Centralisers, complex reflection groups and actions in the Weyl group E6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_6$$\end{document}

The compact, connected Lie group E6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_6$$\end{document} admits two forms: simply connected and adjoint type. As we previously established, the Baum–Connes isomorphism relates the two Langlands dual forms, giving a duality between the equivariant K-theory of the Weyl group acting on the corresponding maximal tori. Our study of the An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_n$$\end{document} case showed that this duality persists at the level of homotopy, not just homology. In this paper we compute the extended quotients of maximal tori for the two forms of E6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_6$$\end{document}, showing that the homotopy equivalences of sectors established in the An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_n$$\end{document} case also exist here, leading to a conjecture that the homotopy equivalences always exist for Langlands dual pairs. In computing these sectors we show that centralisers in the E6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_6$$\end{document} Weyl group decompose as direct products of reflection groups, generalising Springer’s results for regular elements, and we develop a pairing between the component groups of fixed sets generalising Reeder’s results. As a further application we compute the K-theory of the reduced Iwahori-spherical C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra of the p-adic group E6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_6$$\end{document}, which may be of adjoint type or simply connected.


Introduction
The structure of element centralisers in Weyl groups plays a key role in our understanding of the representation theory of reductive groups of p-adic type.For the narrow class of regular elements Springer showed that the centralisers have the structure of a complex reflection group inherited directly from the action on the corresponding regular eigenspace [12].Building on the work of Springer, Brewer [3] classified the irreducible, rank n complex reflection groups in a rank 2n real reflection group, exhibiting these as subgroups of centralisers in the automorphism group of the root graph.In the case of elliptic elements Reeder studied the centralisers by constructing a symplectic form on the coinvariant representation [7].
As remarked by Reeder, there is no general theory for the structure of centralisers in Weyl groups, though the classical cases A n , B n , C n , D n are well understood.In this paper we determine the structure for all the centralisers in the Weyl group of type E 6 .We provide a description in terms of generators and relations and give a classification of each centraliser as a product of reflection groups.In the case of the seven conjugacy classes of regular elements this recovers Springer's identification of the centralisers as complex reflection groups, though our approach is somewhat different to Springer.
Theorem 1.Let W be the Weyl group of type E 6 .For each w ∈ W the centraliser Z W (w) decomposes as a product of reflection groups.
For all the elements of order 1 or 2 in the Weyl group, the centralisers are real reflection groups.In all but one of the remaining cases they are truly complex reflection groups.The one exception is the centraliser of the element of type A 2 .This is the product of the cyclic group C 3 with the group (S 3 C 2 ): the latter of which is not a complex reflection group.This factor however is a reflection group over the finite field F 3 .The structure as reflection groups is outlined in Figure 3.
In the case of an elliptic element w in a Weyl group W , Reeder studied the centraliser via its actions on the (finite) group of coinvariants Γ/(I − w)Γ where Γ is the root lattice, see [7].Reeder defined a pairing on Γ/(I − w)Γ, for an elliptic element w, and showed that for Γ a self-dual lattice the pairing is non-degenerate.Dual to the action on coinvariants one may consider the action of W on the invariant part of the Pontryagin dual of Γ, which is the maximal torus T ∨ in the Langlands dual of the simply connected form of the Lie group.For non-elliptic elements, the coinvariants and invariants become infinite, leading us to study the finite component groups of the fixed sets in maximal tori T and T ∨ of Langlands dual groups.The centraliser of an element w ∈ W can be represented by automorphisms of these two finite groups, and we show that these groups and actions are dual.
We will establish this duality in a more general context: let W be a finite group acting orthogonally on a Euclidean vector space t, preserving a lattice Γ in t.Let Γ ∨ denote the dual lattice in t * and let T = t/Γ, T ∨ = t * /Γ ∨ .The fixed sets T w and (T ∨ ) w each consist of a product of a sub-torus with a finite subgroup: T w ∼ = T w 1 × F w and (T ∨ ) w ∼ = (T ∨ ) w 1 × F ∨ w where T w 1 and (T ∨ ) w 1 denote the identity components of the w-fixed sets in T and T ∨ respectively.(In the case where W is a Weyl group, Γ the root lattice and w ∈ W an elliptic element, the group F ∨ w is the fixed set for the action of w on T ∨ and the Pontryagin dual of F ∨ w is identified with Reeder's group of coinvariants.)We will construct a non-degenerate pairing between F w , F ∨ w for all elements of the group W , allowing us to prove: Theorem 2. Let W be a finite group acting orthogonally on a Euclidean vector space t, preserving a lattice Γ in t.Then the dual component group F ∨ w is (canonically) isomorphic to the Pontryagin dual of F w .Therefore F ∨ w is (non-canonically) isomorphic to F w .Theorem 3. The actions of Z W (w) on F w and F ∨ w are dual.Moreover, by exploiting the relationship between invariants and coinvariants we establish the following formula for the cardinality of the component group.Theorem 4. Let w ∈ W with w = I and let r be the rank of I − w.Let g be the greatest common divisor of the r × r-minors of the matrix of I − w expressed in coordinates with respect to a basis for the dual lattice Γ ∨ .Then The above results allow us to determine the fixed sets and to show that T w and (T ∨ ) w agree: Theorem 5.For w ∈ W the fixed sets T w and (T ∨ ) w are non-canonically isomorphic as topological groups.In the special case of Weyl groups acting on maximal tori, this gives an isomorphism between the w-fixed sets for the Langlands dual forms.
For a Weyl group W and maximal torus T = t/Γ, the extended affine Weyl group is defined to be Γ W .The W -equivariant K-theory of the maximal torus T ∨ in the Langlands dual is identified with the K-theory of C * r (Γ W ), while the W -equivariant K-homology of T is identified with K Γ W * (t) (see [5]).Hence the Baum-Connes isomorphism for Γ W corresponds to a pairing between the equivariant K-theory groups K * W (T ) and K * W (T ∨ ) which we showed in [5] is given by a Poincaré duality.In particular the pairing makes these two groups isomorphic up to torsion.
Returning to the case of E 6 , our description of the centralisers and the fixed sets allows us to compute the action of the Weyl group W on the inertia space (cf.Theorem 8) associated to the action of W on the maximal torus, thereby obtaining a description of the extended quotient.Using the equivariant Chern character, we compute (up to torsion) the W -equivariant K-theory for the maximal tori of both the simply connected and adjoint-type Lie groups of type E 6 , exhibiting the isomorphism between these Ktheory groups.
The isomorphism between these K-theory groups corresponds to a cohomological relation between the corresponding extended quotients for the actions of the Weyl group W on the maximal tori.We refine this in the case of E 6 by showing that the isomorphism in cohomology arises from a homotopy equivalence at the level of sectors.Theorem 6.Let W be the Weyl group of type E 6 and let T and T ∨ denote the corresponding (real) maximal tori.For each w ∈ W there is a homotopy equivalence T w /Z W (w) ∼ (T ∨ ) w /Z W (w).This stratification phenomenon was introduced and studied in the case of A n in [6], and we conjecture that this demonstrates a general principle for any compact connected semisimple Lie group: Conjecture (Comparison of sectors).For any compact connected semisimple Lie group, the homeomorphisms between T w , (T ∨ ) w provided by Theorem 5 descend to homotopy equivalences In Section 5, we provide two main applications.The first is the computation of the K-theory for the group C * -algebras of the two extended affine Weyl groups of type E 6 showing that these agree.The second is a geometric description of the set of tempered representations in the Iwahori-spherical block of the p-adic adjoint group E 6 .For each of these applications, full use is made of our results in Table 2.
The paper is organised as follows: 1. Component groups of fixed sets 3. Fixed sets and their quotients: the simply connected type 20 4. Fixed sets and their quotients: the group of adjoint type 32 5. Applications 37 6.Power relations between conjugacy classes and reflection structures of centralisers 39

Acknowledgement
The authors would like to thank the referee for their comments and the community of developers and mathematicians who write and maintain the GAP computer algebra system.While none of the arguments given here rely on computer calculations, and the conjugacy class representatives were built carefully by hand to simplify the construction of centralisers and their actions, we found the Lie Algebra data package provided by Willem Adriaan de Graaf and Thomas Breuer very helpful in our initial investigations.

