Transformation Groups, Vol. ?, No. ?, ??, pp. 1–?? \begin{matrix}c \\ ◯ \end{matrix}Birkhäuser Boston (??)

SPLITTING FIELDS FOR EXTENDED COMPLEX REFLECTION GROUPS AND HECKE ALGEBRAS

GUNTER MALLE

Fachbereich Mathematik Universität Kaiserslautern Postfach 3049 D–67653 Kaiserslautern Germany malle@mathematik.uni-kl.de

Abstract. Finite complex reflection groups have the remarkable property that the character field k of their reflection representation is a splitting field, that is, every irreducible complex representation can be realized over k. Here we show that this statement remains true for extensions of finite complex reflection groups by elements in their normalizer. Also, we generalize the corresponding result for cyclotomic Hecke algebras to Hecke algebras attached to extended finite complex reflection groups.

1. Introduction

Let W ∈ GL(V) be a finite group generated by complex reflections, where V is a finite dimensional complex vector space. It is a well-known and important fact, first proved by Benard (see Bessis [2] for a shorter proof), that the field k generated by the character values of W on V is a splitting field for W, i.e., every complex representation of W can be realized over the field k. The only known proof of this result is case by case and character by character.

Attached to the complex reflection group W there are (cyclotomic) Hecke algebras H(W), deformations of the group algebra of W. By Tits’ deformation theorem, the irreducible characters of H(W) over a splitting field are in bijection with the irreducible complex characters of W. The nature of the splitting field for H(W) was first considered by Benson–Curtis [1] in the case of Weyl groups (see also Lusztig [14] for the explicit construction of an isomorphism to the group algebra of W), and then completely solved by the author [16] for arbitrary complex reflection groups. Here it turns out that a splitting field can be obtained by adjoining suitable roots of monomials in the parameters of H(W) to the splitting field k of W.

When studying representations of twisted as well as disconnected groups of Lie type (see for example [15]), and also generalizations of these corresponding to complex reflection groups (see [4]), one is naturally lead to consider cosets W ϕ of W with respect to elements ϕ in the normalizer N_{GL(V)}(W) of W. Again, the question about realizability of representations of these cosets arises. This was first studied by Digne–Michel [8] both for Weyl groups and their associated Hecke-algebras, and later completed by Geck [9]. See Geck–Kim–Pfeiffer [10] for explicit character tables in this situation.

Received ??. Accepted ??.

In the present paper we construct splitting fields in the general setting of cosets of complex reflection groups as well as associated extended cyclotomic Hecke algebras. In fact we show that the natural analogue of the statement for the trivial coset, as explained above, continues to hold (see Theorem 2.16 for cosets of reflection groups, and Theorem 4.11 for cosets of cyclotomic Hecke algebras). In order to derive an action of ϕ ∈ N_{GL(V)}(W) on a cyclotomic Hecke algebra attached to W, we have to investigate the question of existence of 1-regular elements in cosets W ϕ, see Theorem 3.4. This result seems to be of independent interest. We obtain that automorphisms of W constructed in this way always lift to automorphisms of the same order of the corresponding braid group, which then in turn give rise to the desired automorphisms of cyclotomic Hecke algebras.

Acknowledgement: The computations with primitive complex reflection groups were considerably facilitated by using the functions for complex reflection groups implemented by Jean Michel in the computer algebra package Chevie; see

http://www.math.jussieu.fr/˜ jmichel/chevie.html

for the download of this free GAP 3 package.

2. Cyclic extensions of finite complex reflection groups

We start with an easy lemma comparing character fields for groups and for cosets of normal subgroups:

Lemma 2.1. Let G be a finite group, H ◁ G a normal subgroup with cyclic factor group and ϕ ∈ G such that G = ⟨H, ϕ⟩. Let χ be an irreducible character of G restricting irreducibly to H. Then

ℚ(χ(g) | g ∈ G) = ℚ(χ(hϕ) | h ∈ H).

Proof. Write k_{G}, k_{Hϕ} for the respective fields. The containment k_{Hϕ} ≤ k_{G} is clear. Now let σ ∈ Gal(ℚ̄/k_{Hϕ}). Then χ^{σ} is an irreducible character of G which agrees with χ on the coset Hϕ. The orthogonality relations on the coset show that then also χ^{σ}|_{H} = χ|_{H}, hence χ^{σ} = χ. Thus k_{G} lies in the fixed field of σ. This shows that k_{G} ≤ k_{Hϕ}.

