by simple root reflections s , i I from (2.7) as generators subjected to the { i ∈ } following relations:

2 si =1, (2.23) BR0

sisjsi = sjsisj , i = j I, ··· ··· 6 ∈ (2.24) BR mij mij

∨ where mij = 2, 3, 4, 6 for |aija{zji =} 0, 1|, 2, 3,{z respectively.} Here aij = αj, αi is the Cartan matrix entry (2.4). Equivalently these relations may be written in theh Coxeteri form:

s2 =1, (s s )mij =1 , i = j I. (2.25) CoxeterRel i i j 6 ∈

The exact sequence (2.22) deﬁnes the canonical action of WG on HG so that the corresponding action on the Lie algebra h = Lie(HG) is provided by (2.7):

s (h ) = h α ,α∨ h = h a h . (2.26) WeylGroupAction i j j −h i j i i j − ji i This action preserves the scalar product , in h, which allows to identify Weyl group W = W (Φ ) with a subgroup in the orthogonalh i group O h, , . G G h i Let us stress that the situation is bit diﬀerent in the case of non-co nnected groups. Thus in the following we encounter an example of non-connected group, the orthogonal group O2ℓ having two connected components. In this case the quotient NO2ℓ (H)/H is larger then the

Weyl group WO2ℓ deﬁned by the generators and relations (2.23), (2.24) and contains the outer automorphism of the root system of the simple connected Lie groups SO2ℓ.

The important fact is that the exact sequence (2.22) does not split in general i.e. NG is not necessarily isomorphic to the semi-direct product WG ⋉ HG (for various details see [D],[T2], [CWW], [AH]). Thus in general we may only pick a suitable section of the projection map preliminary draft 6 December 5, 2019 1:46 p in (2.22). One such universal section for arbitrary reductive Lie group we constructed by T Tits [T2] saying that for general G there exists a larger subgroup WG NG, containing WG as a quotient and ﬁtting into the following exact sequence: ⊂

/ (2) / T / / 1 HG WG WG 1 . (2.27) Titsgroup

(2) Z Z ℓ Here HG ( /2 ) is the 2-torsion subgroup of the maximal torus HG in the reductive ≃ T complex Lie group G. In case the group G is semisimple the Tits group WG allows for the following explicit presentation by the following generators [AH] (see the notations of (2.12))1

π J s˙ = e 2 i , i I, (2.28) Titsgen i ∈ and relations (compare with (2.23), (2.24)):

(s ˙ )2 = eπıhi , s˙ s˙ =s ˙ s˙ , i = j , i i j ··· j i ··· 6 mij mij (2.29) Tits

Ads˙ (h) =| s{zi(h}) , |h{z h}. i ∀ ∈ T Using this presentation we readily observe that the Tits group WG is not only a subgroup of the normalizer N , but is a subgroup of the corresponding compact group G G, and G c ⊂ more precisely of the normalizer subgroup NGc (Hc) of the maximal torus Hc in the compact group Gc. The true meaning of this construction is rather elusive; in [GLO] we have proposed some underlying reasons for the existence of this Tits construction. Below we will follow another direction and consider only the case of classical Lie groups G. Recall that complex classical Lie group is a simple reductive Lie group allowing embedding in the general Lie group as a ﬁxed point subgroup of an involution. Equivalently the classical groups may be deﬁned as stabilizer subgroup of bilinear forms. Essentially they are exhausted by the following families of the groups C C C C GLℓ+1( ),O2ℓ+1( ), Sp2ℓ( ),O2ℓ( ), (2.30) ClassicalGroups0 corresponding to the series Aℓ, Bℓ, Cℓ and Dℓ of Dynkin diagrams. Let us stress that we dis- tinguish the classical Lie groups per se from their various cousins like SLℓ+1(C), P SLℓ+1(C), C C Spinℓ+1( ), the metaplectic group Mp2ℓ( ) et cet. In the case of the classical Lie groups the results of [D], [T2], [CWW], [AH] might be formulated as the following statement:

In the case of general linear group GLℓ+1(C) there is a section to the exact sequences • T C (2.22) and (2.27), so that NGLℓ+1( ) (and the Tits subgroup WGLℓ+1 ) contains the Weyl

group W (Aℓ)= WGLℓ+1 associated with the root system Aℓ;

1In the following expressions we write the group elements as exponential of the linear combinations of Lie algebra generators using the canonical exponential map exp : Lie(G) G. Note that map has a non-trivial 2πJi 2→πPi 2πıhi kernel so that for instance in case G = GLℓ+1(C) we have e = e = e = 1 for each i I. ∈ preliminary draft 7 December 5, 2019 1:46 In the case of orthogonal groups Oℓ+1(C) there exits a section to the corresponding • T exact sequence (2.27), so that WOℓ+1 contains a subgroup isomorphic to WOℓ+1 ; In the case of the symplectic group Sp (C) no section of the corresponding exact se- • 2ℓ quence (2.27) exists.

3 Classical Lie groups via involutive automorphisms

The deﬁnition of classical Lie groups as simple Lie groups isomorphic to ﬁxed point subgroups with respect to certain involutive automorphism in GLℓ+1(C) goes back to E. Cartan and a relevant expositions of the subject can be found in [H] and [L]. Below we recall basics of this construction.

Given the general linear group GLℓ+1(C) and a maximal torus Hℓ+1 identiﬁed with the subgroup of diagonal elements, let us describe its group of outer automorphisms Out(GLℓ+1). Given Dynkin diagram Γ(Aℓ), let us choose natural ordering of the set of its vertices I = 1, 2,...,ℓ , and consider the outer automorphism of the root system induced by the Dynkin { } diagram symmetry:

ι : I I , i ℓ +1 i . (3.1) OutDynkin −→ 7−→ −

It deﬁnes the automorphism of the set ΠAℓ of simple roots, and therefore can be extended to an (outer) automorphism of the root system Φ(Aℓ). Thus extended automorphism ι obviously preserves the lattice hZ and can be lifted to the following involutive automorphism of the Lie group GLℓ+1(C) (we keep the same notation ι):

ι : g (gτ )−1 , g GL (C) , (3.2) CanonicalInv 7−→ ∈ ℓ+1 where τ is the reﬂection at the opposite diagonal:

(gτ ) = g , g = g GL (C). (3.3) TranspOpp ij ℓ+2−j,ℓ+2−i k ijk ∈ ℓ+1

Note that the automorphism ι respects the maximal torus Hℓ+1 GLℓ+1(C) given by invertible diagonal matrices. Given the involutive automorphism ι Out(⊂ GL ) we have a family ∈ ℓ+1 of involutive automorphisms still respecting the maximal torus Hℓ+1. Such automorphisms are obtained by combining ι with arbitrary inner automorphisms AdS Int(GLℓ+1), where ι ∈ N C (H ) subjected to the condition (GL ). Each such θ = Ad ι S ∈ GLℓ+1( ) ℓ+1 SS ∈ Z ℓ+1 S ◦ deﬁnes the corresponding ﬁxed points subgroups

θ Gθ = (GLℓ+1(C)) . (3.4)

Classiﬁcation of the corresponding subgroups is due to E. Cartan (see [H] Chapter IX, and [L] Chapter VII for details). Crucial fact following from his theory is that the requirement of G being a simple reductive Lie group imposes strong restrictions on possible choice of the inner automorphism AdS . The corresponding list of simple reductive groups is exhausted by

preliminary draft 8 December 5, 2019 1:46 the following cases

O = g GL : θ (g)= g , 2ℓ+1 ∈ 2ℓ+1 B Sp = g GL : θ (g)= g , (3.5) ClassicalGroups 2ℓ ∈ 2ℓ C O = g GL : θ (g)= g , 2ℓ ∈ 2ℓ D } where θG are the involutive automorphisms of general linear group

θ : g gι −1 , (3.6) CanonicalInv1 G 7−→ SG SG with the diagonal matrices SG given by

= diag(1, 1,..., 1) GL , det( )=( 1)ℓ ; SB − ∈ 2ℓ+1 SB − = diag(1, 1,..., 1, 1) GL , det( )=( 1)ℓ ; (3.7) CanonicalInv2 SC − − ∈ 2ℓ SC − = η , η = diag(Id , Id ) , det( )=1 . SD SC · ℓ − ℓ SD

There is also trivial case of θG = id corresponds to the general linear Lie group GLℓ+1(C) itself. In the following we will use the term “classical group” in the narrow sense by excluding the trivial case of general linear groups. Let us note that the construction of classical Lie groups via involutive automorphims described above differs from more traditional one where instead of the reflection at the opposite diagonal (3.3) the standard transposition is used. For instance elements of the orthogonal subgroup O GL (C) are defined by the condition g =(gt)−1 where ℓ+1 ⊂ ℓ+1 (gt) = g , g = g GL (C). (3.8) TRST ij j,i k ijk ∈ ℓ+1 Although two constructions are equivalent (one may be obtained from another by adjoint action of a certain group element of the general linear group) the construction (3.5) is more convenient when dealing with symmetries of the Dynkin diagrams and abstract root data [DS]. Thus in the following we use (3.5) as our basic definition.

4 The normalizer of maximal torus in GLℓ+1

In the case of general linear Lie groups GLℓ+1 the exact sequence (2.22) is split by simple reasons. Indeed each element g Nℓ+1 induces an (inner) automorphism Adg of the Lie algebra gl (C), preserving the Cartan∈ subalgebra h as well as the scalar product , in it. ℓ+1 h i This deﬁnes a group homomorphism from N into the orthogonal group O hC, , , so ℓ+1 h i that its image is the Weyl group: N / (GL ) W O (C) GL (C) . (4.1) splitProjGL ℓ+1 Z ℓ+1 −→ GLℓ+1 ⊂ ℓ+1 ⊂ ℓ+1 This yields a canonical embedding of the Weyl group into the orthogonal group, which gives rise to homomorphism W GL (C). GLℓ+1 → ℓ+1 preliminary draft 9 December 5, 2019 1:46 This argument implies the following explicit presentation of the lifts of the simple root generators s W (A )= W , 1 i ℓ as elementary permutation matrices i ∈ ℓ GLℓ+1 ≤ ≤ Idi−1 s S = 0 1 N , S2 = 1 , (4.2) permat i 7−→ i 1 0 ∈ ℓ+1 i Idℓ−i satisfying the standard Coxeter relations (S S )2 = 1 , i j > 1 , (S S )3 = 1 , i j =1 , (4.3) WeylPerm i j | − | i j | − | of the generators of permutation group Sℓ+1.

The lifts Si of the generators si of the Weyl group WGLℓ+1 should be compared with T the generators of the corresponding Tits group WGLℓ+1 deﬁned as follows. For the reductive T group GLℓ (C) the Tits group W Nℓ is given by the extension +1 GLℓ+1 ⊂ +1

/ (2) C / T / / (4.4) TitsgroupGL 1 Hℓ+1( ) WGLℓ+1 WGLℓ+1 1 , where H(2) is the 2-torsion subgroup of the maximal torus H . Let e , 1 k (ℓ +1) ℓ+1 ℓ+1 { kk ≤ ≤ } be the bases of the coweight lattice of GLℓ+1(C) corresponding the diagonal matrices with only one non-zero element being a unit on the diagonal (it might be expressed through the ∨ ∨ (2) fundamental coweights as follows ekk = pk pk−1, see (10.14)). Then Hℓ+1 is generated by the elements − T := eπıekk , 1 k (ℓ + 1) . (4.5) 2torsGL k ≤ ≤ T The group WGLℓ+1 is generated by (2.28) for the roots system Aℓ

Idi−1 π J s˙ = e 2 i = 0 −1 N , (s ˙ )2 = T T −1 , (4.6) i 1 0 ∈ ℓ+1 i i i+1 Idℓ−i together with an additional central element given by product of all Tk, k =1,...,ℓ+1. Here we write down matrix form of generators ˙i using standard faithful representation. Comparing the matrix forms we arrive at the following relations: S = T s˙ =s ˙ T , 1 i ℓ . (4.7) permat0 i i i i i+1 ≤ ≤ This leads to another presentation of the Tits group.