Component groups of fixed sets
In this section we will consider the general case of a finite group W acting orthogonally on a Euclidean vector space t of dimension n, preserving a lattice Γ.The action thus descends to the torus T = t/Γ.Let w be an element of W and T w the fixed set of the action of w on T .Since the action is by automorphisms of the group T , in each case the fixed set is a (closed) subgroup of T and is therefore isomorphic to the direct product of a finite abelian group with the identity component of T w , which we denote T w 1 .We note that T w 1 is the image under the exponential map of the 1-eigenspace of w acting on the Lie algebra t.Let F w denote the group of components of the w-fixed set, that is We use the inner product to identify t with t * .Note that this identifies the W -action on t with the dual action on t * , preserving the dual lattice This induces an action of W on the dual torus T ∨ = t/Γ ∨ .Let F ∨ w denote the component group for the action on the dual torus: Let W be a finite group acting orthogonally on a Euclidean vector space t, preserving a lattice Γ in t.Then the dual component group F ∨ w is (canonically) isomorphic to the Pontryagin dual of F w .Therefore F ∨ w is (non-canonically) isomorphic to F w .
To prove the theorem we will construct a T 1 -valued pairing between the Pontryagin duals F w and F ∨ w thus establishing an isomorphism The Pontryagin dual F w is canonically isomorphic to the set of characters on T w which are trivial on the subgroup T w 1 .The group of characters on T is identified with the lattice Γ ∨ via χ x (y + Γ) := e 2πi x,y .
The characters on T w are given by the quotient: Hence the set of characters on T w vanishing on T w 1 is given by {x ∈ Γ ∨ : χ x ≡ 1 on T w 1 }/((I − w)Γ ∨ ).Note that χ x ≡ 1 on T w 1 if and only if x is orthogonal to the fixed set t w , so we obtain the identification: Likewise for the dual F ∨ w we have We introduce the notation Following Reeder, let m denote the minimal polynomial of w and let ṁ Reeder restricts to the elliptic case, i.e.where m(1) = 0.In the non-elliptic case we will define m(t) = ṁ(t) − ṁ(1) t − 1 and in general we set Definition 1.1.For x ∈ Γ ∨ and y ∈ Γ we define a twisted pairing x, y w := x, m(w)y .
We will consider this integer valued pairing modulo µ where Note that µ is always non-zero since m is the minimal polynomial, and w is a normal operator.
The following Lemma shows that m(w) provides an inverse in the elliptic case, and a partial inverse in general, to I − w.Proof.Clearly m(w)(I − w) = 0 on the fixed set t w which is the orthogonal complement of x w , so it suffices to show that m(w)(I − w)y = µy for y ∈ x w .
In the elliptic case the result is immediate from the fact that ṁ(t)(1 − t) = µ − m(t) and m(w) = 0.
Lemma 1.3.The twisted pairing −, − w descends to a well-defined pairing Hence the pairing is well defined on the quotients.
We note that since x w (Γ) contains (I − w)Γ it spans the space x w = (I − w)t.In particular x w (Γ) is a lattice in the space x w , and similarly for x w (Γ ∨ ).
We consider the lattices (x w (Γ) Proof.If x ∈ Γ ∨ then x, y ∈ Z for all y ∈ x w (Γ).Now Clearly this map has kernel t w (Γ ∨ ).It remains to check surjectivity.Since x w (Γ) is a lattice in x w and is the intersection of x w with Γ it follows that x w is complemented in Γ. Picking a complement for x w (Γ), let π : Γ → x w (Γ) denote the retraction obtained by killing the complement.
Given any x ∈ (x w (Γ)) ∨ consider the homomorphism from Γ to Z defined by y → x, π(y) .
Since there is a perfect pairing between Γ and Γ ∨ , there exists x ∈ Γ ∨ such that x, π(y) = x , y .Now for all y ∈ x w (Γ) we have Thus x = p xw x which is in the image.
Exchanging the roles of Γ and Γ ∨ gives the dual case.
Proposition 1.5.The pairing . is left and right non-degenerate.
Proof.Let x ∈ x w (Γ ∨ ) and suppose that x, y w ∼ = 0 mod µ for all y ∈ x w (Γ), that is x, 1  µ m(w)y ∈ Z for all y in x w (Γ).Hence 1 µ m(w) * x pairs integrally with all y in x w (Γ), and lies in the space x w , so Thus the pairing is non-degenerate on the left.Now let y ∈ x w (Γ) and suppose that x, y w ∼ = 0 mod µ for all x ∈ x w (Γ ∨ ), that is and as m(w) is injective on x w we have y ∈ (I − w)Γ.Hence the pairing is also nondegenerate on the right.