Thus, in what follows, we may and will switch between character fields for a group and a coset of a normal subgroup.

2.2. The general setup. Let W ≤ GL(V) be a finite complex reflection group on V = ℂ^{n} and

k := ℚ(tr_{V} (w) | w ∈ W)

the character field of this reflection representation. This is a finite abelian extension of ℚ. By a result of Benard and Bessis (see [2]) k is a splitting field for W.

We assume that W is essential, that is, W does not fix a non-trivial subspace of V pointwise. Clearly, by the Theorem of Maschke, by restricting the W -action to a suitable W -invariant subspace of V this can always be achieved. Let ϕ ∈ N_{GL(V)}(W) be an element of GL(V) of finite order normalizing W. We propose to study the splitting field of the extension W̃ := ⟨W, ϕ⟩. The following consequence of the Theorem of Benard–Bessis shows that in order to find splitting fields we can restrict ourselves to the investigation of character fields:

Proposition 2.3. Let W ≤ GL(V) be a finite complex reflection group, ϕ ∈ N_{GL(V)}(W) of finite order, W̃ := ⟨W, ϕ⟩ and k = ℚ(tr_{V} (wϕ) | w ∈ W) the character field of W̃ on V. Then for any ϕ-stable irreducible complex character χ of W we have:

(a) χ extends to W̃,

(b) any extension χ̃ of χ can be realized over the composite of k with the character field of χ̃.

Proof. The first part is immediate since W̃ is a cyclic extension of W [12, §6]. For the second part note that a representation affording χ can be realized over k by the main result of [2]. But χ̃ occurs with multiplicity 1 in the induced of χ to W̃. Thus by a well-known criterion [12, §6] it can be realized over the composite of k with the character field of χ̃.

2.4. The irreducible complex reflection groups. The results in this paper all build on the knowledge of the finite irreducible complex reflection groups, which were classified by Shephard–Todd [17]. We follow their notation for the irreducible types. They show that there are 34 exceptional groups, denoted G_{4}, …, G_{37}, which all occur in dimension at most 8, the infinite series of symmetric groups 𝔖_{n}, n ≥ 2, and of cyclic groups Z_{d}, d ≥ 3, also denoted G(d, 1, 1), and an infinite series of imprimitive groups.

We recall the construction and some properties of these imprimitive complex reflection groups. Let n, d ≥ 2. We write ζ_{d} := exp(2πi/d) for a primitive dth root of unity. The group of all monomial n × n matrices with entries dth roots of unity is denoted G(d, 1, n). It is an irreducible imprimitive complex reflection group, isomorphic to the wreath product Z_{d} ≀ 𝔖_{n} of the cyclic group of order d with the symmetric group 𝔖_{n}. Define a homomorphism p : G(d, 1, n) → ℂ^{×} by sending a monomial matrix to the product of its non-zero entries. For any divisor e|d we let G(d, e, n) be the kernel of p composed with taking the d/eth power. That is, G(d, e, n) consists of those monomial matrices whose non-zero entries have product a d/eth root of unity. It is a normal reflection subgroup of G(d, 1, n), irreducible unless d = e = n = 2. The imprimitive Weyl groups of type B_{n} respectively D_{n} are recovered as the special cases d = 2, e = 1, respectively d = 2, e = 2.

2.5. Normalizers of irreducible reflection groups. It turns out that when W is irreducible, not very many different possibilities for ϕ do exist. They were first classified in [6] (see [4, Prop. 3.13] for the correct list), building on the classification of finite irreducible reflection groups. Nevertheless, some interesting phenomena may occur, as illustrated by the following examples:

Example 2.6. (a) Let W = W (D_{4}), the Weyl group of type D_{4}, and ϕ an element realizing the exceptional graph automorphism of order 3 of W. This can be chosen to be rational (see (2.9)(3)), so the character field of W̃ on V is ℚ. But clearly every invariant character has some extension whose values are non-rational. Thus we can only hope for the following: given some extension of the reflection representation, with character field k, any invariant character has some extension defined over k.

(b) Let W = G(e, e, 2) with e even, the dihedral group of order 2e. The reflection representation of dimension 2 of W is defined over ℚ(ζ_{e} + ζ_{e}^{−1}), with an eth primitive root of unity ζ_{e}. Clearly it extends to the reflection representation of the dihedral group W̃ = G(2e, 2e, 2), defined over the totally real field ℚ(ζ_{2e} + ζ_{2e}^{−1}). On the other hand W is normal in the 2-dimensional reflection group G(e, 1, 2), defined over ℚ(ζ_{e}). The outer cosets of the two extensions differ by the scalar ζ_{2e} inside N_{GL(V)}(W). Thus even when considering extensions of the same degree which only differ by central elements, the character fields can behave very differently.