Lemma 4.1 The elements S , 1 i ℓ and T , 1 k (ℓ + 1) , (4.8) 2torsGLN i ≤ ≤ k ≤ ≤ satisfying the relations 2 2 Si = Tk = 1 , TkTn = TnTk , (S S )2 = 1 , i j > 1 , (S S )3 = 1 , i j =1 , (4.9) WeylExt1 i j | − | i j | − |

SiTk = Tsi(k)Si ,

T provide a presentation of the Tits group WGLℓ+1 . preliminary draft 10 December 5, 2019 1:46 Proof : The only non-obvious relation in the last line of (4.9) follows from explicit matrix computation:

T S T −1 = T S T −1 = S T T = T T S , 1 i ℓ . (4.10) i i i i+1 i i+1 i i i+1 i i+1 i ≤ ≤ ✷ In Section 3 we have introduced involutive automorphisms θ Aut(GL (C)) (3.5), G ∈ ℓ+1 (3.6), (3.7) deﬁning the classical Lie groups. The involutions θG do not act on the lift of

WGLℓ+1 deﬁned by the generators Si but do act on its extension given by the Tits group. Explicitly we have

θ :s ˙ s˙ , T T , B i 7−→ 2ℓ+1−i i 7−→ 2ℓ+2−i θ :s ˙ s˙ , T T , (4.11) CINV C i 7−→ 2ℓ−i i 7−→ 2ℓ+1−i θ :s ˙ s˙ , T T . D i 7−→ 2ℓ−i i 7−→ 2ℓ+1−i T Therefore, thus deﬁned automorphism θG Aut(W ) preserves the normal subgroup ∈ GLℓ+1 H(2) = T , 1 k ℓ +1 . ℓ+1 h k ≤ ≤ i Let us describe the action of the involutions θG in terms of the another set of generators T of WGLℓ+1 ; introduce new elements given by

Idi−1 −1 −1 0 −1 −1 Si = TiSiTi = Ti+1SiTi+1 = −1 0 , Si = (Si) . (4.12) permat1 Idℓ−i !

T Lemma 4.2 The elements Si, Si W satisfy the following relations: ∈ GLℓ+1 2 S2 = S = 1 , S S = S S = T T , 1 i ℓ ; i i i i i i i i+1 ≤ ≤ (S S )2 = (S S )2 = (S S )2 = 1 , i j > 1 , (4.13) WeylExt2 i j i j i j | − | (S S )3 = (S S )3 = 1 , S S S = S S S , i j =1 . i j i j i j i j i j | − | Proof : The relations follow from the similar relations for the symmetric group (2.24), (2.23) and (2.25). The latter (braid) relation can be checked using

Si =s ˙iTi = Ti+1s˙i , (4.14) and the braid relation (2.29) in the Tits group. ✷

The action of the involutions θG (4.11) may be written as follows

θB(Si) = S2ℓ+1−i , θC (Si) = S2ℓ−i , 1 i ℓ , ≤ ≤ (4.15) permatAction θ (S ) = S , 1 i<ℓ, θ (S ) = S . D i 2ℓ−i ≤ D ℓ ℓ

Note that the two sets Si and Si of generators define two different embeddings of the { } { } T Weyl group WGLℓ+1 into the Tits group WGLℓ+1 (and thus in the normalizer Nℓ+1) which preliminary draft 11 December 5, 2019 1:46 are interchanged by involutions θG. Let us stress that for example it is possible to find such embedding W GL (C) that the image of W is invariant with respect to GLℓ+1 ⊂ ℓ+1 GLℓ+1 θOℓ+1 . Indeed this fact is obvious in the case of the realization of the orthogonal subgroup Oℓ+1 GLℓ+1(C) based on (3.8) because for Si defined by (4.2) we have Si Oℓ+1. The same fact for⊂ the realization (3.5) follows from the equivalence of the two realizations.∈ However the resulting matrix expressions for the lifts of si are more involved and require the use of complex numbers while in our realization the matrix entries are elements of the set 1, 0, +1 . Thus in the following we will use the pair S , S of lifts related by (4.15). {− } { i} { i}

5 Classical groups and their Weyl groups

For each complex classical Lie group G = GL (C)θG the corresponding involution θ ℓ+1 G ∈ Aut(GLℓ+1) respecting the embedding Hℓ+1 GLℓ+1(C) of the diagonal maximal torus Hℓ+1 C ⊂ naturally acts on the Lie algebra glℓ+1( ), the corresponding root system ΦGLℓ+1 , normalizer

Nℓ+1(Hℓ+1) of Hℓ+1 and the Weyl group WGLℓ+1 = Nℓ+1(Hℓ+1)/Hℓ+1. Both the maximal torus HG of G, normalizer NG(HG) and its quotient WG = NG(HG)/HG can be expressed in terms of the corresponding objects of the underlying linear group GLℓ+1. Let us give explicit expressions for the generators of the group of automorphysms Aut(ΦG) of the root systems and the corresponding Weyl groups in terms of the generators of the Weyl group of the corresponding general linear Lie group.

Proposition 5.1 The following explicit description of the generators of the group Aut(ΦG) of automorphisms of the classical Lie algebra root systems ΦG and the corresponding Weyl groups holds:

For G = O2ℓ+1 one has Aut(Φ(Bℓ)) = W (Φ(Bℓ)), and the simple root generators • B A2 s ℓ W (Φ(B )) can be expressed in terms of s ℓ W (Φ(A )) as follows: i ∈ ℓ i ∈ 2ℓ Bℓ A2ℓ A2ℓ A2ℓ A2ℓ A2ℓ A2ℓ s1 = sℓ sℓ+1sℓ = sℓ+1sℓ sℓ+1 , (5.1) BWeyl0 sBℓ = sA2ℓ sA2ℓ = sA2ℓ sA2ℓ , 1

Cℓ For G = Sp2ℓ one has Aut(Φ(Cℓ)) = W (Φ(Cℓ)), and the simple root generators si • A − ∈ W (Φ(C )) can be expressed in terms of s 2ℓ 1 W (Φ(A )) as follows: ℓ i ∈ 2ℓ−1 sCℓ = sA2ℓ−1 , sCℓ = sA2ℓ−1 sA2ℓ−1 = sA2ℓ−1 sA2ℓ−1 , 1

For G = O2ℓ one has Aut(Φ(Dℓ)) = W (Φ(Dℓ)) ⋊ Out(Φ(Dℓ)). The simple root gen- • Dℓ Z Z erators si W (Φ(Dℓ)) together with the generator R Out(Φ(Dℓ)) = /2 can be ∈ A − ∈ expressed in terms of s 2ℓ 1 W (Φ(A )) as follows: i ∈ 2ℓ−1 Dℓ A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 s1 = sℓ sℓ−1 sℓ+1 sℓ = sℓ sℓ+1 sℓ−1 sℓ , sDℓ = sA2ℓ−1 sA2ℓ−1 = sA2ℓ−1 sA2ℓ−1 , 1

The Weyl group W (Aℓ) allows embedding into the Lie group GLℓ+1 as was discussed in Section 4. Therefore it is now natural to ask for a possibility to lift the presentations (5.1), θG (5.2), (5.3) of Weyl groups WG into the corresponding classical groups G = GLℓ+1(C) via C the pair of lifts of the Weyl group WGLℓ+1 into the general linear group GLℓ+1( ) considered T previously. For the Tits extension WGLℓ+1 the following result is easily checked.

T Proposition 5.2 The explicit presentation for the Tits group WG of the classical Lie group θG G =(GLℓ+1) may be provided by replacement of all si’s by s˙i’s in (5.1), (5.2) and (5.3).

Proof : We give the case by case veriﬁcation in Lemmas 10.1, 10.2 and 10.3 by straightforward computation using standard faithful representations (3.5). ✷ For the Weyl groups we can not expect such simple answer. Indeed we know that although for the groups O2ℓ and O2ℓ+1 the corresponding Weyl groups allow embedding in the corresponding classical group this is not so for Sp2ℓ. So for the case of the Weyl groups we have the following result.

Theorem 5.1 Given a classical group G with its maximal torus HG G, let Si, Si, 1 T ⊂ ≤ i ℓ be the generators (4.2), (4.12) of the Tits group WGLℓ+1 and let Tk, 1 k ℓ +1 be the≤ elements (4.5). Then the following holds: ≤ ≤

In case G = O the generators SB deﬁned by • 2ℓ+1 i B S1 = Sℓ+1SℓSℓ+1 = SℓSℓ+1Sℓ , (5.4) 2torsB0 SB = S S , 1

are θB-invariant and generate a ﬁnite group isomorphic to W (Bℓ)= WO2ℓ+1 . In case G = O the generators SD deﬁned by • 2ℓ i D S1 = SℓSℓ−1Sℓ+1Sℓ , (5.5) 2torsD0 SD = S S , 1

are θD-invariant and generate a ﬁnite group isomorphic to W (Dℓ)= WO2ℓ .

T In case G = Sp the Tits group W is generated by θC -invariant generators • 2ℓ Sp2ℓ C S1 = TℓSℓ , (5.6) SC = S S , 1

6 Description of θG-invariants

The explicit expressions for the generators the Weyl groups/Tits groups of the classical groups presented above implies some general relations between maximal torus normalizers, Weyl groups and Tits groups for general linear groups GLℓ+1(C) and their classical subgroups θG G = (GLℓ+1(C)) . In this Section we prove a set of such relations. First let us note the following property of the involutions θG deﬁning classical Lie groups G.

θG C Lemma 6.1 The centralizer of the ﬁxed point subset Hℓ+1 in GLℓ+1( ) is Hℓ+1.

Proof We shall solve the set of equation for g GL (C) ∈ ℓ+1 gh = hg, h HθG . (6.1) CENTH ∈ ℓ+1 θG ι Due to obvious isomorphism Hℓ+1 = Hℓ+1 we may use the basic autmomorphims ι (3.3) instead of θG in (6.1). We ﬁx Hℓ+1 GLℓ+1 be the diagonal subgroup. Explicitly elements ι ⊂ ι ι g CGLℓ+1 (Hℓ+1) of the centralizer CGLℓ+1 (Hℓ+1) of invariant subtorus Hℓ+1 Hℓ+1 satisfy the∈ conditions ⊂

xigij = gijxj , (6.2) where

g = g , h = diag(x , x ,...,x ) , (6.3) k ijk 1 2 ℓ+1 −1 with the additional condition xℓ+2−i = xi reflecting ι-invariance of h. Since g does not de- pend on xi and there are no universal x-independent relations between entries of h (although they subjected quadratic relations x x = 1) we infer g =0, i = j i.e. g H . ✷ i ℓ+2−i ij 6 ∈ ℓ+1 As we already notice the definition of the Weyl group as a quotient of the normalizer NG(HG) by HG is not equivalent to the definition of the Weyl group via generators and relations (2.23), (2.24) for non-connected classical Lie groups. Precisely the group NG(HG)/HG for all classical Lie groups obtained as fixed point subgroups of the general linear group (we exclude the case of trivial involution) may be identified with the group of automorphims of the corresponding root system

N (H )/H Aut(Φ ) . (6.4) G G G ≃ G This group ﬁts the following exact sequence

1 W Aut(Φ ) Out(Φ ) 1 (6.5) −→ G −→ G −→ G −→ and we have

Aut(B ) W (B ), Aut(C ) W (C ), Aut(D ) W (D ) Z . (6.6) ℓ ≃ ℓ ℓ ≃ ℓ ℓ ≃ ℓ × 2 preliminary draft 14 December 5, 2019 1:46 Let us stress that notation Out(ΦG) is natural for connected simple groups where indeed this group is the factor of all automorphims over inner automorphisms. In the case of non- connected group (see detailed discussion of the group O2ℓ in Section 10.4) the whole group Aut(ΦG) is realized by inner automorphims.