CENTRALISERS, COMPLEX REFLECTION GROUPS AND ACTIONS IN THE WEYL GROUP E 6 8
Proof of Theorem 2. As noted above it suffices to prove that F w is the Pontryagin dual of Fixing an element of F w this gives a character on F ∨ w , hence we have a homomorphism from F w to the Pontryagin dual of F ∨ w .Left non-degeneracy of the pairing −, − w implies injectivity of this homomorphism.
Likewise fixing an element of F ∨ w gives a character on F w , hence we have a homomorphism from F ∨ w to the Pontryagin dual of F w .Right non-degeneracy again implies injectivity.
Combining these two maps we deduce that Since the cardinalities are equal the injections must be isomorphisms.
The following result gives a tool for computing the cardinality of these groups, which is relevant in the computation of K-theory.It may be useful to bear in mind the following example.
Example 1.6.The conjugacy class representative s 0 s 1 s 5 s 3 of type A 4  1 is represented by the matrix with respect to the lattice Γ and its transpose with respect to the lattice Γ ∨ .The rank of I − M is 4 and the greatest common divisors of the 4 × 4 minors is easily seen to be 4 for both I − M and I − M T .As we will establish in the next theorem, this computes the order of the component group of the corresponding fixed set.
This example is simplified by the fact that the operator is self adjoint, ensuring that the dual matrices over Γ and Γ ∨ are transpose.In general, the matrices with respect to the dual bases are related by the formula (I − M ) → (I − M −1 ) T , however the argument given in Theorem 5 below shows that the gcd of the corresponding minors is still the same in these two matrices.Theorem 4. Let w ∈ W with w = I and let r be the rank of I − w.Let g be the greatest common divisor of the r × r-minors of the matrix of I − w expressed in coordinates with respect to a basis for the dual lattice Γ ∨ .Then Let v 1 , . . ., v r be a basis for (I − w)Γ ∨ , and let w 1 . . ., w r be a basis for x w (Γ ∨ ).
Expressing these vectors in coordinates with respect to a basis for Γ ∨ , each element of the basis {v i } lies in the integer column span of I − w, so the greatest common divisor of the r × r-minors of the matrix (v 1 | . . .|v r ) must be divisible by g.But conversely each column of I − w can be written in terms of the basis, hence we deduce that g equals the greatest common divisor of the r × r-minors of the matrix (v 1 | . . .|v r ).
The subgroup x w (Γ ∨ ) = x w ∩Γ ∨ has a complement Λ in Γ ∨ and the index i of (I −w)Γ ∨ in the lattice x w (Γ ∨ ) equals the index of (I − w)Γ ∨ ⊕ Λ in Γ ∨ .Let x 1 . . ., x n−r be a basis of Λ, so w 1 . . ., w r , x 1 , . . .x n−r gives a basis for Γ ∨ and hence the corresponding n × n matrix has determinant 1.
The elements v 1 , . . ., v r , x 1 , . . ., x n−r form a basis for (I − w)Γ ∨ ⊕ Λ which has index i in Γ ∨ , hence the corresponding matrix has determinant i.But the determinant can be built from the minors of (v 1 | . . .|v r ), so must be divisible by g.That is g|i.
Now consider exterior products: The determinants are given by As v 1 , . . ., v r and w 1 , . . ., w r span the same r-dimensional subspace of t * it follows that and from the above equation we see that the coefficient is i: for the exterior algebra of Γ ∨ are precisely the r × r minors.But the above equation tells us that as Therefore i = g as claimed.
Remark 1.7.We note that since |F w | = |F ∨ w | by Theorem 2 we may use the greatest common divisors of minors of the matrix of I − w with respect to a basis for either the lattice Γ or the lattice Γ ∨ to compute the cardinality of the groups F w , F ∨ w .As a consequence of Theorem 2 we have the following: Theorem 5.For w ∈ W the fixed sets T w and (T ∨ ) w are non-canonically isomorphic as topological groups.In the special case of Weyl groups acting on maximal tori, this gives an isomorphism between the w-fixed sets for the Langlands dual forms.
Proof.The fixed sets are isomorphic to T w 1 × F w , (T ∨ 1 ) w × F ∨ w and we know that the component groups are isomorphic by 2 so we only need to show that the identity components are isomorphic.This follows from the fact that these are precisely the image of the fixed sets in t ∼ = t * under the respective quotient maps.
We now turn to the actions of the centraliser Z W (w) on the groups F w and F ∨ w , induced by the actions of Z W (w) on the w-fixed sets T w and (T ∨ ) w .Although the groups F w and F ∨ w are isomorphic, there is in general no Z W (w)-equivariant isomorphism between the component groups.Instead we will show that the actions are dual.Nonetheless even though these dual actions can be very different we will see (cf.Section 4) that the numbers of orbits of the centraliser on F w and F ∨ w are the same.This is an instance of our conjecture and can be seen clearly in the dual actions of G 25 on F 3 3 in case (21) in Section 3.
Theorem 3. The actions of Z W (w) on F w and F ∨ w are dual.
Proof.For s ∈ F w , t ∈ F ∨ w ∼ = F w denote the corresponding pairing by s, t w .We must show: By construction these isomorphisms are equivariant for the action of Z W (w).
The pairing of F w and F ∨ w is defined by and for g ∈ Z W (w) we have χ(gx + (I − w)Γ ∨ , gy + (I − w)Γ) := e 2πi gx,gy w .Now note that gx, gy w = gx, m(w)gy = gx, gm(w)y since g is in the centraliser of w, so as W acts isometrically on t we have gx, gy w = x, y w .Now for s ∈ F w , t ∈ F ∨ w the pairing of gs with gt is defined to be the pairing of gs with the image of gt under the composition The first isomorphism is tautologically equivariant while the second is equivariant by the above calculation.Letting ψ denote the image of t under this composition we have gs, gt w = gψ, gs = ψ, s = s, t w .Bearing in mind the example of E 6 we now suppose that Γ ∨ refines Γ with quotient Z = Γ ∨ /Γ cyclic of prime order.Moreover we assume that the induced action of W on the quotient Z is trivial.It follows that Z lies in each of the fixed groups T w .
The w-fixed set in the dual torus T ∨ = t/Γ ∨ = T /Z contains the image of T w under the quotient map T → T ∨ .In particular (T ∨ ) w 1 is the image of T w 1 under the quotient map, since the identity component of the fixed set is precisely the image of the fixed set t w under the respective exponential map.Hence the quotient map induces a map from We have two cases: either Z lies in T w 1 or Z ∩ T w 1 is trivial.In the former case the map from F w to F ∨ w is injective.It is thus an isomorphism since the groups have the same cardinality by Theorem 2. Hence in this case the fixed set (T ∨ ) w is precisely the image of the fixed set T w under the quotient by Z, that is: In the latter case the quotient map T w → (T ∨ ) w gives an isomorphism of the identity components and takes F w to a subgroup of F ∨ w , which, again by Theorem 2, must have index |Z| in F ∨ w .Hence in this case the image of Remark 1.8.In the case that Z < F w the Z W (w)-orbits in F w include |Z| singletons.In this case, dually the orbits in F ∨ w are partitioned into |Z| sets.To see this we note that the inclusion of Z in F w induces a quotient F w → Z. Since F ∨ w ∼ = F w we have a quotient map F ∨ w → Z, given by the pairing of F ∨ w with Z.By Theorem 3 the pairing is equivariant but Z is fixed by Z W (w), hence the map F ∨ w → Z is invariant under the action of Z W (w).Each Z W (w)-orbit in F ∨ w thus lies in a coset of the kernel of the map F ∨ w → Z.