In order to state the classification of non-trivial cosets, let us denote by ϕ̄ the image of ϕ in PGL(V).

Proposition 2.7. Let W be an irreducible complex reflection group on the n-dimen-sional complex vector space V and ϕ ∈ N_{GL(V)}(W). Then up to elements of W, either ϕ̄ = 1, or W = G(de, e, n), with e > 1, and ϕ̄ of order dividing e comes from the embedding G(de, e, n) ◁ G(de, 1, n), or we are in one of the cases indicated in Table 1.

Table 1. Exceptional ϕ ∈ N_{GL(V)}(W)
W  |ϕ̄| k  origin of ϕ
G(4, 2, 2) 3 ℚ(i)  < G_{6}
G(3, 3, 3) 2 ℚ(ζ_{3})  < G_{26}
G(2, 2, 4) 3 ℚ  graph aut. of D_{4}
G_{5}  2 ℚ(ζ_{3}, \sqrt{−2})  < G_{14}
G_{7}  2 ℚ(ζ_{12})  < G_{10}, < G_{15}
G_{28}  2 ℚ(\sqrt{2})  graph aut. of F_{4}

The third column gives the character field of a corresponding representation. Note that G_{28}, G(2, 2, 4) in Shephard and Todd’s notation are the Weyl groups of type F_{4}, D_{4}, respectively.

This is shown in [4], using the classification of irreducible complex reflection groups. We deduce the following consequence:

Corollary 2.8. Under the assumptions of Proposition 2.7 there exist w ∈ W and ζ ∈ ℂ^{×} such that either ζwϕ is a reflection, or we are in one of the following cases:

– W = G(3, 3, 3) and ζwϕ of order 4 comes from the embedding G(3, 3, 3) < G_{26},

– W = G(2, 2, 4) and ζwϕ of order 3 comes from the embedding W (D_{4}) < W (F_{4}),

– W = G_{28} and ζwϕ realizes the graph-automorphism of W (F_{4}).

2.9. Four exceptional automorphisms. Let us consider the three exceptional cases in the preceding Corollary and one further case in some more detail.

(a) The imprimitive complex reflection group W = G(4, 2, 2) can be generated by the three reflections

( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , ( \begin{matrix}0 & −i \\ i & 0 \end{matrix} )

of order 2 in GL_{2}(ℂ). It has three systems of imprimitivity in V = ℂ^{2}, the standard one consisting of the basis vectors e_{1} = (1, 0), e_{2} = (0, 1), and two exceptional ones, with invariant 1-dimensional subspaces spanned by

\begin{matrix}\{(1, 1), (1, −1)\}, \\ \{(1, i), (1, −i)\}, \end{matrix}

respectively. Any element ϕ ∈ N_{GL(V)}(W) permutes these. If ϕ fixes one of the systems, then (up to scalars) it extends W to the imprimitive group G(4, 1, 2). On the other hand, the element

(1)  ϕ := \frac{1}{2} ( \begin{matrix}i − 1 & i + 1 \\ i − 1 & −i − 1 \end{matrix} )

of order 3 permutes the three systems of imprimitivity transitively. Note that ζ_{3}ϕ is a reflection (of order 3), so ⟨W, ζ_{3}ϕ⟩ is the primitive complex reflection group G_{6}.

(b) The imprimitive complex reflection group W = G(3, 3, 3) can be generated by the three reflections

( \begin{matrix}0 & ζ_{3} & 0 \\ ζ_{3}^{2} & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} )

of order 2 in GL_{3}(ℂ). It has four systems of imprimitivity in V = ℂ^{3}, the standard one consisting of the basis vectors e_{1} = (1, 0, 0), e_{2} = (0, 1, 0), e_{3} = (0, 0, 1), and three exceptional ones, with invariant 1-dimensional subspaces spanned by

{(1, 1, ζ_{3}), (1, ζ_{3}, 1), (ζ_{3}, 1, 1)},
{(1, 1, ζ_{3}^{2}), (1, ζ_{3}^{2}, 1), (ζ_{3}^{2}, 1, 1)},
{(1, 1, 1), (1, ζ_{3}, ζ_{3}^{2}), (1, ζ_{3}^{2}, ζ_{3})}

respectively. Any element ϕ ∈ N_{GL(V)}(W) has to permute these. Assume that ϕ fixes the standard system. Then up to elements of W and scalars ϕ is given by a diagonal matrix

( \begin{matrix}1 & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & b \end{matrix} )

with a, b ∈ ℂ^{×}. In particular, if ϕ fixes a second system, it is the identity. Hence N_{GL(V)}(W) does not act as the full symmetric group on the four systems.