Proposition 6.1 Let θ be the involution in GLℓ+1 deﬁning a classical Lie group G with maximal torus HG and the Weyl group WG. Then the following isomorphisms hold

θG Hℓ+1 = HG, (6.7) P1

θG (Nℓ+1(Hℓ+1)) = NG(HG), (6.8) P2

θG WGLℓ+1 = Aut(ΦG). (6.9) P3 Proof : Any two θ-fixed elements of Hℓ+1 define a pair of mutually commuting elements of G. Thus we have an embedding Hθ H . By Lemma 6.1 any element commuting with Hθ ℓ+1 ⊂ G ℓ+1 shall be in Hℓ+1 i.e. we have embedding HG Hℓ+1 and taking into account that elements of H are θ -invariant we obtain H HθG .⊂ G G G ≃ ℓ+1 θG To establish (6.8) first note that elements g (Nℓ+1(Hℓ+1)) are θ-fixed elements of ∈ θ GL stabilizing H i.e. ghg−1 H for any h H . It is clear that for any h H G ℓ+1 ℓ+1 ∈ ℓ+1 ∈ ℓ+1 ∈ ℓ+1 we have ghg−1 HθG . This defines an embedding (N (H ))θG N (H). In the ∈ ℓ+1 ℓ+1 ℓ+1 ⊂ G following we use the fact that the involution θG on the commutative group Hℓ+1 defines the decomposition Hℓ+1 = H+H− into a product of commutative subgroups, so that for each element h H there exist unique h H such that ∈ ℓ+1 ± ∈ ±

θG θG −1 h = h+h− = h−h+, h+ = h+, h− = h− . (6.10)

Using (6.7) we can make identiﬁcation H H . + ≃ G Now we shall construct an embedding N (H ) N (H )θG i.e. prove that any θ- G G ⊂ ℓ+1 ℓ+1 invariant element g GLℓ+1 stabilizing HG also transforms h Hℓ+1 into some other element ′ ∈ ∈ −1 h Hℓ+1. This is trivial for elements of H+ HG and thus we need to prove that gh−g is ∈ ≃ θ an element of H . The element gh g−1 commutes with all elements in H G H due to ℓ+1 − ℓ+1 ⊂ ℓ+1 the fact that H+ and H− are mutually commute and g stabilizes H+. Moreover for a given −1 −1 g all elements gh−g mutually commute. By Lemma 6.1 this entails that gh−g Hℓ+1. Combining these two embeddings we obtain (6.8). To prove (6.9) let us compare two⊂ groups

θ θ θ Nℓ+1(Hℓ+1) /Hℓ+1 , (Nℓ+1(Hℓ+1)/Hℓ+1) . (6.11) ISO

There is a natural map from the first group to the second one as each element in the l.h.s. coset defines an element of g N (H ) modulo element of Hθ and therefore modulo ∈ ℓ+1 ℓ+1 ℓ+1 Hℓ+1. Note that elements of the second group in (6.11) are defined by the condition

gθ = g h, g N (H ), h H , (6.12) THETA · ∈ ℓ+1 ℓ+1 ∈ ℓ+1 preliminary draft 15 December 5, 2019 1:46 modulo right action of Hℓ+1 on g. Due to involutivity of θG we have

θ g = gθ =(gh)θ = ghhθ, (6.13) and thus

hθ = h−1. (6.14)

˜ ˜ Changing representative g g˜ = gh, h Hℓ+1 for g in the coset Nℓ+1(Hℓ+1)/Hℓ+1 we may also modify (6.12) as follows→ ∈

g˜θ = g hh˜θ =g ˜ hh˜θh˜−1. (6.15) · · Thus one can get rid of h in (6.12) if we manage to solve the equation

h˜θ = hh,˜ (6.16) EQ2 for any h. Using the decomposition

h˜− = h˜+h˜− , (6.17) and taking into account that h H we can easily check that the equation (6.16) can be ∈ − always solved. Hence we conclude that two groups (6.11) are isomorphic and thus the last statement (6.9) follows from the fact that the Weyl groups are given by the quotients

θG θ θ θ WGLℓ+1 =(Nℓ+1(Hℓ+1)/Hℓ+1) = Nℓ+1(Hℓ+1) /Hℓ+1 = NG(HG)/HG = Aut(ΦG). ✷

θG Proposition 6.2 Given a classical complex Lie group G = (GLℓ+1(C)) , the following holds:

T θG T (6.18) TitsG WGLℓ+1 = WG . The explicit presentation for (6.18) may be provided by replacement of all si’s by s˙i’s in (5.1), (5.2) and (5.3).

Proof : To prove the assertion we use the following fact (see [GLO] Proposition 3.1). Let G(R) be the totally split real form of G(C). Then

R T C π0(NG(R)(H( ))) = WG(C)(H( )) . (6.19) TSF

T The connected component of the trivial element of W N R (H(R)) is isomorphic G(R) ⊂ G( ) Rrank(G) to >0 and we have the split exact sequence

rank(G) T 1 R N R (H (R)) W C (H (C)) 1 . (6.20) −→ >0 −→ G( ) G −→ G( ) G −→ preliminary draft 16 December 5, 2019 1:46 Thus we have

θG T C R Rℓ+1 θG (6.21) WGLℓ+1(C)(Hℓ+1( )) = Nℓ+1( )(Hℓ+1)/ >0 , while on the other hand

θ T C R θG Rℓ+1 G WG(C)(HG( )) = (Nℓ+1( )(Hℓ+1)) >0 . (6.22) ISO1 There is an obvious map

θ θ (N (R)(H ))θG Rℓ+1 G N (R)(H )/Rℓ+1 G , (6.23) ISO22 ℓ+1 ℓ+1 >0 −→ ℓ+1 ℓ+1 >0 and we have to show that it is actually isomorphism. The obstruction to the isomorphism (6.23) is given by elements in Nℓ+1(R)(Hℓ+1) invariant with respect to θG only up to multi- plication by an element Λ Rℓ+1 and we shall consider such elements up to the right action ∈ >0 of Hℓ+1. As θG has order two we have the equation

Λ ΛθG =1 . (6.24) EQ1 ·

Now consider another representativeg ˜ := gQ in the same Hℓ+1-coset that

g˜θG = gθG QθG = gΛQθG . (6.25)

Thus we can chose θG-invariant representative if we can solve the equation

Q =ΛQθG . (6.26) and consistency of this equation follows from (6.24). Explicitly we have

q = λ q , Q = diag(q , q , , q ), Λ = diag(λ ,λ , ,λ ) , (6.27) ℓ+2−i i i 1 2 ··· ℓ+1 1 2 ··· ℓ+1 Rℓ+1 where Q, Λ >0 . Direct check shows that such equation is always solvable and thus we have the isomorphism∈

θ θ (N (R)/(H ))θG / Rℓ+1 G N (R)(H )/Rℓ+1 G . (6.28) ISO2 ℓ+1 ℓ+1 >0 ≃ ℓ+1 ℓ+1 >0 This completes the proof of (6.18). Note that the calculations above actually prove that the Z Z Rℓ+1 ﬁrst cohomology group of /2 generated by θG acting in >0 is trivial. T The explicit presentation for the generators of WG is obtained in Lemmas 10.2, 10.5 and 10.9 in the following subsections. ✷

7 Why Sp2ℓ is diﬀerent

According to Theorem 5.1 all classical Lie groups G except of symplectic type allow a lift of the Weyl group WG into the corresponding Lie group G. In other words the exact sequence (2.22) splits, so that the map p has a section. The splitting in these cases allows a simple explanation: the corresponding Weyl group action in hR preserves the standard preliminary draft 17 December 5, 2019 1:46 inner product, and therefore the Weyl group may be identified with a subgroup of O(h), which immediately implies the splitting of the sequence (1.1) in cases of GLℓ+1 and Oℓ+1. In contrast, for symplectic groups the exact sequence (2.22) does not split and as a section of p we encounter a non-trivial extension W T of the Weyl group W = W (C ) Sp2ℓ+2 Sp2ℓ+2 ℓ+1 introduced by Tits. Obviously the peculiarity of the symplectic series of classical Lie groups begs for explanation. In this section we propose an explanation of this phenomena relying on the properties of quaternion numbers H, the unique non-commutative associative normed division algebra over R. Recall that in the case of the classical Lie groups along with the standard approach to the classification of the complex Lie algebras via root data we may use another (Cartan’s) approach based on the analysis of subalgebras of the general linear algebras fixed by certain involutions. Closely related approach is based on the classification of normed division associative algebras. The list of normed division associative algebras A is exhausted by the algebras of real, complex and quaternion numbers; to these algebras we associate three series of general linear groups:

GLℓ+1(R) , GLℓ+1(C) , GLℓ+1(H) . (7.1) NDA The corresponding maximal compact subgroups R Oℓ+1( ) , Uℓ+1 , USpℓ+1 , (7.2) may be deﬁned as stabilizer subgroups of the standard quadratic form over the corresponding division algebra A: n n † 2 (x, x) = x xi = xi , xi A. (7.3) i k k ∈ i=1 i=1 X X Further complexiﬁcation of the compact groups provides a complete list of the classical complex Lie groups:

Oℓ+1(R) R C = Oℓ+1(C) , Uℓ+1 R C = GLℓ+1(C) , ⊗ ⊗ (7.4) USp R C = Sp (C) . ℓ+1 ⊗ 2ℓ+2 Actually the point of view based on matrix groups over non-commutative fields implies some adjustment of standard definitions in the structure theory of Lie groups. An obvious instance is the notion of the maximal torus: for matrix groups over non-commutative fields it is more natural to consider diagonal subgroups of general linear groups over A rather than maximal commutative subgroup. Namely, given a normed division algebra A, let A∗ be its ∗ ℓ+1 multiplicative group of invertible elements; then HA =(A ) will be the diagonal subgroup of GLℓ+1(A). In special case of symplectic groups the underlying normed division algebra A = H is non-commutative and the corresponding diagonal subgroup of GLℓ+1(H),

∗ ∗ HH = H H GLℓ (H) , ×···× ⊂ +1 (7.5) ℓ+1 is a non-commutative Lie group.| The{z modification} of the notion of maximal torus implies the following modification of the definition of the Weyl group. preliminary draft 18 December 5, 2019 1:46 Definition 7.1 Define the Weyl group WGLℓ+1 (A) of GLℓ+1(A) to be the group of inner automorphisms of the diagonal subgroup H GL (A). A ⊂ ℓ+1 Note that while in the commutative case the group of inner automorphisms of the diagonal subgroup Hℓ+1 GLℓ+1 is isomorphic to the quotient group Nℓ+1(Hℓ+1)/Hℓ+1 for non- commutative A this⊂ is not true.

H Lemma 7.1 The group WGLℓ+1 ( ) allows the following description:

H ⋉ ℓ+1 WGLℓ+1 ( ) = Sℓ+1 (SO3) . (7.6) HW

Proof : The group GLℓ+1(H) acts on its diagonal subgroup HH by conjugation, so let us ﬁnd the normalizer subgroup NH = Nℓ+1(H)(HH) which preserves HH. The diagonal group HH is generated by one-parametric subgroups

Id − 0 0 (k) k 1 ∗ HH = 0 d 0 , d H HH , 1 k ℓ +1 . (7.7) 0 0 Idℓ−k ∈ ⊂ ≤ ≤ n o (k) Let us pick a diagonal element D = diag(Id , d , Id ) HH . Then M GL (H) k k−1 k ℓ−k ∈ ∈ ℓ+1 belongs to NH if and only if it satisﬁes the following equations for each 1 k ℓ + 1: ≤ ≤ ′ ′ ′ ′ MD = D M, for some D = diag(d ,...,d ) HH . (7.8) k 1 ℓ+1 ∈ Explicitly using the matrix notation M = g , the above equation reads k ijk g = d′ g , j = k ; g d = d′ g , 1 i, j,k ℓ +1 . (7.9) NormHid ij i ij 6 ik k i ik ≤ ≤

Now for each 1 i ℓ + 1 the former relation in (7.9) implies either gij = 0, j = k, or ′ ≤ ≤ ∀ 6 di = 1. In case gij = 0, j = k we necessarily obtain from the latter relation in (7.9) that ∀ 6 H ′ gik = 0, otherwise M has the i-th zero row and M / GLℓ+1( ). In case di = 1 it follows from the6 latter relation in (7.9) that g d = g , which∈ entails g = 0, since we assume d =1 so ik k ik ik k 6 that Dk = Idℓ+1. Taking into account that gij are subjected to (7.9) for each 1 k ℓ + 1, we deduce6 from the above that there is only one non-zero element in the i-th row≤ of≤M. Since the matrix M is invertible its rows are linearly independent, so for different i the non-zero entries gij have different j’s. This defines the subgroup of monomial matrices M (H) GL (H) consisting of matrices with only one non-zero element in each column ℓ+1 ⊂ ℓ+1 and each row, and yields NH = Mℓ+1(H). Clearly, the quotient group NH/HH is isomorphic to the permutation group:

Mℓ+1(H)/HH = Sℓ+1 , (7.10) so that Deﬁnition 7.1 reads

H ⋉ H ℓ+1 WGLℓ+1 ( ) = Sℓ+1 Int( ) . (7.11) HWeyl

preliminary draft 19 December 5, 2019 1:46 where Int(H) = H∗/R∗ is the group of inner automorphisms of H (note that by Skolem- Noether theorem all automorphisms of H are inner). The norm homomorphism induces the following exact sequence:

Nm 1 / SU / H∗ / R / 1 , 2 >0 (7.12) NormH

Nm(q) = qq¯ , Nm(H) = R>0 .