Elements, conjugacy classes and centralisers
In this section we will provide a list of carefully selected representatives for the conjugacy classes in the Weyl group of type E 6 together with key properties of these elements.We will also introduce a number of special elements of the Weyl group that will play a key role in understanding the centraliser subgroups and their actions.
The elements s 1 , . . ., s 6 denote the standard simple reflections generating the Weyl group W , with corresponding roots denoted r 1 , . . ., r 6 .We define s 0 to be the reflection corresponding to the root which is the negation of the root of highest weight.This is at 120 • to the simple root r 6 , so the group generated by s 0 and s 6 is isomorphic to the symmetric group S 3 .On the other hand, r 0 is orthogonal to the roots r 1 , . . ., r 5 , so s 0 commutes with the reflections s 1 , . . .s 5 .
These relationships can be summarised in the following extension of the Dynkin diagram for E 6 .We remark that the reflection s 0 is the linear part of the additional simple reflection in the affine Weyl group Ẽ6 , and the extended diagram is isomorphic to the Dynkin diagram of this group.This diagram demonstrates additional symmetries of the set of roots for E 6 , which is the reason that including r 0 is so valuable.In choosing representatives for the conjugacy classes we also make use of the reflection T = s s 2 s 4 s 6 3 corresponding to the root r T := −(r 2 + r 3 + r 4 + r 6 ) = 1/2(r 0 + r 3 + r 1 + r 5 ).The name is chosen to reflect the geometry of the roots r 2 , r 3 , r 4 , r 6 in the Dynkin diagram, and the symmetry of this set in the extended diagram again motivates the inclusion of this element.
The roots r 0 , r 1 , r 5 and −r T form a Dynkin diagram of type D 4 with −r T as the central vertex, hence the group s 0 , s 1 , s 5 , T is a Weyl group of type D 4 .We remark in passing that his group contains s 3 as the reflection with root −2r T + r 0 + r 1 + r 5 .
The centralisers will be built from an elementary part along with a small number of additional elements u 1 , u 2 , u 3 , T, T s 3 .
2.1.The special elements.We now define elements u 1 , u 2 in W which generate a copy of S 3 acting (by conjugation) as a dihedral group on the reflections s 0 , . . .s 6 as illustrated on the following figure.The element u 1 is defined as the product of four commuting reflections with roots It is easy to see that this element negates the roots r 0 , r 3 , so s u 1 0 = s 0 and s u 1 3 = s 3 .Acting with u 1 on the root r 6 gives the sum so u 1 conjugates s 2 to s 4 and vice versa.Finally, acting on the root r 1 takes it to so conjugation by u 1 switches s 1 and s 5 as illustrated in the diagram.We note in passing that −u 1 is the so-called non-trivial pinned automorphism for E 6 , as described by Reeder in [8].
Turning to the element u 2 , this is defined as the product of the four commuting reflections with roots {r 5 , r 3 , r 6 + r 3 + r 2 , r 0 + r 6 + r 3 + r 2 + r 1 }.A similar calculation shows that conjugation by this element preserves s 5 , s 4 , s 3 and switches the pairs s 6 , s 2 and s 0 , s 1 .Since the root representation is faithful we see that the product u 1 u 2 has order three, and therefore these elements generate a copy of the dihedral group D 3 and the element u 3 = u 1 u 2 u 1 = u 2 u 1 u 2 is the product of four commuting reflections with roots {r 1 , r 3 , r 4 + r 3 + r 6 , r 5 + r 4 + r 3 + r 6 + r 0 }.In many cases we will see that the u 1 , u 2 dihedral group acts on one or more 2dimensional subtori of the maximal torus of E 6 .There are two natural actions of the dihedral group D 3 on a 2-torus: considering D 3 as the Weyl group W (A 2 ) we have the actions on the maximal tori in the Lie groups SU 3 and P SU 3 .The maximal tori are given respectively by triples (α, β, γ) ∈ T 3 such that αβγ = 1, and the quotient of this by the group C 3 = {(ω, ω, ω) : ω 3 = 1}.The action of the dihedral group simply permutes the three coordinates.In both cases the maximal torus can be described as a hexagon with opposite sides identified, and the two actions correspond to the two conjugacy classes of D 3 in D 6 .The orbifold quotients are an equilaterial triangle in the SU 3 case and a cone for P SU 3 .
Given that u 1 , u 2 act by signed permutations on the roots r 1 . . ., r 6 , r 0 we will see that the action of these on the 2-tori are again given by signed permutations: This has the effect of dualising the actions: the signed permutation action on the SU 3 torus is equivariantly isomorphic to the standard permutation action on the P SU 3 torus, and similarly the signed permutation action on the P SU 3 torus is equivariantly isomorphic to the standard permutation action on the SU 3 torus.
The two equivariant isomorphisms are both given by the map 2.2.The elementary part of the centralisers.Our approach is slightly different to that of Carter, in that we begin with an elementary part of the centraliser that can easily be read off from the extended Coxeter-Dynkin diagram, and then seek additional elements to complete the centraliser.Nonetheless our elementary part lies within Carter's direct product W 1 × W 2 for the conjugacy class representative.Consider an element w = w 1 w 2 . . .w l ∈ W where w 1 , . . .w l are root reflections and l = l(w) is the word length of an element w ∈ W in terms of the generating set of all root reflections in the Weyl group.Associated to such a word there is a root diagram (cf.Carter [4]) where roots are connected by an edge when they are not orthogonal (here we only have single edges as we are in the E 6 case).We write our representatives in such a way that each connected component of the diagram corresponds to a subword.Such subwords g 1 , . . ., g k of course commute with the element w (and with one another) giving a direct product Any roots which are orthogonal to all roots in the diagram will again give root reflections in the centraliser.The group these generate is precisely Carter's group W 2 .The elementary part of the centraliser is Carter determined the order of the centraliser subgroups, and for 11 of the conjugacy classes this shows that the elementary part is the entire centraliser and the structure is obvious.In the remaining 14 cases we are left to discover additional elements of the centraliser.In all cases we will show that these additional generators can be found in the subgroup u 1 , u 2 , T, T s 3 ∼ = D 3 × D 3 , as a consequence of our careful choice of representatives.In determining the centralisers and their actions we will also exploit the fact that the centraliser of an element g is contained in the centraliser of g n , and have chosen conjugacy class representatives accordingly.This relationship between the centralisers is summarised in Figure 3.Our choices of conjugacy class representatives are tabulated below.We list the eigenvalues for each of these representatives for the standard six dimensional representation: these are distinct, confirming that our representatives give distinct conjugacy classes and that the conjugacy classes are characterised by their eigenvalues.We also describe the elementary part of the centraliser, and the index of this group in the full centraliser where it is not 1.2.3.The full centralisers.We begin by stating an elementary lemma which will help to identify elements of the centraliser.
Lemma 2.1.Let A, B be diagonalisable matrices and let V 1 ⊕ • • • ⊕ V n be the eigenspace decomposition for B. If for all but one i, there is an eigenspace of A containing V i then A, B commute.
] the entire centraliser is the elementary part described in the previous section.
1 , A 3 , the representatives are s 0 s 6 , s 0 s 6 s 1 s 5 and s 0 s 6 s 3 respectively, which are all centralised by the element u 1 .In each case the elementary part of the centraliser has index 2 and it is easy to see that this does not contain u 1 .Moreover the action of the u 1 on the elementary part gives the structure of a wreath product as follows: The factors here are all complex reflection groups with the exception of S 3 C 2 .This group is isomorphic to the affine orthogonal group O − 2 (F 3 ) F 2 3 which can be embedded as a reflection group in GL 3 (F 3 ), dual to the embedding as affine orthogonal transformations.Note that this group is also isomorphic to O + 4 (F 2 ) which is unique amongst orthogonal groups as it is not generated by reflections of F 4  2 .

• Similarly in cases
the representatives s 0 s 1 , s 0 s 6 s 1 s 2 , s 0 s 6 s 5 s 1 s 2 are centralised by u 2 and the structures are given by: • In cases A 3 1 with representative s 0 s 1 s 5 and D 4 with representative s 0 s 1 s 5 T , the elementary part of the centraliser has index 6.The elements s 0 , s 1 , s 5 are permuted and T is fixed by the group u 1 , u 2 , which therefore lies in both centralisers.
In the first case, since u 1 , u 2 also commutes with s 3 , the elementary part s 0 × s 1 × s 5 × s 3 is normal, and moreover faithfulness of the u 1 , u 2 action implies these two groups have trivial intersection.Therefore where the notation 3 D 3 indicates the permutation wreath product for the action of D 3 on the three factors.
For the element s 0 s 1 s 5 T the elementary part of the centraliser is simply the cyclic group s 0 s 1 s 5 T ∼ = C 6 , so its intersection with u 1 , u 2 must be central and therefore trivial.Hence we obtain the centraliser as • In the case A 4  1 with representative s 0 s 1 s 5 s 3 , the elementary part of the centraliser is the same as it was in the A 3  1 case above.The −1 eigenspace is the space spanned by r 0 , r 1 , r 5 , r 3 , and in particular it contains r T = 1/2(r 0 + r 1 + r 5 , +r 3 ).
Hence the reflection T lies in the centraliser.As noted in Section 2, the group s 0 , s 1 , s 5 , T is a Weyl group of type D 4 , which has index 6 in the centraliser.The group u 1 , u 2 also centralises s 0 s 1 s 5 s 3 and normalises s 0 , s 1 , s 5 , T .
It remains to consider the intersection which, since u 1 , u 2 centralises T , must lie in the centraliser Z W (D 4 ) (T ) which has order 16 (see [4]).Explicitly this is the Weyl group of type A 4  1 with roots r 0 + r 1 − r T , r 0 + r 5 − r T , r 1 + r 5 − r T , r T , and the first three of these are permuted by u 1 , u 2 , hence the intersection is trivial.
We conclude that The elementary part of the centraliser is s 0 s 6 s 3 × s 1 × s 5 ∼ = C 4 ×C 2 ×C 2 .This can also be written as s 0 s 6 s 3 s 1 s 5 × s 1 × s 5 , where the first factor is, tautologically, central in the full centraliser.Now we consider the −1 eigenspace for our representative, which is 3-dimensional.It is spanned by the root vectors r 1 , r 5 (the −1 eigenspaces of s 1 , s 5 respectively) together with the −1 eigenspace for s 0 s 6 s 3 , which is spanned by r 0 + r 3 .While r 0 + r 3 is not a root vector, the sum r 0 + r 3 + r 1 + r 5 is twice the root vector r T , hence T belongs to the centraliser.Considering the root system {r 1 , r T , r 5 } se see that the group s 1 , T, s 5 ∼ = S 4 and hence has trivial intersection with s 0 s 6 s 3 s 1 s 5 .It follows that the centraliser is • Case A 5 ×A 1 , representative s 0 s 6 s 3 s 4 s 5 s 1 : The elementary part of the centraliser is the same as for the A 5 case, s 0 s 6 s 3 s 4 s 5 × s 1 ∼ = C 6 × C 2 , and has index 3 in the full centraliser.The −1 eigenspace of the representative is 2 dimensional and is spanned by the −1 eigenvector r 0 +r 3 +r 5 of s 0 s 6 s 3 s 4 s 5 and the −1 eigenvector r 1 of s 1 .As in the previous case T is in the centraliser.We note that s 1 , T is a copy of S 3 which has trivial intersection with s 0 s 6 s 3 s 4 s 5 s 1 since s 1 , T has trivial centre.Hence we obtain the centraliser subgroup as The remaining cases are E 6 [a 2 ], D 4 [a 1 ] and A 3 2 .These all correspond to Springer-regular elements, [12], and we can read off their structures from [12, Table 1] along with the Shephard-Todd classification of complex reflection groups [11,Table VII] as the groups G 5 , G 8 , G 25 respectively.
In order to identify the actions on the fixed sets in the maximal torus, we need to identify not just the structure of these groups but their elements which we do as follows.