On the other hand consider the complex reflection group G_{26}, generated by one reflection s of order 2 and two reflections of order 3 satisfying the relations of the B_{3}-diagram. The conjugacy class of s generates a normal reflection subgroup W_{1} of type G(3, 3, 3), with factor group isomorphic to SL_{2}(3). The quotient of G_{26} by W_{1}.Z(G_{26}) is isomorphic to the alternating group 𝔄_{4} on four letters. Clearly this acts faithfully on the four systems of imprimitivity of G(3, 3, 3), so we have shown that N_{GL(V)}(W)/C_{GL(V)}(W) ≅ 𝔄_{4}. The elements of order 3 in this alternating group each fix one of the systems of imprimitivity, hence each (up to scalars) extends W to a maximal imprimitive group G(3, 1, 3). On the other hand, the elements of order 2 act fixed-point freely, so lead to primitive extensions of W. Since all involutions in 𝔄_{4} are conjugate, there is just one isomorphism type up to conjugation. For example, the element

(2)  ϕ := \frac{ζ_{3}}{\sqrt{−3}} ( \begin{matrix}1 & ζ_{3} & 1 \\ ζ_{3} & 1 & 1 \\ ζ_{3} & ζ_{3} & ζ_{3}^{2} \end{matrix} )

permutes the four systems of imprimitivity like a double transposition. Its square is minus a noncentral element of order 2 in W. Since W does not contain elements of order 4 this shows that for any choice of w ∈ W, ζ ∈ ℂ^{×}, ζwϕ has order at least 4.

(c) The imprimitive complex reflection group W = G(2, 2, 4) is the Weyl group of type D_{4}, with standard generators

( \begin{matrix}0 & −1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

and the three permutation matrices corresponding to the transpositions (1, 2), (2, 3), (3, 4). It has three systems of imprimitivity, with invariant 1-dimensional subspaces spanned by

{(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)},
{(1, 1, 1, 1), (1, 1, −1, −1), (1, −1, 1, −1), (1, −1, −1, 1)},
{(1, 1, 1, −1), (1, 1, −1, 1), (1, −1, 1, 1), (1, −1, −1, −1)}

respectively. The Weyl group of type F_{4} contains W as a normal subgroup of index 6 with factor group 𝔖_{3} acting naturally on the set of systems of imprimitivity. The elements of order 2 fix one system, so extend W to the maximal imprimitive group G(2, 1, 4), the Weyl group of type B_{4}. A representative for an automorphism of order 3 is given by

(3)  ϕ := \frac{1}{2} ( \begin{matrix}−1 & −1 & −1 & −1 \\ 1 & 1 & −1 & −1 \\ 1 & −1 & 1 & −1 \\ 1 & −1 & −1 & 1 \end{matrix} ) .

(d) The primitive complex reflection group W = G_{28}, the Weyl group of type F_{4}, can be generated by the reflections

( \begin{matrix}−1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 1 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 1 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & −1 \end{matrix} ) .

The graph automorphism on W is induced by the element

(4)  ϕ := \frac{1}{\sqrt{2}} ( \begin{matrix}0 & 0 & 0 & 2 \\ 0 & 0 & 2 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) .

2.10. Standard elements. For W ≤ GL(V) an irreducible reflection group we say that ϕ ∈ N_{GL(V)}(W) is standard (in its cosetℂ^{×} · W ϕ) if either • ϕ = 1, or

• ϕ is the element defined in (2.9) above in the respective cases W = G(4, 2, 2), W = G(3, 3, 3), W = G(2, 2, 4), W = G_{28}, or

• ϕ is a reflection. Moreover, in the third case we require that

• ϕ is a reflection from G(2de, 1, 2) in the case of W = G(2de, 2e, 2), d > 1,

• ϕ is a reflection from G(4e, 4e, 2) in the case of W = G(2e, 2e, 2), e > 1 (see Example 2.6(b)),

• ϕ is a reflection from G_{10} in the case of W = G_{7}. With this notation we have:

Proposition 2.11. Let W be an irreducible complex reflection group, ϕ ∈ N_{GL(V)}(W) standard. Then the character field k of W̃ := ⟨W, ϕ⟩ on V equals the character field k_{0} of the reflection representation of W on V, or ϕ has order 2 and one of:

(a) W = G(2e, 2e, 2) = W (I_{2}(2e)) (e > 1) and k = ℚ(ζ_{2e}), (k/k_{0}) = 2,

(b) W = G_{5} and k = k_{0}(\sqrt{−2}) = ℚ(ζ_{3}, \sqrt{−2}), or

(c) W = G_{28} = W (F_{4}) and k = k_{0}(\sqrt{2}) = ℚ(\sqrt{2}). Moreover, in the exceptional cases, an extension of a character χ ∈ Irr (W)^{ϕ} has character field not contained in k_{0} if and only if χ is faithful on the center of W (of order 2).

Proof. If W̃ is a reflection group itself, then this follows from the list of fields of definition of complex reflection groups (see Geck–Malle [11, Table 1–2], for example). By Corollary 2.8 and the definition of ϕ being standard this only leaves the groups in (2.9). For those, the element ϕ constructed in (2.9)(a)–(c) visibly lies over the indicated field. This completes the proof of the first part.

Let’s now consider cases (a)–(c) in more detail. For W = G(2e, 2e, 2) the dihedral group of order 4e, W̃ is the dihedral group of order 8e. If χ ∈ Irr (W̃) has the center of W (and W̃) in the kernel, then it is a representation of the factor group W̃ /Z(W) ≅ W, hence has character values in k_{0}, and conversely. For W = G_{5} it can be checked from the known character table that only the 2-dimensional characters are faithful on the center. On the other hand the character table of the coset W ϕ for W = G_{5} in Table 2 shows that only extensions of 2-dimensional characters involve \sqrt{−2}. Finally, for W = W (F_{4}) precisely the two 4-dimensional characters labelled ϕ_{4,1}, ϕ_{4,13}, and the 16-dimensional character ϕ_{16,5} are ϕ-stable and faithfull on the center. The character table of W ϕ in [10, Table VI] shows that precisely these involve \sqrt{2}.

Note that all exceptions in the above statement except for W (F_{4}) live in dimension 2.

Proposition 2.12. Let W ≤ GL_{n}(ℂ) be an irreducible complex reflection group, ϕ ∈ N_{GL(V)}(W) standard, and k the character field of W̃ := ⟨W, ϕ⟩ on V. Then each ϕstable representation of W extends to a representation of W̃ defined over k.

Proof. First note that ϕ-invariant characters extend to W̃ by Proposition 2.3(a). Now, if ϕ is a reflection then W̃ is a reflection group itself, and the result follows from the Theorem of Benard–Bessis [2].

Thus by Corollary 2.8 and the definition of ϕ being standard we are in one of the four particular cases studied in (2.9). By Proposition 2.3(b) it suffices to show that each invariant character has an extension whose character field is contained in k.

Table 2. Character table for coset G_{5}ϕ

±1 ±ζ_{3}^{2} ±ζ_{3} ζ_{8}^{5}, ζ_{8}^{7} ζ_{24}, ζ_{24}^{19} ζ_{24}^{5}, ζ_{24}^{23} ζ_{8}, ζ_{8}^{3} ζ_{24}^{7}, ζ_{24}^{13} ζ_{24}^{11}, ζ_{24}^{17}
ϕ_{1,0}  1  1  1  1  1  1  1  1  1
ϕ_{1,8}  1 ζ_{3} ζ_{3}^{2}  1  ζ_{3}  ζ_{3}^{2}  1  ζ_{3}  ζ_{3}^{2}
ϕ_{1,16} 1 ζ_{3}^{2} ζ_{3}  1  ζ_{3}^{2}  ζ_{3}  1  ζ_{3}^{2}  ζ_{3}
ϕ_{2,9}  .  .  . −\sqrt{−2}  \sqrt{−2}  −\sqrt{−2} \sqrt{−2}  −\sqrt{−2}  \sqrt{−2}
ϕ_{2,5}  .  .  . \sqrt{−2} −\sqrt{−2}ζ_{3}  \sqrt{−2}ζ_{3}^{2} −\sqrt{−2} \sqrt{−2}ζ_{3} −\sqrt{−2}ζ_{3}^{2}
ϕ_{2,1}  .  .  . −\sqrt{−2} \sqrt{−2}ζ_{3}^{2} −\sqrt{−2}ζ_{3}  \sqrt{−2} −\sqrt{−2}ζ_{3}^{2}  \sqrt{−2}ζ_{3}
ϕ′_{3,6}  1  1  1  −1  −1  −1  −1  −1  −1
ϕ_{3,2}  1 ζ_{3} ζ_{3}^{2}  −1  −ζ_{3}  −ζ_{3}^{2}  −1  −ζ_{3}  −ζ_{3}^{2}
ϕ_{3,4}  1 ζ_{3}^{2} ζ_{3}  −1  −ζ_{3}^{2}  −ζ_{3}  −1  −ζ_{3}^{2}  −ζ_{3}