∗ ∗ Taking into account R = (H) H = R>0 µ2 with µ2 = 1 being a group of roots of 1 in R, we obtain Z ∩ × {± }

Aut(H) = H∗/R∗ = SU(2)/µ SO . (7.13) IntH 2 ≃ 3

Thus Aut(H) is identiﬁed with the adjoint group SO3 of the group SU2 of unite quaternions. This complete the proof of (7.6). ✷

Let AutC(H) Aut(H) be the subgroup of automorphisms preserving the subalgebra C =(R Rı) H⊂. ⊕ ⊂ Lemma 7.2 The following holds:

∗ ∗ ∗ AutC(H)=(C C  )/R , (7.14) AUT ⊔ and the following central extension splits

∗ ∗ 1 C /R AutC(H) Gal(C/R) 1 . (7.15) AUT1 −→ −→ −→ −→

Proof : Consider the tautological faithful two-dimensional representation H Mat2(C), q qˆ: → 7→

ı 0 0 1 0 ı 1ˆ = Id2 , ˆı = , ˆ = , kˆ = . (7.16) 0 −ı −1 0 ı 0 The norm homomorphism (7.12) is given by the matrix determinant:

q = q + ıq + q + kq H , q 2 = qq¯ = q2 + q2 + q2 + q2 , 0 1 2 3 ∈ | | 0 1 2 3 q0 + ıq1 q2 + ıq3 (7.17) 2 qˆ = , q = detq ˆ R>0 . (q2 ıq3) q0 ıq1! | | ∈ − − − Then elements z of the subalgebra C = R Rı H are identiﬁed with the diagonal matrices ⊕ ⊂ z 0 z zˆ = , z 2 = zz¯ = detz ˆ , (7.18) 7−→ 0z ¯ | | while the general quaternion has representation for a, b C ∈ a b q qˆ = , detq ˆ = a 2 + b 2 . (7.19) 7−→ −¯b a¯ | | | | preliminary draft 20 December 5, 2019 1:46 Now a quaternion q belongs to centralizer of subalgebra C H if and only ifq ˆzˆ =z ˆ′ qˆ for any z C, which reads: ⊂ ∈ az bz¯ z′a z′b = . (7.20) −¯bz a¯z¯ −z¯′¯b z¯′a¯ Since z C is arbitrary, either z = z′ and b = 0, or z =z ¯′ and a = 0. Obviously, the case b = 0 corresponds∈ to q C∗ H∗, and in case a = 0 we have q C∗ H∗. Taking into account that q R∗ implies∈ (7.14).⊂ The second statement (7.15) follows∈ ⊂ from the fact that ∈ ˆ acts by complex conjugation:

z−1 =z ¯ , (7.21) Jaction and its image in the quotient group (C∗ C∗ )/R∗ is precisely the generator of Gal(C/R) given by the complex conjugation. ✷ ⊔

Recall that standard Weyl group of the symplectic Lie group Sp2ℓ+2 is given by ⋉ Z Z ℓ+1 WSp2ℓ+2 = W (Cℓ+1) = Sℓ+1 ( /2 ) . (7.22) Here Z/2Z in r.h.s. may be identiﬁed with the Galois group Gal(C/R) and thus allowing via splitting of (7.15) an embedding

W W (H). (7.23) Sp2ℓ+2 ⊂ GLℓ+1 H The quaternionic analog WGLℓ+1 ( ) of the standard Weyl group has obvious obstruction for the embedding into the general Lie group GLℓ+1(H). Indeed this lift implies in particular transition from the adjoint action of quaternions on itself to its left action. On the other hand we have the following standard exact sequence

π 1 / R∗ / H∗ / Aut(H) / 1 , (7.24) SU20 or taking into account the identiﬁcations SU(2) = H∗/Nm(H) (for the notations see (7.12)) and Aut(H)= SO3

π / / / / 1 / µ2 / SU2 / SO3(R) / 1 . (7.25) SU

2 The generator of µ2 above might be identiﬁed with the square  of the quaternionic unity. Indeed, the group Aut(H) SO acts in the subspace R3 of purely imaginary quaternions ≃ 3

xˆ = ıx1 + x2 + kx3 , (7.26) via standard three-dimensional rotations. Then the section of projection π may be iden- tiﬁed with the lift of the orthogonal rotations to the conjugation action of the unit norm quaternions q SU : ∈ 2 Ad :x ˆ qxqˆ −1 . (7.27) q −→ preliminary draft 21 December 5, 2019 1:46 Let is pick a rotation given by the diagonal element g = diag( 1, +1, 1) SO . (7.28) SAMPL − − ∈ 3 Solving the equation gx = qxq−1 for (7.28), that is

ıx1 + x2 kx3 = q ıx1 + x2 + kx3 q¯ , (7.29) −c b − we derive that q =  and thus gˆ =  , gˆ2 = 1 . (7.30) − The fact that the square is equal minus unite element implies that indeed we have a central extension with generator which may be identified with 2. Now taking into account (7.11) we arrive at the following result. H H Proposition 7.1 The quaternionic Weyl group WGLℓ+1 ( ) allows the extension WGLℓ+1 ( ) 1 (µ )ℓ+1 W (H) W (H) 1 . (7.31) QWEYL −→ 2 −→ GLℓ+1 −→ GLℓ+1 −→ f The extended group W (H) allows a natural embedding GLℓ+1 f W (H) GL (H) . (7.32) f GLℓ+1 ⊂ ℓ+1 The Tits group extension fℓ+1 T 1 (µ2) W WSp 1 , (7.33) QTITS −→ −→ Sp2ℓ+2 −→ 2ℓ+2 −→ is then obtained by restriction of (7.31) to the subgroup W W (H). Sp2ℓ+2 ⊂ GLℓ+1 Thus the underlying reason for the appearance of the Tits groups in the case of the symplectic Lie groups may be traced back to the extensions (7.24), (7.25). The non-triviality of this extension is closely related with the basic µ2-extension of the Galois group Gal(C/R) characterizing quaternions. Actually the cohomology class measuring non-triviality of this extension is directly related with the basic invariant characterizing the real central simple division algebras. Indeed such algebras are characterized by the invariant taking values in the 2-torsion of the Brauer group (see e.g. [S], [PR]), which coincides with the whole Brauer group in case of R: Br(R) = H2(Gal(C/R), C∗) H2(Gal(C/R),µ ) µ . (7.34) BRR ≃ 2 ≃ 2 The corresponding cohomology group describes the extensions of the form 1 C∗ A Gal(C/R) 1 , (7.35) AEXT −→ −→ −→ −→ and the case of the semidirect product C ⋊ Gal(C/R) with the natural action of Gal(C/R) via complex conjugation corresponds to the trivial cocycle. The only other non-trivial case corresponds to the algebra A = H of quaternions so that the lift of the generator σ ∈ Gal(C/R) may be identified with the quaternionic unite , 2 = 1. This provides a link between the non-trivial extension (7.25) and the construction of −quaternions via the non- trivial extension (7.35) (for more conceptual explanations of the relation between universal µ2-extension of the orthogonal groups and the second cohomology of the Galois group of the base field see [BD] and references therein). In particular this directly relates the fact that 2 = 1 in the algebra of quaternions with the unavoidable appearance of the Tits extension of the− Weyl group for classical symplectic Lie groups. preliminary draft 22 December 5, 2019 1:46 8 Lie groups closely related with classical Lie groups

In this section we consider several examples of construction of suitable sections of the extensions (1.1) for the Lie groups closely relate with classical Lie groups. Precisely we will treat the cases of unimodular subgroup SL GL of the general linear group, of the groups ℓ+1 ⊂ ℓ+1 Pin2ℓ+1, Pin2ℓ, Spin2ℓ+1 and Spin2ℓ. In all cases we provide explicit construction of sections of (1.1) realized as central extensions of the corresponding Weyl groups. Let us stress that these sections diﬀer from the Tits lifts (which are not central extensions of the corresponding Weyl groups in general). Although the possibility to deﬁne sections via central extensions is obvious for Pin groups (they are central extensions of the corresponding classical orthogonal groups) it is a bit less obvious for unimodular and spinor Lie groups.

8.1 Construction of a section for G = SLℓ+1

We will freely use the notations for root data of type Aℓ defined in the Appendix. Recall that the unimodular subgroup SLℓ+1 GLℓ+1 is defined as a kernel of the determinant map thus fitting the following exact sequence:⊂

ϕ det / / / ∗ / (8.1) DETes 1 / SLℓ+1(C) / GLℓ+1(C) / C / 1 .

The group center of unimodular group is identiﬁed with the cyclic group: (SLℓ+1(C)) = πı̟∨ Z Z/(ℓ + 1)Z generated by ζ = e2 ℓ . The image ϕ(ζ) belongs to the center (GL )= C∗ Z ℓ+1 and is given by:

2πı p∨ ϕ(ζ) = e ℓ+1 ℓ+1 (GL (C)) . (8.2) ∈ Z ℓ+1

In Section 4 we construct a lift of the Weyl group WGLℓ+1 = W (Aℓ) corresponding to the root system Aℓ to the group GLℓ+1. As the Weyl groups for SLℓ+1 and GLℓ+1 coincide to construct a section of (1.1) for SLℓ+1(C) we shall invert the homomorphims ϕ on the subgroup W GL . Formally the inverse of ϕ may be written as follows GLℓ+1 ⊂ ℓ+1 −1 ϕ−1(g) = g det (−1)(det g) , g GL (C) , (8.3) · ∈ ℓ+1 where det(−1) is a multi-valued inverse of the determinant map det in (8.1), which can be deﬁned by

∨ (−1) ∗ (−1) 1 p log(z) ∗ det : C (GL ) , det (z) = e ℓ+1 ℓ+1 , z C . (8.4) −→ Z ℓ+1 ∈ To make this map single-valued we shall choose a branch of the log-function. Note that all generators S of W (C) deﬁned in (4.2) satisfy the relation det S = 1. Thus we pick i GLℓ+1 i − log( 1) = ıπ and introduce the following lift of generators S in SL (C): − i ℓ+1 πı p∨ σ = e ℓ+1 ℓ+1 S , 1 i ℓ . (8.5) LIFTSL i i ≤ ≤ preliminary draft 23 December 5, 2019 1:46 Proposition 8.1 The group W (Aℓ) generated by the elements (8.5) provides a central extension f 1 / (SL ) / W (A ) / W (A ) / 1 , (8.6) Z ℓ+1 ℓ ℓ of the Weyl group W (Aℓ) with the deﬁningf relations σ2 = ζ , for each 1 k ℓ ; k ≤ ≤ σkσj = σjσk , if mkj =2, (8.7) CR1

and σkσjσk = σjσkσj , if mkj =3 ,

πıω∨ where ζ = e2 ℓ is the generator of the center (SL ). The group W˜ (A ) allows an Z ℓ+1 ℓ embedding in NSLℓ+1 (HSLℓ+1 ) compatible with the exact sequence 1 H N (H ) W (A ) 1 , (8.8) −→ SLℓ+1 −→ SLℓ+1 SLℓ+1 −→ ℓ −→ and the embedding (SL ) H . Z ℓ+1 ⊂ SLℓ+1

ıπ ∨ +1 pℓ+1 Proof . The element e ℓ is in the center of GLℓ+1(C) and its square is equal to ζ. The relations (8.7) then follow from the relations (4.2) and (4.3) for the generators of the Weyl group W (Aℓ). ✷

8.2 Construction of a section for SOℓ+1

The problem of lifting the Weyl groups W (Bℓ) and W (Dℓ) into special orthogonal subgroups is similar to that for SL GL . In (5.4), (5.5) we introduce a lift of Weyl groups W (B ) ℓ+1 ⊂ ℓ+1 ℓ and W (Dℓ) to the corresponding orthogonal groups. It is natural to split this construction into two parts depending on the parity of the rank. In case of the odd orthogonal group the following sequence splits,

ϕ det / / / / (8.9) DetOrthogonal 1 / SO2ℓ+1 / O2ℓ+1 / µ2 / 1 , so that O (O ) SO . (8.10) 2ℓ+1 ≃ Z 2ℓ+1 × 2ℓ+1 Here the generator z of the center (O ) = Z/2Z is given by (see Section 10.2 for Z 2ℓ+1 notations):

∨ z = T BT B T B = T Beπı̟1 = Id (O ) , (8.11) CENG 0 1 ··· ℓ 0 − 2ℓ+1 ∈ Z 2ℓ+1 B The Weyl group generators Si (5.4) satisfy det(SB) = 1 , det(SB)=1 , 1

2 Proof : Follows from the fact that z = 1 in SO2ℓ+1. ✷ Note that here we follow the analogous construction for the case of unimodular group SL GL by using (8.11) as a lift of the generator of µ in (8.9). ℓ+1 ⊂ ℓ+1 2 D In the case of the even orthogonal group generators Si (5.5) of W (Dℓ) already belong to SO due to det(SD) = 1, and therefore provide an embedding W (D ) N . 2ℓ i ℓ ⊂ SO2ℓ

8.3 Construction of a section for Pinℓ+1

The group Pin(V ) is a central extension of O(V ), which can be deﬁned as follows. Let V be real vector space supplied with the standard quadratic form

dim(V ) v 2 = v2, v =(v ,...,v ) V. (8.14) QuadraticSpace k k i 1 dim(V ) ∈ i=1 X Let (V ) be the corresponding Cliﬀord algebra, deﬁned as a quotient of the tensor algebra T (VC) as follows:

(V ) = T (V ) v v v 2 1 . (8.15) CliffordAlgebra C · − k k · . The standard Z-grading on the tensor algebra induces a Z/2Z-grading α : (V ) (V ) of C → C the Cliﬀord algebra, thus splitting it into the direct sum:

(V ) = (V )+ (V )− . (8.16) C C ⊕ C Let : (V ) (V ) be the transposition anti-automorphism of the Clifford algebra, defined ⊤ C ⊤ → C by (v1 v2) = v2 v1 for any v1, v2 V (V ). Define the conjugation anti-automorphism x x¯ ·= α(x)⊤ on· (V ). The spinor∈ norm⊂ C is then given by: → C Nm : (V ) R∗ , x Nm(x) = x x¯ , (8.17) C −→ 7−→ so that for all u, v V we have ∈ 1 u v + v u (u, v) = Nm(u) + Nm(v) Nm(u + v) = · · , 2 − 2 (8.18) SpinorNorm Nm(v) = v v = v 2 = (v, v) . − · −k k − Definition 8.1 (see [ABS]) The Clifford group Γ(V ) is the subgroup of the group of invertible elements of (V ) that respects the linear subspace V (V ) i.e. C ⊂ C Γ(V ) = x ( (V ) 0 ) xV α(x)−1 V . (8.19) ∈ C \{ } | ⊆ preliminary draft 25 December 5, 2019 1:46 The action of Γ(V ) on V defined above respects the norm (8.18) and thus allows homomorphism to O(V ) for all u V (V ): ∈ ⊂ C (u, v) u s (v) := u v α(u)−1 = v 2 u , v V. (8.20) ReflectionAction 7−→ u · · − (u,u) ∀ ∈ Moreover, since the orthogonal group is generated by reflections with respect to elements u V , the group Γ(V ) maps onto O(V ). Clearly, R∗ Γ(V ) acts trivially and this leads to ∈ ⊂ the following exact sequence:

π 1 / R∗ / Γ(V ) / O(V ) / 1 . (8.21) CliffordGroup

Restriction to the subgroup of elements with the norm in µ2 = 1 results in taking a ∗ {± } quotient over the subgroup Nm(R ) = R>0, which yields the central extension Pin(V ) of O(V ):

π / / / / 1 / µ2 / Pin(V ) / O(V ) / 1 , (8.22) PinOrthogonalCentralExt where µ = 1 is a subgroup of the center (Pin(V )) of the pinor group Pin(V ). 2 {± } Z Taking into account (8.20) we may define a properly normalized liftu ˆ of a simple reflection s with respect to a vector u V into Pin(V ) as follows u ∈ u 2 uˆ := , uˆ = 1 , Nm(û) = 1 , u V (V ) . (8.23) LIFT Nm(u) − ∀ ∈ ⊂ C | | p Lemma 8.2 Let ǫ1,...,ǫdim(V ) V be an orthonormal basis, let Tk, 1 k dim(V ) be reflections at ǫ {V and let S , 1} ⊂ i (dim(V ) 1) be reflections at simple≤ root≤ ǫ ǫ . k ∈ i ≤ ≤ − i − i+1 Then the following elements of Pin(V ) represent liftings of the reflections S , T O(V ): i k ∈ =ǫ ˆ , ( )2 = 1, 1 k dim(V ) , Tk k Tk ≤ ≤ (8.24) PINGL0 ǫî ǫî+1 2 i = − , ( i) = 1 , 1 i ℓ . S √2 S ≤ ≤

Proof : One shall check that the action of the lifts i and i on V according to (8.20) coincides with the action of T and S . Thus for weT have theS following: i i Ti π( )(v) = v α( )−1 =ǫ ˆ v ( ǫˆ ) Tk Tk · · Tk k · · − k =ǫ ˆ (x ǫˆ + ... + x ˆǫ + ... + x ǫˆ ) ( ǫˆ ) = T v . k · 1 1 k k dim(V ) dim(V ) · − k k Similarly the action of can be veriﬁed. ✷ Si Lemma 8.3 The elements (8.24) satisfy the following relations (compare with (4.9)): = , i = j ; = , i j > 1 , TiTj −Tj Ti 6 SiSj −SjSi | − | = , i j =1 , (8.25) TitsExt SiSjSi SjSiSj | − | = , TiSi −SiTi+1 preliminary draft 26 December 5, 2019 1:46 T and therefore provide a central extension of the Tits group WGL(V ) (and of its subgroup W generated by , 1 i dim(V ) 1) by µ (Pin(V )). GL(V ) Si ≤ ≤ − 2 ⊆ Z Proof : The relations can be easily checked by straightforward computation. For example let us verify the 3-move braid relation: one has ǫˆ ǫˆ ǫˆ ǫˆ +ˆǫ ǫˆ 1 = i i+1 − i i+2 i+1 i+2 − , SiSi+1 2 (8.26) ǫˆ ǫˆ +ˆǫ ˆǫ ǫˆ ǫˆ 1 ( )2 = − i i+1 i i+2 − i+1 i+2 − . SiSi+1 2 3 This implies the Coxeter relation ( i j) = 1 for i j = 1, which is equivalent to the 3-move braid relation due to −1 = S S. ✷ | − | Si Si Now let us construct a lift of the generators of Weyl group WO(V ) into Pin(V ). We use a version of the embedding that is based on the expressions given in Theorem 5.1 for the lift A A of WO(V ) to O(V ). Note that the conjugated generators Si are expressed thorough Si and A A A Ti according to (4.12) and the lift of the generators Si and Ti is provided by Lemma 8.3. We will consider the cases of W (Bℓ) and W (Dℓ) separately.

In case odd orthogonal group we have O(V ) = O2ℓ+1. By (5.4) Weyl group W (Bℓ) can T θB be imbedded into the θB-invariant subgroup (WGL2ℓ+1 ) via SB = S S S = S S S , SB = S S , 1

In case odd orthogonal group we have O(V )= O2ℓ. By (5.5) Weyl group W (Dℓ) can be T θD imbedded into the θD-invariant subgroup (WGL2ℓ ) via SD = S S S S , SD = S S , 1

Proposition 8.2 The elements of pinor group Pin(V ) introduced in (8.28), (8.30) satisfy the following relations for any i, j I: ∈ ( B)2 = 1 , ( B)2 = 1 , 1

Proof : Relations in the first lines of (8.31),f (8.32) follow fromǫ ˆ2 =1 andǫ ˆ ǫˆ = ǫˆ ǫˆ i i · j − j · i for i = j. The other relations may be verified by writing explicitly from (8.24): 6 B ˆǫℓ ǫˆℓ+2 −1 ǫî +ˆǫi+1 = − , i i = , S1 √2 T S Ti √2 (8.34) ǫˆ ǫˆ +ˆǫ ǫˆ +ˆǫ ǫˆ ˆǫ ǫˆ D = ℓ−1 ℓ ℓ−1 ℓ+2 ℓ ℓ+1 − ℓ+1 ℓ+2 . S1 2 In particular, one can verify in straightforward way that ǫˆ ǫˆ +ˆǫ ǫˆ +ˆǫ ˆǫ ǫˆ ǫˆ D D D = ℓ−2 ℓ ℓ ℓ+1 ℓ+1 ℓ+3 − ℓ−2 ℓ+3 = D D D . (8.35) S1 S3 S1 2 −S3 S1 S3 The other 3-move braid relations from the last lines of (8.31), (8.32) follow by (8.25). ✷

8.4 Construction of a section for Spinℓ+1

In this section we describe a lift of the Weyl group WO(V ) into the spinor group Spin(V ) combining the results of previous Sections 8.2 and 8.3 with the constructions of Section 5. The construction will be compatible with the following commutative diagram:

µ2 µ2 .

ϕ det / / / / 1 Spin(V ) Pin(V ) µ2 1 (8.36) SPin

π π ϕ det / / / / 1 / SO(V ) / O(V ) / µ2 / 1

Note that the elements of Spin(V ) Pin(V ) are single out by the condition that they are represented by a product of even⊂ number of elementsu ˆ Pin(V ), or equivalently, ∈ Spin(V ) = Pin(V ) (V )+. Moreover we have epimorphism Spin(V ) SO(V ). As we already manage to construct∩ C inverse image under ϕ : SO(V ) O(V ) of→ the generators of the corresponding Weyl groups we can use (8.23) to lift generator→ s (8.13) and generators D Si of W (Dℓ) into the corresponding spinor groups. We again consider the cases of odd and even orthogonal groups separately. preliminary draft 28 December 5, 2019 1:46 Lemma 8.4 Introduce the following elements B Spin Si ∈ 2ℓ+1 B =z ˆ B , B = B , 1

( B)2 = ( 1)ℓ , ( B)2 = 1 , 1

1 / (Spin ) / W (B ) / W (B ) / 1 . (8.40) Z 2ℓ+1 ℓ ℓ f Proof : By (8.25) one hasz ˆ B = B z,ˆ i I, therefore using (8.31) one ﬁnds out, Si Si ∀ ∈ ( B)2 =z ˆ2( B)2 =z ˆ2 = ( 1)ℓ . (8.41) S1 S1 − ✷ The same argument impliese the remaining relations in (8.39). The case of even orthogonal group is already covered by the Proposition 8.2 as the elements D entering (8.32) are already in Spin . Si 2ℓ

T 9 Adjoint action of the Tits groups WG

In this section we compute the action of the elements SG W T in the corresponding i ∈ G Lie algebras g = Lie(G) for classical groups Lie groups G, including G = GLℓ+1. Given a classical group G, the calculation of adjoint action can be done using the appropriate faithful representation. We provide explicit description of the action using the description of T the groups WG from Sections 4 and 5 above. Note that the resulting formulas easily follow from the explicit expressions for Ad , i I obtained in [GLO]. s˙i ∈ T Proposition 9.1 The adjoint action of the Tits group WG on the Lie algebra g = Lie(G) via homomorphism (2.29) is given by

s˙ e s˙−1 = f , s˙ f s˙−1 = e , (9.1) ADT1 i i i − i i i i − i

−1 −1 s˙i ej s˙i = ej, s˙i fj s˙i = fj, aij =0 , (9.2) ADT11 preliminary draft 29 December 5, 2019 1:46 1 s˙ e s˙−1 = e , [... [e , e ] ...] , i j i a ! i i j | ij| |aij | (9.3) ADT2 ( 1)|aij | | {z } s˙ f s˙−1 = − f , [... [f , f ] ...] , i = j . i j i a ! i i j 6 | ij| |aij |

Let us emphasize that we keep the| ordering{z } I = 1,...,ℓ of the corresponding set { } of vertices of Dynkin diagram for each classical G, and use the explicit realization of the Cartan-Weyl generators φ(e ), φ(f ), i I via the standard faithful representation φ. i i ∈ T Proposition 9.2 (The case Aℓ) The adjoint action of elements Si WGLℓ+1 for i 1,...,ℓ is given by ∈ ∈ { } fj , i = j

 ej , i j > 1 | − | AdSi (ej) =  , (9.4)  ad (e ) , i j = 1  ei j − −

adei (ej) , i j =1  − −  and 

ej , i = j

 fj , i j > 1 | − | AdSi (fj) =  . (9.5)  ad (f ) , i j = 1  − fi j − −

adfi (fj) , i j =1  −   B T Proposition 9.3 (The case Bℓ) The adjoint action of elements Si WSO2ℓ+1 for i 1,...,ℓ is given by ∈ ∈ { } fj , i = j

 ej , i j > 1  | − | Ad B (ej) =  δi,1 , (9.6) Si  ( 1)  − ad−aij (e ) , i j = 1  − a ! ei j − − | ij|

 adei (ej) , i j =1  −  and 

ej , i = j

 fj , i j > 1  | − | Ad B (fj) =  . (9.7) Si  1 −aij  ad (fj) , i j = 1  a ! fi − − | ij|  adf (fj) , i j =1  − i −   preliminary draft  30 December 5, 2019 1:46 Proposition 9.4 (The case C ) The adjoint action of elements SC W T for i 1,...,ℓ ℓ i Sp2ℓ is given by ∈ ∈{ } ( 1)δi,1 f , i = j − j  ej , i j > 1  | − | Ad C (ej) =  , (9.8) Si  ad (e ) , i j = 1  ei j  − − − 1 −aij adei (ej) , i j =1  aij ! −  | | and   ( 1)δi,1 e , i = j − j  fj , i j > 1  | − |  −a Ad C (fj) =  ij . (9.9) Si  ad (fj) , i j = 1  fi − −  |aij | ( 1) −aij − adfi (fj) , i j =1  aij ! −  | |   D T Proposition 9.5 (The case Dℓ) The adjoint action of elements S W for i 1, 2; 3,...,ℓ i ∈ O2ℓ ∈{ } is given by

fj , i = j

 ej , aij =0 , ι(i)= j  −  AdSD (ej) =  ej , aij =0 (9.10) i    , , ι(i) = j ad (e ) , a = 1 , ij  −  and  ej , i = j

 fj , ι(i)= j , ι(i)= j  −  AdSD (fj) =  fj , aij =0 , ι(i) = j . (9.11) i  6   ad (f ) , a = 1 , ij  − −   10 Appendix: General linear and classical Lie groups