CENTRALISERS, COMPLEX REFLECTION GROUPS AND ACTIONS IN THE WEYL GROUP E 6 18
The representative s 1 T s 5 s T 0 corresponds to the block diagonal matrix R π/2 ⊕ R π/2 which has π/2 rotations in coordinates 1, 2 and 3, 4. We equip R 4 with the structure of a complex vector space, declaring that the operator s 1 T s 5 s T 0 is multiplication by i. Hence the elements of W (D 4 ) centralising s 1 T s 5 s T 0 are precisely those which are C-linear in this space.Specifically the intersection of our centraliser with the subgroup W (D 4 ) is the Pauli group P generated by the matrices: These correspond to the elements s T 1 s T 5 , T (s 1 s 5 s 0 ) T s 5 and s 0 s 5 respectively.The group P = Z W (s 1 T s 5 s T 0 ) ∩ W (D 4 ) has index 6 in the full centraliser, hence as the centraliser is contained in W (D 4 ) S 3 it fits into an extension We now identify lifts of the generators of S 3 .The conjugation action of u 3 corresponds to interchanging ρ 5 , ρ 0 and fixing ρ 1 , ρ T .In particular, the conjugation action of u 3 on R π/2 ⊕ R π/2 inverts the second rotation, as does the element s 5 , and therefore the element s 5 u 3 is in the centraliser of s 1 T s 5 s T 0 .It is represented as a complex 2 × 2 matrix it is the diagonal matrix with entries 1, i: we denote this complex matrix by α.Now we lift u 1 , conjugation by which corresponds to exchanging ρ 1 , ρ 5 while preserving ρ T , ρ 0 .Conjugating R π/2 ⊕ R π/2 by this we obtain the matrix 0 R π/2 R π/2 0 which can be conjugated back to R π/2 ⊕ R π/2 by the transposition (24), given by s T 5 .Hence s T 5 u 1 lies in the centraliser of s 1 T s 5 s T 0 .This is represented by the complex matrix To summarise, the centraliser of s 1 T s 5 s T 0 is generated by the Pauli matrices σ 1 , σ 2 , σ 3 along with α, β.
Note that the lifts α and β each have order 4, indeed α 2 = σ 3 and β 2 = σ 1 .Moreover ασ 1 α −1 = σ 2 , hence the generators α, β are sufficient.The centraliser is thus the group s 5 u 3 , s T 5 u 1 which is faithfully represented as the matrix group α, β .These complex reflections generate the symmetry group of the complex polygon 4(96)4 (see section 1.6 of [10]), identifying Z W (s 1 T s 5 s T 0 ) with the complex reflection group G 8 in the Shephard-Todd classification.
• Case A 3 2 , representative s 0 s 6 s 1 s 2 s 5 s 4 : The elementary part of the centraliser is This has index 24 in the full centraliser, and it is evident that the missing elements include u 1 , u 2 , which permute the direct factors of the elementary part, giving a subgroup of index 4 in the centraliser isomorphic to C 3 3 D 3 .
Applying Lemma 2.1 we see that T s 3 is in the centraliser.We will show that s 0 s 6 , T s 3 , s 5 s 4 is the complex reflection group G 25 and hence is the full centraliser.
We start with the six dimensional real representation of the Weyl group W (E 6 ) on the real vector space spanned by the E 6 root system.On this space define the operator Note that 4(s 0 s 6 s 1 s 2 s 5 s 4 ) 2 + 4s 0 s 6 s 1 s 2 s 5 s 4 + I = −I since (s 0 s 6 s 1 s 2 s 5 s 4 ) 2 +s 0 s 6 s 1 s 2 s 5 s 4 +I = 0. Hence we can define a complex scalar multiplication on the Lie algebra using J as multiplication by i.An element of the Weyl group W (E 6 ) centralises s 0 s 6 s 1 s 2 s 5 s 4 if and only if it commutes with J, i.e. is C-linear for this structure.Moreover the elements s 0 s 6 , T s 3 , s 5 s 4 are complex reflections: taking r 0 , r 1 , r 5 as a basis of this complex space these elements are given by: where η = e πi/6 .These generating reflections satisfy the braid relations and also α 1 α 3 = α 3 α 1 .Since the geometry of the fixed hyperplanes of the complex reflections is determined by the above relations, the reflection group is the Shephard-Todd group G 25 = s 0 s 6 , T s 3 , s 5 s 4 .
• Case E 6 [a 2 ].The representative s 6 s 2 s T 0 s T 1 s 4 s 3 squares to the representative s 0 s 6 s 1 s 2 s 5 s 4 above, therefore its centraliser is a subgroup of G 25 .In the above generators α 1 , α 2 , α 3 we have This evidently commutes with α 2 , α 3 which is which is the binary tetrahedral group SL 2 (3), that is G 4 in the Shephard-Todd classification.Its centre is a copy of C 2 generated by the element (α ) 2 intersects trivially with α 2 , α 3 .The group C 3 × SL 2 (3) is the Shephard-Todd group G 5 , so we have the found the full centraliser