Conjugacy classes are indicated by the eigenvalues in the reflection representation.

For W = G(4, 2, 2) and W = G(3, 3, 3) the character tables of the respective cosets are given in Tables 3 and 4; the entries clearly lie in k. For W = G(2, 2, 4) the Weyl group of type D_{4}, by our choice of ϕ we have k = ℚ. Since all characters of W are rational and ϕ has order three, each ϕ-invariant character χ has (precisely) one rational extension.

Finally let W = G_{28} with ϕ standard of order 2, so k = ℚ(\sqrt{2}). The character table of W̃ in [10, Table VI] shows that k is the character field of W̃.

Table 3. Character table for coset G(4, 2, 2)ϕ

−ζ_{3}, −ζ_{3}^{2} ζ_{12}, ζ_{12}^{5} ζ_{3}, ζ_{3}^{2} ζ_{12}^{7}, ζ_{12}^{11}
ϕ_{1,0}  1  1  1  1
ϕ_{1,6}  1  −1  1  −1
ϕ_{2,1}  1  ζ_{4}  −1  −ζ_{4}
ϕ_{2,3}  1  −ζ_{4}  −1  ζ_{4}

Table 4. Character table for coset G(3, 3, 3)ϕ

1, ±ζ_{4} −ζ_{3}^{2}, ±ζ_{12}^{5} −ζ_{3}, ±ζ_{12} ζ_{3}, ±ζ_{12} −1, ±ζ_{4} ζ_{3}^{2}, ±ζ_{12}^{5}
ϕ_{1,0}  1  1  1  1  1  1
ϕ_{1,9}  1  −1  −1  1  −1  1
ϕ_{3,1}  1  −ζ_{3}^{2}  −ζ_{3}  ζ_{3}  −1  ζ_{3}^{2}
ϕ_{3,2}  1  −ζ_{3}  −ζ_{3}^{2}  ζ_{3}^{2}  −1  ζ_{3}
ϕ_{3,4}  1  ζ_{3}^{2}  ζ_{3}  ζ_{3}  1  ζ_{3}^{2}
ϕ_{3,5}  1  ζ_{3}  ζ_{3}^{2}  ζ_{3}^{2}  1  ζ_{3}

Remark 2.13. Let G be a finite group with two subgroups G_{0}, C such that C is cyclic, [G_{0}, C] = 1 and G = G_{0}C. Thus G is the extension of the normal subgroup G_{0} by a central cyclic group. It is well-known that then all irreducible complex representations of G_{0} extend to G. We give an explicit construction of such an extension.

Note that G is the central product of G_{0} with C over Z := G_{0} ∩ C. Let n := |G|/|G_{0}| = |C|/|Z|. Thus, if ϕ denotes a generator of C then n is minimal with ϕ^{n} ∈ G_{0}. Let D : G_{0} ⟶ GL_{m}(ℂ) be an irreducible representation of G_{0}. Since ϕ^{n} is central in G_{0}, D(ϕ^{n}) = μI_{m} is a scalar matrix, where μ is a root of unity of order at most |Z|. Fix an nth root μ^{1/n} of μ in ℂ, a root of unity of order at most |C|. Then clearly

D(gϕ^{i}) := D(g) · (μ^{1/n})^{i}, for g ∈ G_{0}, i ∈ ℤ,

extends D to G.

The situation is even simpler if Z lies in the kernel of D. Then also

D(gϕ^{i}) := D(g), for g ∈ G_{0}, i ∈ ℤ,

extends D to G.