10.1 General linear group GLℓ+1(C) C Let glℓ+1( ) = gl(V ) be the Lie algebra induced by endomorphisms φ : V V of the complex vector space V Cℓ+1 via the commutator Lie bracket: → ≃ [φ , φ ] = φ φ φ φ , (10.1) APPA 1 2 1 ◦ 2 − 2 ◦ 1 preliminary draft 31 December 5, 2019 1:46 C where is a composition of linear maps V V . Let h glℓ+1( ) be its maximal commuta- ◦ → ⊂ C tive subalgebra. Choosing a basis ε1,...,εℓ+1 V of vector space V we identify glℓ+1( ) with the Lie algebra of matrices with{ the basis}e ⊂, 1 i, j ℓ + 1 deﬁned by the following ij ≤ ≤ endomorphisms:

e : V V, ε δ ε , 1 k ℓ +1 . (10.2) Matunits ij −→ k 7−→ ik j ≤ ≤ The corresponding Lie brackets is given by

e , e = δ e δ e . (10.3) ij kl jk il − il kj The Cartan subalgebra h gl (C) is identiﬁed with a subalgebra of diagonal matrices. ℓ+1 ⊂ ℓ+1 ∗ ∗ ℓ+1 Fix an orthonormal basis ǫ1,...,ǫℓ+1 h in the Euclidean space h C ; , and identify h and h∗ via the{ standard scalar} ⊂ product , on Cℓ+1. Thus we≃ can considerh i h i the diagonal entries ǫk := ekk as functions on h, so that they provide a (complex) linear coordinate system on h. Then any linear functional λ h∗ has the form ∈

λ = λ1ǫ1 + ... + λℓ+1ǫℓ+1 . (10.4)

The general linear group GLℓ+1(C) is isomorphic to the group of invertible rank ℓ +1 matrices, and maximal torus H GL (C) is identiﬁed with the subgroup of diagonal ℓ+1 ⊂ ℓ+1 matrices. The diagonal entries of t = t (g), g H provide coordinates on the maximal k k ∈ ℓ+1 torus Hℓ+1 and generate the group of (rational) characters

∗ ∗ ∗ ℓ+1 X (H ) = Hom(H , C ) hZ = Z . (10.5) GLℓ+1 ℓ+1 ≃ Namely, any λ h∗ gives rise to a homomorphism ∈ λ ∗ h 2πλ(h) 2πhλ, hi e : Hℓ+1 C , e e = e , h h ; −→ 7−→ ∈ (10.6) WeightGL eλ = tλ1 tλ2 ... tλℓ+1 . 1 2 · · ℓ+1 ∗ ∗ ℓ+1 The elements λ hZ consisting of λ h with integral components (λ1,...,λℓ+1) Z are called the weights;∈ they span the weight∈ lattice: ∈

∗ Λ = Zǫ ... Zǫ hZ . (10.7) GLweights W 1 ⊕ ⊕ ℓ+1 ≃ C The adjoint action of Hℓ+1 in Lie algebra glℓ+1( ) provides the Cartan decomposition of C glℓ+1( ) with respect to Hℓ+1

gl (C) = h gl , ℓ+1 ⊕ ℓ+1 ij α ∈Φ(A ) ijM ℓ (10.8) (gl ) = X gl : ad (X) = α (h)X, h h = Ce , i = j , ℓ+1 ij ∈ ℓ+1 h ij ∀ ∈ ij 6 where the corresponding root system Φ(Aℓ) of type Aℓ reads

∗ Φ(A ) = α = ǫ ǫ , i = j hZ . (10.9) RootdataGL ℓ ij i − j 6 ⊂ preliminary draft 32 December 5, 2019 1:46 The simple root system Π Φ (A ) is given by Aℓ ⊂ + ℓ

ΠAℓ = αi = ǫi ǫi+1 Φ+(Aℓ) , − ⊂ (10.10) ∗ Φ (A ) = α = ǫ ǫ , i

∗ Λ = Zα ... Zα Λ hZ . (10.11) GLroots R 1 ⊕ ⊕ ℓ ⊂ W ≃

∗ Let X∗(Hℓ+1) = Hom(C ,Hℓ+1) be the group of co-characters (i.e. one-parametric subgroups) in the maximal torus:

χ∨ : C∗ H , t diag ta1 ,...,taℓ+1 . (10.12) GLcochars a −→ ℓ+1 7−→ { } ℓ+1 ∗ The group of co-characters X (H ) hZ = Z is dual to X (H ) via ∗ ℓ+1 ≃ ℓ+1 ∨ λ ∨ a1λ1+...+aℓ+1λℓ+1 ha , λi e χa (t) = t = t . (10.13)

The dual basis in hZ is referred to as fundamental co-weights:

p∨ = e + ... + e , p∨, α = δ , 1 k, i ℓ +1 , (10.14) GLcoweights k 11 kk h k ii ki ≤ ≤ that span the co-weight lattice:

Λ∨ = h h : λ(h) Z , λ Λ = Zp∨ ... Zp∨ . (10.15) W ∈ ∈ ∀ ∈ W 1 ⊕ ⊕ ℓ+1 ∨ The lattice ΛW can be identiﬁed with the kernel of the map exp : h Hℓ+1, so that ∨ → Hℓ+1 = h/ΛW . ∗ The determinant map det : GLℓ+1(C) C may be deﬁned as a unique homomorphims such that →

ℓ+1 ℓ+1 P tieii P ti (10.16) det : ei=1 ei=1 . −→ We have the following exact sequence

π det / / / ∗ / (10.17) 1 / SLℓ+1(C) / GLℓ+1(C) / C / 1 , where SLℓ+1(C) is the unimodular subgroup. Its Lie algebra slℓ+1 has the standard faithful representation given by the following homomorphism of associative algebras:

ℓ+1 φ : slℓ+1(C) End(C ) , U −→ (10.18) StandardrepA φ(Xα ) = ei,i , φ(X α ) = ei ,i , φ(hα ) = ei,i ei ,i , i +1 − i − +1 i − +1 +1 where eij are the matrix units (10.2). The Lie algebra slℓ+1 may be identiﬁed with the Lie subalgebra of matrices with zero trace in glℓ+1, so that gl = sl C . (10.19) ℓ+1 ℓ+1 ⊕ preliminary draft 33 December 5, 2019 1:46 The Lie algebras slℓ+1 and glℓ+1 share the Dynkin diagram ΓAℓ with the set of vertices I of size ℓ, the reduced root system Φ(Aℓ) and the Weyl group WAℓ . However, in case of slℓ+1 its rank equals to the rank of the root lattice, contrary to the glℓ+1 case. More precisely, let ε ,...,ε be an orthonormal basis in Cℓ+1. Then the fundamental weights p of gl { 1 ℓ+1} k ℓ+1 are linear forms deﬁned by pk(hαi )= δki (see (10.7), (10.14)):

p = ǫ + ... + ǫ , 1 k ℓ +1 . (10.20) k 1 k ≤ ≤ Deﬁne a collection of vectors in Rℓ+1, ǫ + ... + ǫ ǫAℓ = ǫ 1 ℓ+1 ; 1 k ℓ +1 , (10.21) k k − ℓ +1 ≤ ≤

Aℓ Aℓ which span a codimension one Euclidean subspace, due to ǫ1 + ... + ǫℓ+1 = 0. Then the simple roots and fundamental weights of slℓ+1 can be written as follows: k αAℓ = ǫAℓ ǫAℓ , ̟Aℓ = p p = ǫAℓ + ... + ǫAℓ . (10.22) RootDataA k k − k+1 k k − ℓ +1 ℓ+1 1 k

Aℓ Aℓ ∨ Aℓ −1 Aℓ The Cartan matrix A = aij with aij = αj , (αi ) , and its inverse A = cij take the form: k k h i k k

ℓ ℓ−1 ℓ−2 ... 2 1 2 −1 0 ... 0 . . . ℓ−1 2(ℓ−1) 2(ℓ−2) ... 2·2 2 −1 2 .. .. . −1 1  ℓ−2 2(ℓ−2) 3(ℓ−2) ... 3·2 3  A =  .. ..  , A = . (10.23) ACartanMat 0 . . −1 0 ℓ +1 ......    . .. −1 2 −1   2 2·2 3·2 ... (ℓ−1)2 ℓ−1   0 ... 0 −1 2       1 2 3 ... ℓ−1 ℓ      In these terms one obtains the following expressions for the coroots and coweights:

hAℓ = (αAℓ )∨ = αAℓ , (αAℓ )∨ (̟Aℓ )∨ = aAℓ (̟Aℓ )∨ , i i h j i i j ij j j∈I j∈I X X (10.24) Acoweights (̟Aℓ )∨ = cAℓ (αAℓ )∨ , i I. i ij j ∈ j I X∈

10.2 The case of type Bℓ root system

The Lie algebra so2ℓ+1 can be identiﬁed with the θB-ﬁxed subalgebra: C τ −1 so2ℓ+1 = X gl2ℓ+1( ): X = θB(X) , θB(X) = SBX SB , ∈ − (10.25) Bfaithful S = diag 1, 1,..., 1 GL , B − ∈ 2ℓ+1 where τ is the transposition along the opposite diagonal (3.3). This provides the faithful representation,

φ : so gl (C) , (10.26) StandardrepB 2ℓ+1 −→ 2ℓ+1 preliminary draft 34 December 5, 2019 1:46 ∨ given by the following presentation of the Chevalley-Weyl generators hk = αk , ek, fk, k I C ∈ of so2ℓ+1 in terms of the generators of gl2ℓ+1( ):

φ(e1) = √2(eℓ,ℓ+1 + eℓ+1,ℓ+2) , φ(f1) = √2(eℓ+1,ℓ + eℓ+2,ℓ+1);

φ(ek) = eℓ+1−k,ℓ+2−k + eℓ+k,ℓ+1+k , φ(fk) = eℓ+2−k,ℓ+1−k + eℓ+1+k,ℓ+k , 1

The representation (10.27) implies the following presentation of the type Bℓ root system in terms of the root system of type A2ℓ

αBℓ = αA2ℓ + αA2ℓ , i I. (10.28) BAroots i ℓ+1−i ℓ+i ∈

Explicitly the symmetry of the Dynkin diagram of type A2ℓ is given by ι : i 2ℓ +1 i , i 1, 2,..., 2ℓ , 7−→ − ∈{ } (10.29) αA2ℓ αA2ℓ . i 7−→ 2ℓ+1−i Then (10.28) reads

Bℓ A2ℓ A2ℓ αi = αℓ+1−i + ι αℓ+1−i . (10.30)

2ℓ+1 Introduce the Euclidean space R with a standard orthonormal basis ǫ1,...,ǫ2ℓ+1 , acted on by the involution ι as follows: ι : R2ℓ+1 R2ℓ+1 , ǫ ǫ . (10.31) −→ i 7−→ − 2ℓ+2−i Consider the ι-ﬁxed Euclidean subspace Rℓ R2ℓ+1 spanned by ⊂ ǫBℓ = ǫ + ι ǫ = ǫ ǫ , 1 k ℓ , (10.32) Bfolding k ℓ+1−k ℓ+1−k ℓ+1−k − ℓ+1+k ≤ ≤ then the root data of type Bℓ is given by

Bℓ Bℓ B B ℓ ℓ Bℓ ǫ1 + ... + ǫℓ α1 = ǫ1 ̟ = 1 2 Bℓ Bℓ Bℓ   α = ǫ ǫ , B B B k k k−1  ̟ ℓ = ǫ ℓ + ... + ǫ ℓ ,  −  k k ℓ (10.33) RootDataB  1

αBℓ , (̟Bℓ )∨ = (αBℓ )∨, ̟Bℓ = δ , (10.34) h i j i h i j i ij preliminary draft 35 December 5, 2019 1:46 and have the following form:

Bℓ ∨ Bℓ α1 = 2ǫ1 Bℓ ∨ Bℓ Bℓ ̟i = ǫi + ... + ǫℓ , Bℓ ∨ Bℓ Bℓ  αk = ǫk ǫk−1 ,  − ( i I (10.35)  1