Fixed sets and their quotients: the simply connected type
In this section we will determine the fixed sets of each conjugacy class representative in the standard maximal torus T in the simply connected form of the Lie group of type E 6 .We will analyse in detail the actions of the centralisers on the fixed sets for each conjugacy class representative.In each case we will see that the quotient, which we refer to as a sector, has the homotopy type of a finite union of points or a finite union of circles.These results are summarised in Table 2.  • G 5 , G 8 , G 25 , G 28 exceptional complex reflection groups (cf.[11]) • rep is an abbreviation for the chosen representative of the conjugacy class.
• Z the centre of the simply connected Lie group of type E 6 .
The fixed sets are identified as topological spaces using Theorem 4 combined with the following well known lemma.Recall that l(w) denotes the length of the element w with respect to the generating set of all root reflections.Lemma 3.1 ([4, Lemma 2]).l(w) is the number of eigenvalues with multiplicity of w which are not equal to 1.
Using this lemma, for each w ∈ W the torus T w 1 has dimension 6 − l(w), where l(w) denotes the word length of w with respect to the generating set consisting of all root reflections in W .
Combining this with Theorem 5 determines the topology of the fixed sets in both the simply connected and adjoint forms: Theorem 7.For each w ∈ W the fixed sets T w and (T ∨ ) w are both homeomorphic to the disjoint union of g copies of a 6 − l(w)-torus where g is the greatest common divisor of the r × r-minors of the matrix of I − w and r is the rank of I − w.
We now turn to the actions of centralisers on the corresponding fixed sets.Since the action of the Weyl group W on the maximal torus T is given by conjugation, all of the fixed sets contain the centre of the Lie group as a pointwise fixed subset.For example in the 3 elliptic cases of Carter-type E 6 , E 6 [a 1 ], E 6 [a 2 ], the fixed sets, and hence the quotients are identified with the centre Z.
In cases A 4 × A 1 , D 5 and D 5 [a 1 ] the fixed set consists of a single circle containing the (pointwise fixed) centre so any element of W preserving the circle must fix it pointwise.In particular the centralisers pointwise fix the circle identifying the quotient with the fixed set.

Alternative Coordinate Systems.
To understand the actions of centralisers on the fixed sets it will be helpful to consider alternative coordinate systems on the torus.
When considering the A n case it is often helpful to use coordinates where the roots are (1, −1, 0, . . ., 0), (0, 1, −1, 0, . . ., 0) etc. and for which the maximal torus is correspondingly given by tuples with product 1 (as is the case for the standard maximal torus in SU n+1 ).We obtain these coordinates on the above 5-torus by the change of coordinates: Under this identification the A 5 Weyl group generated by s 1 , . . ., s 5 acts by permuting the coordinates.(2) su 3  3 coordinates: Another alternative coordinate system is obtained by considering the roots r 1 , r 2 , r 5 , r 4 , r 0 , r 6 as a basis for the Lie algebra.The advantage of this approach is that the dihedral group u 1 , u 2 acts by signed permutations on the pairs (r 1 , r 2 ), (r 5 , r 4 ) and (r 0 , r 6 ).Moreover the actions of s 1 , s 5 , s 0 will remain relatively simple in our new coordinates.The omitted root vector r 3 can be obtained from hence the root lattice is the refinement of the lattice generated by r 1 , r 2 , r 5 , r 4 , r 0 , r 6 by the addition of the vector 1  3 (−r 0 + r 6 − r 1 + r 2 − r 5 + r 4 ).Motivated by this we take the ordered basis r 1 , −r 2 , r 5 , −r 4 , r 0 , −r 6 for the Lie algebra, for which coordinates the lattice is Z 6 extended by the vector ( 13 , . . ., 1  3 ).The maximal torus is given by T 6 /C 3 where the C 3 acts by multiplication by (ω, . . ., ω) for ω a cube root of unity.We denote these new coordinates (α : β : : The action of u 1 , u 2 on T 6 /C 3 is to permute and invert pairs of coordinates as follows.
The conversion from these coordinates to the standard r 1 , . . ., r 6 coordinates is given by the map (α : β : : which is well defined on C 3 -cosets.Now applying s 1 changes the first of these coordinates to (η (α ) −1 ) −1 (η ) 2 β = η α β while leaving the other coordinates unchanged.Thus in the new coordinates s 1 takes (α : Similarly s 5 changes to ( δ ) −1 leaving the other coordinates unchanged, and s 0 changes η to (η ζ ) −1 leaving the other coordinates unchanged.
The action of T in the new coordinates is much simpler than in the original coordinates.In standard coordinates T is given by the matrix:

3.2.
The fixed sets and their quotients.We now proceed to compute the fixed sets and sectors.
(1) Case ∅: The fixed set is T and the centraliser is W (E 6 ), which acts on T with quotient a 6-simplex.
(2) Case A 1 : The fixed variety is a single 5-torus (α, β, γ, δ, , 1).This is canonically identified with the maximal torus for the Lie group of type A 5 .The centraliser of s 0 is the direct product of s 0 acting trivially with the Weyl group s 1 , s 2 , s 3 , s 4 , s 5 of type A 5 acting on the A 5 maximal torus.The quotient of the fixed set by the centraliser is a 5-simplex.
The fixed set of our element is then identified with the subset of the SU 6 -torus The centraliser is the direct product of s 0 u 2 with the symmetric group S 4 = s 3 , s 4 , s 5 , where s 0 acts trivially and S 4 acts by permutation of the last four coordinates.First we factor out the action of the S 4 which we will do in logarithmic coordinates (x 1 , x 1 , x 3 , . . ., x 6 ) subject to the constraint x 1 + x 1 + x 3 + x 4 + x 5 + x 6 = 0. We take x 1 to vary from − 1 /2 to 1 /2.
In these coordinates the elements of the S 4 factor act by permuting the coordinates x 3 , . . ., x 6 , and have fundamental domain D given by See [6,Equation 11] for the details.
As noted in Example 7.2 ibid the quotient of the 4-torus (1) under the action of this S 4 is an orientable twisted 3-simplex bundle over the circle parameterised by α with monodromy defined by Now we consider the action of the element u 2 .Since this commutes with the S 4 we can act directly on the quotient bundle described above.Recall that u 2 CENTRALISERS, COMPLEX REFLECTION GROUPS AND ACTIONS IN THE WEYL GROUP E 6 24 negates r 3 , r 4 , r 5 and exchanges r 0 with −r 1 and r 2 with −r 6 , so the action of u 2 on the fixed set is: Changing coordinates as above this becomes In logarithmic coordinates this is While this does not preserve the fundamental domain D for the S 4 action, composing with the permutation that reverses the last 4 coordinates returns us to D, so we have the map In the x 1 = 0 fibre the vertex set of the simplex is and in a general fibre the vertex set {v i (x 1 )} is given by translating each of these vectors by ( In each simplex two of the vertices are fixed by the u 2 action, namely v 0 (x 1 ) and v 2 (x 1 ) while the other two are interchanged.We think of the fibres as given by the join of two 1-simplices corresponding to the edge e 0 = [v 0 (x 1 ), v 2 (x 1 )] and the edge e The action of u 2 on the simplex bundle preserves the fibres and moreover preserves the join structure on each, yielding again a simplex bundle with fibres given by the join of e 0 with the quotient of e 1 by inversion.
The quotient of the fixed set by the centraliser is therefore a simplex bundle over the circle, where the monodromy is given by the reflection of the 3-simplex which inverts the e 0 factor and fixes the e 1 factor.This yields a Seifert fibred 4-orbifold: the base space is the quotient of the join of e 0 and e 1 by the Klein 4 group inverting both of these edges, [9].
The centraliser has the form s 0 s 6 × ( s 1 , s 2 u 1 ) ∼ = C 3 × (S 3 C 2 ) with the element s 0 s 6 acting trivially.The subgroups s 1 , s 2 and s 5 , s 4 = s 1 , s 2 u 1 act as the Weyl groups on these 2-torus factors so the direct product S 3 × S 3 acts in the natural way on T 2 × T 2 with quotient a product of two equilateral triangles.
The element u 1 acts to swap these, so the quotient is the symmetric product of two copies of an equilateral triangle.
(5) Case A 3 1 : The fixed set is a single 3-torus (α, α 2 , γ, 2 , , 1), and is the intersection of the fixed sets of each of the elements s 0 , s 1 , s 5 .The centraliser is the direct product of s 3 with the wreath product s 0 u 1 , u 2 , where the conjugates of s 0 are s 1 , s 5 .Since these act trivially the centraliser acts as u 1 , u 2 × s 3 .
Recall that u 1 exchanges r 1 with −r 5 , r 2 with −r 4 and negates r 3 , r 6 , so on the fixed set the action of u 1 exchanges the values of α and −1 , and inverts γ.On the other hand u 2 is given by 1 case.We decompose our 3-torus as the product }. and note that both factors are preserved by the actions of u 1 , u 2 .Moreover s 3 pointwise fixes the first factor and inverts the second.Since u 1 , u 2 also invert the second factor, the products u 1 s 3 , u 2 s 3 (which also generate a dihedral group) pointwise fix the second factor, and act on the first as which is the standard dihedral action with quotient an equilateral triangle.
Setting I = {φ ∈ T 1 : Im φ ≥ 0} which we identify in the natural way as the quotient of T 1 by the map φ → φ −1 , the quotient of the torus by s 1 , s 5 is given by This is a square bundle over the circle, whose monodromy map is rotation by π.
Now the action of u 1 preserves β (and hence x), and exchanges α with −1 .In primed coordinates so descending to the square bundle the action simply exchanges α with .
The quotient of each fibre is a right-isosceles triangle, hence the quotient of the fixed set by the centraliser is a triangle bundle over the circle with monodromy map reflecting the triangle.This is a Seifert bundle over the (4, 4, 2) triangle, [9].The centraliser is We will now change to su 3  3 coordinates in which the actions of the generators of the centraliser are all relatively straightforward.
Recall that the tuple (α : β : : δ : η : ζ ) in new coordinates corresponds to (η (α ) −1 , (η ) 2 β , (η ) 3 , (η ) 2 δ , η ( ) −1 , (η ) 2 ζ ) in the original coordinates so in the new coordinates a point of the fixed set must satisfy where η, ζ are ±1 giving the element of the Klein 4 group and the other variables lie in the circle.This simplifies to so for (η, ζ) in the Klein 4 group V 4 , the corresponding component of the fixed set is parametrised as where ω is a non-trivial cube root of 1.
In these coordinates the subgroup u 1 , u 2 simply acts by the signed permutations: and preserve η, ζ.As noted in Section 2.1 this action on the P SU 3 torus is equivariantly isomorphic to the permutation action on the SU 3 torus, for which the quotient is an equilateral triangle.
The element T acts by multiplication by φ where φ is a cube root of Since η = ±1 the action simply multiplies the projective coordinates α , , η by η and hence as α η = ζη the coordinate ζ is also multiplied by η.
We now turn to the actions of s 1 , s 5 and s 0 .It suffices to consider the element s 1 as the group u 1 , u 2 conjugates s 1 , s 5 and s 0 .For a general point we have: by the signed permutation action of u 1 , u 2 .As noted above this can be identified with the quotient of the SU 3 torus by its Weyl group, giving an equilateral triangle.
Turning to the components indexed by the non-identity elements of the Klein 4 group we see that these are permuted transitively by s 1 , T , hence there is just one more component in the quotient.We obtain this by taking the quotient of the (η, ζ) = (1, −1) component by its stabiliser in the centraliser, noting that this component is fixed pointwise by T .This component is parametrised by and we now determine its stabiliser.
As noted above the elements u 1 , u 2 preserve all the components so we begin by considering the action of the subgroup W (D 4 ) on the set of the three nonidentity components.This action gives a map from W (D 4 ) = s 1 , s 5 , s 0 , T to the permutation group S 3 in which s 1 , s 5 , s 0 all map to one transposition and T maps to another.Thus the component stabiliser in W (D 4 ) is generated by T along with the kernel of this map.Clearly the elements s 1 s 5 , s 0 s 5 , (s 1 s 5 ) T and (s 0 s 5 ) T lie in the kernel.We note that s 1 s 5 (s 0 s 5 ) T has order 4 (it corresponds to a pair of rotations by π/2 in the standard representation of W (D 4 )) hence s 1 s 5 , (s 0 s 5 ) T is a dihedral group of order 8, and similarly for s 0 s 5 , (s 1 s 5 ) T .All the other pairs from s 1 s 5 , s 0 s 5 , (s 1 s 5 ) T and (s 0 s 5 ) T have product of order 2 and hence pairwise commute.The intersection of the two dihedral groups is precisely the centre of each group (which has order 2) and hence together they generate a group of order 32 which is thus the whole of the kernel.
Hence the component stabiliser in W (D 4 ) is the semidirect product It remains to take the quotient of this by the action of u 1 , u 2 which acts by signed permutation of the coordinates (α ) 2 , ( ) 2 , (ζ ) 2 .The result therefore agrees with the quotient of the identity component (though geometrically one is 4 times the area of the other), giving the quotient T w /Z W (w) as the union of two equilateral triangles.(9) Case A 2 ×A 2 1 : The fixed set is the single 2-torus with coordinates (α, α 2 , 1, 2 , , 1).The centraliser is s 0 s 6 × ( s 1 u 1 ) and the elements s 0 s 6 , s 1 both act trivially on the fixed set in the Lie algebra, and hence on its fixed torus.This leaves the action of u 1 to consider, which exchanges and inverts α, .Replacing the variable by = −1 , the quotient is therefore the symmetric product of two copies of T 1 , which is a Möbius band.
(10) Case A 2 2 : The fixed set is a family of three 2-tori indexed by the centre as follows: Since the centre is pointwise fixed by the Weyl group, the quotient is 3 copies of the quotient of the identity component.
We write the centraliser as ( s 1 s 2 u 2 )× s 4 , s 5 .Note that s 1 s 2 acts trivially on the fixed set, hence the action is just by the quotient group u 2 × s 4 , s 5 = s 4 , s 5 u 2 ∼ = D 6 .
Hence the quotient of the fixed set by the centraliser is three copies of the triangle.
The centraliser is s 0 s 6 s 3 s 4 × s 1 with the first factor acting trivially.The action of s 1 takes α to α −1 β and leaves β unchanged.Setting β = α −1 β this action simply exchanges α, β hence the quotient is the symmetric product of two copies of T 1 (a Möbius band).
(13) Case D 4 : The fixed set is a single 2-torus (α, α 2 , α , 2 , , 1), which is the ζ = η = 1 component of the fixed set of s 0 s 1 s 5 s 3 (case 8).Indeed our representative is chosen so that its cube equals s 0 s 1 s 5 s 3 so of course the fixed sets must be nested.The centraliser is s 0 s 1 s 5 T × u 1 , u 2 with the first factor acting trivially.As in case 8 the action of u 1 , u 2 can be equivariantly identified with the permutation action of D 3 on the SU 3 torus yielding the quotient as an equilateral triangle.
(14) Case D 4 [a 1 ]: The fixed set is the same single 2-torus (α, α 2 , α , 2 , , 1) as for the previous case of type D 4 , and here we note that the square of the representative yields the element s 0 s 1 s 5 s 3 .The centraliser is the complex reflection group G 8 given by s 5 u 3 , s T 5 u 1 .Again we will use the calculations from case 8, with the fixed set described as As noted in that case, the elements s 5 , T act trivially on this set, hence we again reduce to the action of the u 3 , u 1 = u 1 , u 2 giving an equilateral triangle.
(15) Case A 2 2 × A 1 : The fixed set consists of 3 circles indexed by a cube root of unity β with coordinates (β 2 , β, 1, 2 , , 1).Each of these circles contains one of the elements of the centre, indeed we can write elements of the fixed set in the form The first factor is central, hence fixed by the whole Weyl group, and the second lies in the identity component of the fixed set, and hence is fixed by s 0 , s 6 , s 5 , s 1 , s 2 .The centraliser is ( s 0 s 6 u 2 )× s 5 where only the element u 2 acts non-trivially.This acts on the second factor by inverting .The quotient is therefore 3 intervals.
(16) Case A 3 × A 2 1 : The fixed set is a family of four circles, indexed by ζ, η with This factorisation is chosen so that for each choice of η, ζ we obtain a circle lying in the corresponding 2-torus appearing in case 8.
The centraliser is s 0 s 6 s 3 s 1 s 5 × s 1 , T, s 5 with the first factor acting trivially.The second factor lies in the centraliser from case 8, which contains the group W (D 4 ) = s 1 , T, s 5 , s 0 .The subgroup s 1 , T, s 5 is sufficient to transitively permute the three non-identity components in case 8 and hence transitively permutes the circles in this case.
Since the centre lies in the circle ζ = η = 1 it follows that this circle is fixed by the whole of s 0 s 6 s 3 s 1 s 5 × s 1 , T, s 5 .
For the non-identity components, as in case 8 consider the component (η, ζ) = (1, −1) which is pointwise fixed by T .This component is stabilised by T, s 1 s 5 ∼ = D 4 and this must be the whole stabiliser as it has index 3 in s 1 , T, s 5 ∼ = S 4 .As the element T acts trivially it remains to take the quotient by the action of s 1 s 5 .This fixes ( −1 η, −2 , η, 2 , , 1) and takes hence it has the effect of negating .Thus the quotient of the non-identity components yields a circle.
The quotient of the fixed set by the centraliser is therefore two circles.
(17) Case A 4 ×A 1 : The fixed set is a single circle with coordinates (α, α −2 , 1, α −2 , α −4 , 1).As noted after Theorem 7 this circle contains the centre Z of the Lie group, hence the action of the centraliser must be trivial.
(18) Case A 5 : The fixed set is a family of three circles indexed by a choice of a cube root of unity and with coordinates (α, , 1, 2 , , 1).As in case 15 this can be written as a product of a circle with the centre so that points of the fixed set take the form ( 2 , , 1, 2 , , 1)(α , 1, 1, 1, 1, 1).The centraliser is s 0 s 6 s 3 s 4 s 5 × s 1 with the first factor acting trivially.The second factor inverts α hence gives a reflection on each of the three circles.
Hence the quotient is three copies of the interval.
(19,20) Cases D 5 ,D 5 [a 1 ]: In both cases the fixed set is a single circle with coordinates (δ −2 , δ −1 , 1, δ, δ 2 , 1).As in case 17, this must contain the three elements of the centre so is fixed pointwise, yielding a circle as the quotient.
We now consider the action of the three generators s 0 s 6 , T s 3 , s 5 s 4 on V .Starting with s 0 s 6 , this fixes e 1 and e 3 , and takes e 2 to e 1 + e 2 + e 3 .Similarly s 5 s 4 fixes e 1 and e 3 , and takes e 2 to 2e 1 + e 2 + e 3 .
The element T s 3 fixes e 1 , e 2 , and takes e 3 to e 3 + 2e 2 .
In matrix form the generators s 0 s 6 , T s 3 , s 5 s 4 act on V as: It is not hard to see that these generate the Hessian group, i.e. the group of all orientation preserving affine maps on A.
Dualising the action back to V is therefore all orientation preserving linear maps which fix the vector e 1 .The quotient is therefore 4 points, corresponding to the three points on the line spanned by e 1 (the centre of the Lie group) and a single orbit for all the remaining vectors in V .where is a cube root of unity and ζ 2 = η 2 = 1.We choose this factorisation to the the copy of the Klein group V 4 matches with that appearing in cases 8 and 16.
The centraliser is s 0 s 6 s 3 s 4 s 5 s 1 × s 1 , T with the first factor acting trivially.The fixed set lies in the fixed set from case 16, and the group s 1 , T is a subgroup of the centraliser from that case.
The η = ζ = 1 part is the centre of the Lie group and is thus fixed by the centraliser.As noted in case 16, the three other possible values for (η, ζ) are transitively permuted by s 1 , T .The triple with (η, ζ) = (1, −1) is pointwise fixed by T and its stabiliser in the group s 1 , T has index 3 and is thus T .Hence the 6 points given by (η, ζ) = (1, 1) and (η, ζ) = (1, −1) form a strict fundamental domain for the action of the centraliser.
The quotient is therefore 6 points.
Let G denote a split group of type E 6 over a p-adic field; the group may be of adjoint type or simply connected.Let C * r (G) denote the reduced C * -algebra of G.The reduced C * -algebra admits the Bernstein decomposition where B(G) is the Bernstein spectrum of G.Each point s in the Bernstein spectrum is an equivalence class of cuspidal pairs (M, σ) where M is a Levi subgroup of G and σ is an irreducible cuspidal representation of M. In particular, we have the Iwahori point i defined by the pair (T p , 1) where T p is a maximal torus of (the p-adic group) G and 1 is the trivial representation of T p .We will write A = C * r (G) i .This is called the reduced Iwahori-spherical C * -algebra.The spectrum of the C * -algebra A comprises all irreducible tempered representations of G which admit a nonzero Iwahorifixed vector.
Let G ∨ denote the (complex) Langlands dual of G, let T C denote a maximal torus in G ∨ , and let T denote the maximal compact subgroup of T C .
According to [1, Eqn.(4.9)], we have The results in §5.1 now lead immediately to the following answer for the K-theory of A: From the point of view of noncommutative geometry, the C * -algebra A behaves, at the level of K-theory (after tensoring by C), as if its spectrum was equal to the extended quotient T //W .