We can now determine the character fields in the irreducible case:

Proposition 2.14. Let W ≤ GL(V) be a finite irreducible complex reflection group, ϕ ∈ N_{GL(V)}(W) of finite order,

k := ℚ(tr_{V} (wϕ) | w ∈ W)

the character field of W̃ := ⟨W, ϕ⟩ on V. Then every ϕ-stable complex irreducible character of W has an extension to W̃ afforded by a representation defined over k.

Proof. By Proposition 2.3(b) it suffices to show that every ϕ-invariant character χ ∈ Irr (W) has an extension χ̃ to W̃ with character field contained in k.

After changing ϕ by an element of W we may assume without loss that ϕ′ := ζ^{−1}ϕ is standard for some ζ ∈ ℂ^{×}. Let W′ := ⟨W, ϕ′⟩ and k′ the character field of tr_{V} |_{W′}.

We first assume that (W, ϕ′) is not one of the three exceptional cases in Proposition 2.11. Then k′ = k_{0} in the notation of Proposition 2.11. Let w ∈ W with tr_{V} (wϕ′) ≠ 0. Then

tr_{V} (wϕ) = tr_{V} (wζϕ′) = ζtr_{V} (wϕ′) ≠ 0.

Since tr_{V} (wϕ′) ∈ k′ = k_{0} and tr_{V} (wϕ) ∈ k we conclude that ζ ∈ k, so k ≥ k_{0}(ζ). Now let χ be an arbitrary ϕ-invariant character of W. Clearly χ is also ϕ′-invariant, so by Proposition 2.12 there exists an extension χ′ of χ to W′ defined over k′ = k_{0}. By Remark 2.13 there exists an extension χ̃ to ⟨W′, ζ⟩ = ⟨W̃ , ζ⟩ with values χ̃(wϕ′) = ζ^{−1}χ′(wϕ). In particular all values of χ̃ are contained in k_{0}(ζ), hence in k.

It remains to consider the cases (a)–(c) in Proposition 2.11 above. Note that in all three cases (W′ : W) = 2 and k′ is a quadratic extension of k_{0}, generated by \sqrt{y} for some y ∈ k_{0}. First assume that χ ∈ Irr (W)^{ϕ} has Z(W) in its kernel. By the second part of Proposition 2.11 there exists an extension χ′ to W′ with values in k_{0}. By Remark 2.13 χ′ extends to ⟨W′ζ⟩ = ⟨W̃ , ζ⟩ with values in k_{0}.

So finally let χ ∈ Irr (W)^{ϕ} have two extensions χ_{1}, χ_{2} to W′ with values not contained in k_{0}. Then

χ_{2}(wϕ′) = σ(χ_{1}(wϕ′)) = −χ_{1}(wϕ′)

with the generator σ of Gal(k′/k_{0}). Thus, χ_{1}(wϕ′) is a k_{0}-multiple of \sqrt{y}. By Proposition 2.12 at least one of χ_{1}, χ_{2} can be realized over k′ = k_{0}(\sqrt{y}). By Remark 2.13 there exists an extension χ̃ of χ to W̃ whose values are k_{0}-multiples of ζ\sqrt{y}. In particular, all values of this extension lie in k_{0}(ζ\sqrt{y}). Since this applies in particular to χ = tr_{V} |_{W}, we deduce that k = k_{0}(ζ\sqrt{y}), and moreover for all such characters there exists an extension with values in k. This completes the proof.

2.15. Splitting fields for cosets of reflection groups.

Theorem 2.16. Let W ≤ GL(V) be a finite complex reflection group, ϕ ∈ N_{GL(V)}(W) of finite order,

k := ℚ(tr_{V} (wϕ) | w ∈ W)

the character field of W̃ := ⟨W, ϕ⟩ on V. Then every ϕ-stable complex irreducible character of W has an extension to W̃ afforded by a representation defined over k.

Proof. Again by Proposition 2.3(b) it suffices to show that every ϕ-invariant character χ ∈ Irr (W) has an extension χ̃ to W̃ with character field contained in k. We argue by induction on dim V, the induction basis being given by the case of irreducible W in Proposition 2.14.

First assume that there are ϕ-invariant non-trivial decompositions V = V_{1} ⊕ V_{2}, W = W_{1} × W_{2} such that W_{1} acts trivially on V_{2} and vice-versa, and W_{1}, W_{2} are reflection subgroups of W. If we let W̃_{i} := ⟨W, ϕ|_{V_{i}}⟩ for i = 1, 2, then W̃ ≤ W̃_{1} × W̃_{2}. If χ = χ_{1} ⊗ χ_{2} ∈ Irr (W) = Irr (W_{1} × W_{2}) is ϕ-invariant, then χ_{i} is ϕ-invariant for i = 1, 2, hence by induction extends to a character χ̃_{i} of W̃_{i} over the character field of tr_{V} |W̃_{i}. This shows that χ extends to a character χ̃ of W̃ realizable over the composite of the character fields of tr_{V} |W̃_{i}. But clearly this is the character field k of tr_{V} |W̃, hence we are done in this case.