Lemma 10.1 The Weyl group W (Bℓ) is isomorphic to a subgroup in W (A2ℓ) via

Bℓ A2ℓ A2ℓ A2ℓ A2ℓ A2ℓ A2ℓ s1 = sℓ sℓ+1sℓ = sℓ+1sℓ sℓ+1 , (10.37) BWeyl sBℓ = sA2ℓ sA2ℓ = sA2ℓ sA2ℓ , 1

A2ℓ A2ℓ A2ℓ A2ℓ = λ1ǫℓ + ... + λkǫℓ+2−k + λk−1ǫℓ+1−k + ... + λℓǫ1 (10.38) λ ǫA2ℓ ... λ ǫA2ℓ λ ǫA2ℓ ... λ ǫA2ℓ − 1 ℓ+2 − − k ℓ+k − k−1 ℓ+1+k − − ℓ 2ℓ+1 = λ λ, (αA2ℓ )∨ αA2ℓ λ, (αA2ℓ )∨ αA2ℓ − h ℓ+1−k iA2ℓ ℓ+1−k − h ℓ+k iA2ℓ ℓ+k A2ℓ A2ℓ = sℓ+ksℓ+1−k(λ) . Similarly, for k = 1 one has sBℓ (λ) = λ λ, (αBℓ )∨ αBℓ = λ λ, 2ǫBℓ ǫBℓ 1 − h 1 iBℓ 1 − h 1 iBℓ 1 Bℓ Bℓ Bℓ = λ1ǫ1 + λ2ǫ2 + ... + λℓǫℓ − (10.39) = λ ǫA2ℓ + ... + λ ǫA2ℓ λ ǫA2ℓ + λ ǫA2ℓ λ ǫA2ℓ ... λ ǫA2ℓ ℓ 1 2 ℓ−1 − 1 ℓ 1 ℓ+2 − 2 ℓ+3 − − ℓ 2ℓ+1 A2ℓ A2ℓ A2ℓ = sℓ sℓ+1sℓ (λ) , preliminary draft 36 December 5, 2019 1:46 since sA2ℓ sA2ℓ sA2ℓ simply swaps ǫA2ℓ ǫA2ℓ . ✷ ℓ ℓ+1 ℓ ℓ ↔ ℓ+2 The Lie group O2ℓ+1 may be presented as θB-invariant subgroup of GL2ℓ+1:

O = g GL : g = gθB . (10.40) FaithulRepB 2ℓ+1 ∈ 2ℓ+1 T T Let us describe the corresponding Tits group WO2ℓ+1 in terms of generators of WGL2ℓ+1 .

B π J Lemma 10.2 Let s˙ = e 2 i , i I be the Tits generators (2.28) then the following holds: i ∈ T T 1. The Tits group WOℓ+1 is isomorphic to a subgroup of WGL2ℓ+1 via

s˙B =s ˙A2ℓ s˙A2ℓ s˙A2ℓ , s˙B =s ˙A2ℓ s˙A2ℓ , 1

T θB 2. The elements (10.41) belong to the θB-ﬁxed subgroup (WGL2ℓ+1 ) . 3. Presentation (10.41) matches with (10.37) due to

B AdsB h = s . (10.42) ˙i | i

B Proof : (1) Sinces ˙i SO2ℓ+1, i I, one might verify (10.41) using the standard faithful representation φ : SO∈ GL ∈ . For i = 1, we have 2ℓ+1 → 2ℓ+1

Idℓ−1 π B J 0 0 1 A2 A2 A2 φ(s ˙ ) = φ(e 2 1 ) = 0 −1 0 = φ(s ˙ ℓ s˙ ℓ s˙ ℓ ) , (10.43) TitsgenB1 1 1 0 0 ℓ ℓ+1 ℓ Idℓ−1 ! and similarly, for 1

Idℓ−k 0 −1 π 1 0 B J A2ℓ A2ℓ 2 k Id − φ(s ˙k ) = φ(e ) =  2k 3  = φ(s ˙ℓ+1−ks˙ℓ+k) . (10.44) TitsgenB2 0 −1 1 0  Id −   ℓ k    (2) From (4.11) one reads

θ :s ˙A2ℓ s˙A2ℓ , 1 i 2ℓ +1 , (10.45) B i 7−→ 2ℓ+1−i ≤ ≤ and using the Tits relations (2.29) one infers that (10.41) are θB-invariant. (3) Follows by (2.29) and (10.37). ✷ By analogy with the case of general linear group there is another way to lift simple B T T root generators si W (Bℓ) into the Tits group WO2ℓ+1 . Recall that WGL2ℓ+1 contains the following elements: ∈

πıekk Tk := e , 1 k (2ℓ + 1) ; ≤ ≤ (10.46) TitsGL S = T s˙ =s ˙ T , S = T s˙ =s ˙ T , 1 i 2ℓ . i i i i i+1 i i+1 i i i ≤ ≤ preliminary draft 37 December 5, 2019 1:46 T θB Lemma 10.3 The following elements belong to the ﬁxed point subgroup (WGL2ℓ+1 ) :

B B S1 = Sℓ+1SℓSℓ+1 , Sk = Sℓ+1−kSℓ+k , 1

θB(Si) = S2ℓ+1−i , θB(Si) = S2ℓ+1−i , θB(Tk) = T2ℓ+2−k . (10.48)

Using (4.13) this implies that (10.47) are invariant under θB. ✷

Corollary 10.1 The elements (10.41) and (10.47) can be identiﬁed via

SB = T B s˙B , SB = T Bs˙B , 1

Idℓ−k Idℓ−1 0 1 0 0 1 1 0 B B Id − φ(S ) = 0 1 0 , φ(Sk ) =  2k 3  . (10.50) 1 1 0 0 0 −1 Idℓ−1 ! −1 0  Id −   ℓ k    Identifying this with (10.43), (10.44) one deduces (10.49). ✷

Proposition 10.1 The elements SB W T , i I deﬁned in (10.47) satisfy the relations i ∈ O2ℓ+1 ∈ (2.23), (2.24) of type Bℓ:

B 2 B 2 (S1 ) = ... = (Sℓ ) = 1 , B B B B Si Sj = Sj Si , i, j I such that aij = 0 ; ∈ (10.51) BCentExt SBSBSB = SBSBSB , i, j I such that a = 1 , i j i j i j ∈ ij − B B B B B B B B S1 S2 S1 S2 = S2 S1 S2 S1 .

Proof : One might prove (10.51) applying faithful representation (10.40). Alternatively, since the relations (10.51) are exactly the relations (2.24), they can be derived from (2.29) using substitution (10.49). For the ﬁrst line of (10.51) we have:

B 2 B B B B B 2 B 2 πıh1 2πı(eℓℓ−eℓ+2,ℓ+2) (S1 ) = T0 s˙1 T0 s˙1 = (T0 ) (s ˙1 ) = e = e = 1 , (10.52)

B B B B and sinces ˙k Tk = Tk−1s˙k we derive

B 2 B B B B B B B 2 B B 2 (Sk ) = Tk s˙k Tk s˙k = Tk Tk−1(s ˙k ) = (Tk Tk−1) = 1 . (10.53) preliminary draft 38 December 5, 2019 1:46 For i, j I such that a = 0, sinces ˙BT B = T Bs˙B we have ∈ ij i j j i B B B B B B B B B B B B Si Sj = Ti s˙i Tj s˙j = Tj s˙j Ti s˙i = Sj Si , (10.54) B B B B and similarly, sinces ˙1 T2 = T2 s˙1 , we obtain B B B B B B B B B B B B B B 2 B B B B S1 S2 S1 S2 = T0 s˙1 T2 s˙2 T0 s˙1 T2 s˙2 = (T0 T2 ) s˙1 s˙2 s˙1 s˙2 (10.55) B B B B B B B B =s ˙2 s˙1 s˙2 s˙1 = S2 S1 S2 S1 . The 3-move braid relation for i, j = i + 1 we have: B B B B B B B B B B B B B B B Si Si+1Si = Ti s˙i Ti+1s˙i+1Ti s˙i = Ti Ti+1s˙i Ti+1s˙i+1s˙i B B 2 B B B B B 2 B B B B B B B B B = Ti (Ti+1) s˙i s˙i+1s˙i = Ti (Ti+1) s˙i+1s˙i s˙i+1 = Ti Ti+1s˙i+1Ti s˙i s˙i+1 (10.56) B B B B B B B B B = Ti+1s˙i+1Ti Ti+1s˙i s˙i+1 = Si+1Si Si+1 , B B B ✷ since Ti+1s˙i+1 =s ˙i+1Ti .

10.3 The case of type Cℓ root system

The Lie algebra sp2ℓ is identiﬁed with the θC -ﬁxed subalgebra of gl2ℓ: τ −1 sp2ℓ = X gl2ℓ : X = θC (X) gl2ℓ , θC (X) = SC X SC , ∈ ⊆ − (10.57) StandardrepC SC = diag 1, 1,..., 1, 1 , − − where τ is the matrix transposition with respect to the opposite diagonal (3.3). This provides the standard faithful representation: φ : sp gl , (10.58) StandardrepC0 2ℓ −→ 2ℓ given by the following presentation of the Chevalley-Weyl generators h , e , f , i I: i i i ∈ φ(h ) = e e , 1 ℓℓ − ℓ+1,ℓ+1 φ(h )=(e + e ) (e + e ) , k ℓ+1−k,ℓ+1−k ℓ+k−1,ℓ+k−1 − ℓ+2−k,ℓ+2−k ℓ+k,ℓ+k 1

The representation (10.59) yields the following presentation of the type Cℓ root system: αCℓ = 2αA2ℓ−1 , αCℓ = αA2ℓ−1 + αA2ℓ−1 , 1

αCℓ = αA2ℓ−1 + ι αA2ℓ−1 , αCℓ = αA2ℓ−1 + ι αA2ℓ−1 , 1

Cℓ Cℓ α1 = 2ǫ1 Cℓ Cℓ Cℓ ̟i = ǫi + ... + ǫℓ , Cℓ Cℓ Cℓ  αk = ǫk ǫk−1 ,  − ( i I ;  1

Lemma 10.4 The Weyl group W (Cℓ) is isomorphic to a subgroup of W (A2ℓ−1) via sCℓ = sA2ℓ−1 , sCℓ = sA2ℓ−1 sA2ℓ−1 = sA2ℓ−1 sA2ℓ−1 , 1

Cℓ Cℓ Proof : For k = 1 given λ = λ1ǫ1 + ... + λℓǫℓ one has sCℓ (λ) = λ λ, (αCℓ )∨ αCℓ = λ λ, ǫCℓ 2ǫCℓ 1 − h 1 iCℓ 1 − h 1 iCℓ 1 Cℓ Cℓ Cℓ = λ1ǫ1 + λ2ǫ2 + ... + λℓǫℓ − (10.68) = λ ǫA2ℓ−1 + ... + λ ǫA2ℓ λ ǫA2ℓ−1 + λ ǫA2ℓ−1 λ ǫA2ℓ−1 ... λ ǫA2ℓ−1 ℓ 1 2 ℓ−1 − 1 ℓ 1 ℓ+1 − 2 ℓ+2 − − ℓ 2ℓ A2ℓ−1 = sℓ (λ) , preliminary draft 40 December 5, 2019 1:46 For 1

Sp = g GL : g = gθC GL . (10.69) FaithulRepC 2ℓ ∈ 2ℓ ⊂ 2ℓ Let us describe the Tits group W T , which is the extension of the Weyl group W (C ) (10.67), Sp2ℓ ℓ T and identify it with the ﬁxed point subgroup in WGL2ℓ .