Lemma 1 . 2 .
The operator m(w)(I − w) is µp xw where p xw denotes the orthogonal projection onto x w .

Figure 2 .
Figure 2. The action of the special elements u i .

2 D 3 • D 4
twisted bundle over the circle with fibre a k-simplex • SP 2 (X) the 2-fold symmetric product of a space X • C m the cyclic group of order m • D 3 the dihedral group of order 6 generated by the special elements u 1 , u 2 • C 2 3 D 3 is the permutation wreath product C 3 a dihedral group of order 8 of the form s i u j • S m a symmetric group on m letters generated by a subset of the 36 root reflections • V 4 a Klein 4 group.

) Case A 2 × A 1 :
The fixed set is the single 3-torus (α, α 2 , 1, δ, , 1) and the centraliser is s 0 s 6 × s 1 × s 4 , s 5 ∼ = C 3 × C 2 × S 3 .The first two factors act trivially so we are left with a residual action of S 3 = s 4 , s 5 .These generators act as the matrices −parameters (δ, ), fixing the parameter α.The non trivial part of the action is therefore the standard dihedral action on a 2-torus with quotient an equilateral triangle, crossed with the trivial action on the remaining circle.Hence the quotient is T 1 × ∆ 2 .

6 .Figure 3 .
Figure 3. Inclusions among the centralisers induced by power relations of conjugacy class representatives.Layers show the order of the representatives, indicating the relevant power relation.
CENTRALISERS, COMPLEX REFLECTION GROUPS AND ACTIONS IN THE WEYL GROUPE 6 4

Table 1 .
The conjugacy class representatives and the elementary part of their centralisers 36 s 0 s 6 s 3 s 4 s 5 , s 1 C 6 × C 2 3 E 6 s 1 s 2 s 3 s 4 s 5 s 6 e s 2 s 3 s 4 s 5 s 6 C 12 E 6 [a 1 ] s 1 s 2 s 3 s 4 s 5 s s 3 s 2 s 3 s 4 s 5 s s 3 CENTRALISERS, COMPLEX REFLECTION GROUPS AND ACTIONS IN THE WEYL GROUP E 6 16 5 3 πi , −1