C π J Lemma 10.5 Let s˙ = e 2 i , i I be the Tits generators (2.28) then the following holds: i ∈ 1. The Tits group W T is isomorphic to a subgroup of W T via Sp2ℓ GL2ℓ

s˙C =s ˙A2ℓ−1 , s˙C =s ˙A2ℓ−1 s˙A2ℓ−1 , 1

T θC 2. The elements (10.70) belong to the θC -ﬁxed subgroup (WGL2ℓ ) . 3. Presentation (10.70) matches with (10.67) due to

C AdsC h = s , i I. (10.71) ˙i | i ∈

Proof : (1) Sinces ˙C Sp , i I, one might verify (10.70) using the standard faithful i ∈ 2ℓ ∈ representation φ : Sp GL . For i = 1, we have 2ℓ → 2ℓ

Idℓ−1 π A2 −1 C 2 J1 0 −1 ℓ φ(s ˙1 ) = φ(e ) = 1 0 = φ(s ˙ℓ , (10.72) TitsgenC1 Idℓ−1 and similarly, for 1

Idℓ−k 0 −1 π 1 0 C J A2ℓ−1 A2ℓ−1 2 k Id − φ(s ˙k ) = φ(e ) =  2k 4  = φ(s ˙ℓ+1−ks˙ℓ−1+k) . (10.73) TitsgenC2 0 −1 1 0  Idℓ−k      (2) From (4.11) one reads

θ :s ˙A2ℓ−1 s˙A2ℓ−1 , 1 i 2ℓ , (10.74) C i 7−→ 2ℓ−i ≤ ≤ so that using (2.29) the expressions (10.70) are θC -invariant. (3) Follows by (2.29) and (10.67). ✷ By analogy with the case of general linear group there is another way to lift simple root generators sC W (C ) into the Tits group W T . Recall that W T contains the following i ℓ Sp2ℓ GL2ℓ elements: ∈

πıekk Tk := e , 1 k (2ℓ); ≤ ≤ (10.75) TitsGL2L S = T s˙ =s ˙ T , S = T s˙ =s ˙ T , 1 i 2ℓ . i i i i i+1 i i+1 i i i ≤ ≤ preliminary draft 41 December 5, 2019 1:46 T θC Lemma 10.6 The following elements belong to the ﬁxed point subgroup (WGL2ℓ ) :

C C S1 = TℓSℓ = Tℓ+1Sℓ , Sk = Sℓ+1−kSℓ−1+k , 1

Proof : The explicit action of θC on generators (10.75) reads (4.15):

θC (Si) = S2ℓ−i , θC (Si) = S2ℓ−i , θC (Tk) = T2ℓ+1−k . (10.77)

This implies that (10.76) are invariant under θC . ✷

Corollary 10.2 The elements (10.70) and (10.76) can be identiﬁed via

SC =s ˙C , SC = T C s˙C , 1

Idℓ−1 C 0 −1 A2ℓ−1 φ(S1 ) = 1 0 = φ(s ˙ℓ ) , Idℓ−1 Idℓ−k 0 1 (10.79) 1 0 C Id − φ(Sk ) =  2k 4  . 0 −1 −1 0  Id −   ℓ k    Identifying this with (10.72), (10.73) one deduces (10.78). ✷

Proposition 10.2 The elements SC , i I of the Tits group W T satisfy the following i Sp2ℓ relations: ∈

C C 2 C πıh1 C 2 C 2 (S1 ) = T1 = e , (S2 ) = ... = (Sℓ ) = 1 , C C C C Si Sj = Sj Si , i, j I such that aij =0 , ; ∈ (10.80) CCentExt SC SC SC = SC SC SC , i, j I such that a = 1 , i j i j i j ∈ ij − C C 4 C C 4 (S1 S2 ) = (S2 S1 ) = 1 , and they generate the whole Tits group W T . Sp2ℓ

Proof : One might verify (10.80) using the standard fathful representation (10.69). Alterna- tively, similarly to the proof of Proposition 10.1 the relations (10.80) can be deduced from (2.29) and (10.78). ✷

preliminary draft 42 December 5, 2019 1:46 10.4 The case of type Dℓ root system

The Lie algebra so2ℓ is identiﬁed with the a subalgebra of gl2ℓ as follows:

τ −1 so2ℓ = X gl2ℓ : X = θD(X) gl2ℓ , θD(X) = SDX SD , ∈ ⊆ − (10.81) StandardrepD S = diag 1, 1,..., ( 1) ℓ−1; ( 1)ℓ−1, ( 1)ℓ,..., 1 , D − − − − where τ is the matrix transposition with respect to the reverse diagonal (3.3). This provides the standard faithful representation:

φ : so gl , (10.82) StandardrepD0 2ℓ −→ 2ℓ which implies the following presentation of the Chevalley-Weyl generators h , e , f , i I: i i i ∈ φ(h )=(e + e ) (e + e ) , 1 ℓ−1,ℓ−1 ℓℓ − ℓ+1,ℓ+1 ℓ+2,ℓ+2 φ(h )=(e + e ) (e + e ) , k ℓ+1−k,ℓ+1−k ℓ+k−1,ℓ+k−1 − ℓ+2−k,ℓ+2−k ℓ+k,ℓ+k 1

The representation (10.83) yields the following presentation of the type Dℓ root system:

Dℓ A2ℓ−1 A2ℓ−1 A2ℓ−1 α1 = αℓ−1 + 2αℓ + αℓ+1 , (10.84) DAroots αDℓ = αA2ℓ−1 + αA2ℓ−1 , 1

Consider the root system of type A2ℓ−1 endowed with the automorphism ι of its Dynkin diagram:

ι : i 2ℓ i , i 1, 2,..., 2ℓ 1 ; 7−→ − ∈{ − } (10.85) AinvOdd αA2ℓ αA2ℓ−1 . i 7−→ 2ℓ−i so that (10.84) reads

Dℓ A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 α1 = αℓ−1 + αℓ + ι αℓ−1 + αℓ , (10.86) αDℓ = αA2ℓ−1 + ι αA2ℓ−1 , 1

Then the root data of type Dℓ reads

D D D αDℓ = ǫDℓ + ǫDℓ ǫ ℓ + ǫ ℓ + ... + ǫ ℓ 1 2 1 ̟Dℓ = 1 2 ℓ 1 2     ǫDℓ + ǫDℓ + ... + ǫDℓ  D D D  Dℓ 1 2 ℓ  α ℓ = ǫ ℓ ǫ ℓ ,  ̟2 = − (10.89) RootDataD  k k − k−1  2 1

Lemma 10.7 The Weyl group W (Dℓ) is isomorphic to a subgroup of W (A2ℓ−1) via

Dℓ A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 s1 = sℓ sℓ−1 sℓ+1 sℓ = sℓ sℓ+1 sℓ−1 sℓ , (10.91) DWeyl sDℓ = sA2ℓ−1 sA2ℓ−1 = sA2ℓ−1 sA2ℓ−1 , 1

sDℓ (λ) = λ λ, (αDℓ )∨ αDℓ = λ λ, ǫDℓ + ǫDℓ (ǫDℓ + ǫDℓ ) 1 − h 1 iDℓ 1 − h 1 2 iDℓ 1 2 = λ ǫDℓ λ ǫDℓ + λ ǫDℓ + ... + λ ǫDℓ − 2 1 − 1 2 3 3 ℓ ℓ = λ ǫA2ℓ−1 + ... + λ ǫA2ℓ λ ǫA2ℓ−1 λ ǫA2ℓ−1 (10.92) ℓ 1 3 ℓ−2 − 1 ℓ−1 − 2 ℓ + λ ǫA2ℓ−1 + λ ǫA2ℓ−1 λ ǫA2ℓ−1 ... λ ǫA2ℓ−1 2 ℓ+1 1 ℓ+2 − 3 ℓ+3 − − ℓ 2ℓ A2ℓ−1 A2ℓ−1 A2ℓ−1 A2ℓ−1 = sℓ sℓ−1 sℓ+1 sℓ (λ) , preliminary draft 44 December 5, 2019 1:46 since sA2ℓ−1 sA2ℓ−1 sA2ℓ−1 sA2ℓ−1 simply swaps ǫA2ℓ−1 ǫA2ℓ−1 and ǫA2ℓ−1 ǫA2ℓ−1 . ✷ ℓ ℓ−1 ℓ+1 ℓ ℓ ↔ ℓ+2 ℓ−1 ↔ ℓ+1 The orthogonal group O2ℓ may be identiﬁed with the ﬁxed point subgroup of general linear group:

O = g GL : g = gθD GL . (10.93) FaithulRepD 2ℓ ∈ 2ℓ ⊂ 2ℓ The group O2ℓ is not simply connected and there is the following exact sequence:

det / / / / 1 / SO ℓ / O ℓ / 1 / 1 , 2 2 i {± } (10.94) SOeven z with the non-trivial action of 1 on SO . {± } 2ℓ Lemma 10.8 The even orthogonal group allows for the following decomposition:

Idℓ−1 O = SO S SO , S = 0 0 1 , 2ℓ 2ℓ ⊔ ℓ · 2ℓ ℓ 0 1 0 Idℓ−1 (10.95) Ad : αDℓ αDℓ . Sℓ 1 ←→ 2 so that AdSℓ represents the symmetry of Dynkin diagram of type Dℓ. Moreover, O2ℓ has a structure of a semidirect product: O2ℓ = SO2ℓ ⋊ Out(SO2ℓ).

Proof : we have

θD θC (HGL2ℓ ) = HO2ℓ = HSp2ℓ = (HGL2ℓ ) , (10.96) hence H = H . It is convenient to pick z( 1) = M O as a representative of section O2ℓ SO2ℓ − ℓ ∈ 2ℓ z : 1 O , since its represents the outer automorphism of SO . ✷ {± }→ 2ℓ 2ℓ

D π J Lemma 10.9 Let s˙ = e 2 i , i I be the Tits generators (2.28) then the following holds: i ∈ T T 1. The Tits group WSO2ℓ is isomorphic to a subgroup of WGL2ℓ via

D A2ℓ−1 A2ℓ−1 A2ℓ−1 −1 A2ℓ−1 −1 s˙1 =s ˙ℓ−1 s˙ℓ (s ˙ℓ+1 ) (s ˙ℓ−1 ) , (10.97) TitsgenD s˙D =s ˙A2ℓ−1 s˙A2ℓ−1 , 1

T θD 2. The elements (10.97) belong to the θD-ﬁxed subgroup (WGL2ℓ ) . 3. Presentation (10.97) matches with (10.91) due to

D AdsD h = s , i I. (10.98) ˙i | i ∈

preliminary draft 45 December 5, 2019 1:46 T 4. The Tits group WO2ℓ ﬁts into the following exact sequence:

/ T / T / / 1 WSO2ℓ WO2ℓ Out(Φ(Dℓ)) 1 , (10.99) DNorm

T T ⋊ that actually splits, so that WO2ℓ = WSO2ℓ Out(Φ(Dℓ)).

Proof : (1) Sinces ˙D SO , i I, one might verify (10.97) using the standard faithful i ∈ 2ℓ ∈ representation φ : SO GL . For i = 1, we have 2ℓ → 2ℓ Idℓ−2 0 0 −1 0 π A2 −1 D 2 J1 0 0 0 −1 ℓ φ(s ˙1 ) = φ(e ) = 100 0 = φ(s ˙ℓ ) , (10.100) TitsgenD1  010 0  Idℓ−2   and similarly, for 1

πıekk Tk := e , 1 k 2ℓ ; ≤ ≤ (10.103) TitsGL2LL S = T s˙ =s ˙ T , S = T s˙ =s ˙ T , 1 i 2ℓ . i i i i i+1 i i+1 i i i ≤ ≤

T θD Lemma 10.10 The following elements belong to the ﬁxed point subgroup (WGL2ℓ ) : D S1 = SℓSℓ−1Sℓ+1Sℓ , SD = S S , 1

Proof : The explicit action of θC on generators (10.103) reads (4.15):

θD(Si) = S2ℓ−i , θD(Si) = S2ℓ−i , θD(Tk) = T2ℓ+1−k . (10.105)

This implies that (10.104) are invariant under θD. ✷ preliminary draft 46 December 5, 2019 1:46 Corollary 10.3 The elements (10.97) and (10.104) can be identiﬁed via

SD = T Ds˙D , SD = T Ds˙D , 1

Idℓ−2 0010 D 0 0 0 −1 φ(S1 ) = 1000 ,  0 −1 0 0  Idℓ−2   Idℓ−k (10.107) 0 1 1 0 D Id − φ(Sk ) =  2k 4  . 0 −1 −1 0  Id −   ℓ k    Identifying this with (10.72), (10.73) one deduces (10.78). ✷

Proposition 10.3 The elements SD, i I from (10.104) satisfy the relations (2.23), (2.24) i ∈ of type Dℓ:

D 2 D 2 D 2 (S1 ) = (S2 ) = ... = (Sℓ ) = 1 , SDSD = SDSD , i, j I such that a =0 , ; (10.108) DCentExt i j j i ∈ ij SDSDSD = SDSDSD , i, j I such that a = 1 . i j i j i j ∈ ij − Proof : One might verify (10.108) using the standard faithful representation (10.93). Al- ternatively, similarly to our proof of Proposition 10.1 the relations (10.108) can be deduced from (2.29) and (10.106). ✷

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A.A.G. Laboratory for Quantum Field Theory and Information, Institute for Information Transmission Problems, RAS, 127994, Moscow, Russia; E-mail address: [email protected] D.R.L. Laboratory for Quantum Field Theory and Information, Institute for Information Transmission Problems, RAS, 127994, Moscow, Russia; Moscow Center for Continuous Mathematical Education, 119002, Bol. Vlasyevsky per. 11, Moscow, Russia; E-mail address: [email protected] S.V.O. School of Mathematical Sciences, University of Nottingham , University Park, NG7 2RD, Nottingham, United Kingdom; Institute for Theoretical and Experimental Physics, 117259, Moscow, Russia; E-mail address: [email protected]

preliminary draft 48 December 5, 2019 1:46