It remains to consider the case that W = W_{1} × … × W_{r} with W_{i} ≅ W_{1} mutually isomorphic reflection subgroups, permuted transitively by ϕ. Then ϕ^{r} stabilizes each factor W_{i}, hence W′ := ⟨W, ϕ^{r}⟩ is as in the previous case. So each ϕ-invariant character of W extends to a character χ′ of W′ defined over the character field of W′. Now let D : W′ → GL_{mr}(k) be a (block-diagonal) representation with character χ′, and let ϕ^{r}|_{V_{i}} be represented by the matrix A ∈ GL_{m}(k). Then it is easily checked that D extends to a representation D̃ of W̃ over k by setting

D̃(ϕ) := (\begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & & & ⋮ \\ 0 & 0 & 0 & … & 1 \\ A & 0 & 0 & … & 0 \end{matrix}) .

This gives the desired extension of χ.

3. Existence of 1-regular elements in cosets of reflection groups

In this section we show that cosets of a reflection group in its normalizer always contain so-called regular elements in the sense of Springer. Apart from its intrinsic interest, this result will be used in the next section to construct extended cyclotomic Hecke algebras.

3.1. 1-regular elements in cosets. Let W ≤ GL(V) be a complex reflection group and ϕ ∈ N_{GL(V)}(W). In extension of the terminology of Springer [18] we say that an element wϕ ∈ W ϕ having an eigenvector v not contained in any reflecting hyperplane of W is a regular element of W ϕ with respect to W. If the corresponding eigenvalue has order d we say that wϕ is d-regular and that v is a d-regular eigenvector for W ϕ wrt. W.

We’ll be particularly interested in 1-regular elements. Clearly, the coset W ϕ does not always contain a 1-regular element: for example, take ϕ a scalar matrix whose order is not equal to the order of any element of W.

Note that in dimension 2, a 1-regular element ϕ ≠ 1 is nothing else but a reflection. In this case, ⟨W, ϕ⟩ is again a reflection group.

By an important result of Steinberg (see Lehrer [13] for a short proof) the only 1regular element of W wrt. W is the identity. Thus, if W̃ = ⟨W, ϕ⟩ is again a reflection group, with the same set of reflecting hyperplanes as W, then W ϕ does not contain 1-regular elements with respect to W.

1-regular elements have the following nice property:

Lemma 3.2. Let W ≤ GL(V) be a complex reflection group, ϕ ∈ N_{GL(V)}(W) a 1regular element. Then the extension of W by ⟨ϕ⟩ is split.

Proof. Indeed, let m be minimal with ϕ^{m} ∈ W. Clearly ϕ^{m} is also 1-regular, so ϕ^{m} = 1 by the Theorem of Steinberg cited above.

Apart from this, 1-regular elements also play a crucial role in the construction of extended cyclotomic Hecke algebras in the next section.

We first investigate the existence of 1-regular elements in the irreducible case:

Proposition 3.3. Let W ≤ GL_{n}(ℂ) be an irreducible complex reflection group, ϕ ∈ N_{GL(V)}(W) standard. Then W ϕ contains 1-regular elements with respect to W unless one of

(a) W = G(d, e, n) with d > e > 1, n > 1,

(b) W = G(4, 2, 2), ϕ of order 3,

(c) W = G_{7}, ϕ of order 2. In each of the latter three cases there exists ζ ∈ ℂ^{×} such that W ζϕ contains 1-regular elements.

Proof. Let W̃ := ⟨W, ϕ⟩. We consider the cases in Proposition 2.7 one by one. If ϕ is a reflection whose reflecting hyperplane H is different from those of reflections inside W, then ϕ is the desired 1-regular element, since H cannot be contained in the union of the finitely many reflection hyperplanes H_{s} for s ∈ W. Now note that G(e, e′, n) with e′ < e contains reflections (of order e/e′) outside G(e, e, n), and G_{14} contains reflections of order 2 not contained in G_{5}. This deals with these two cases from Proposition 2.7.

If W is a Weyl group and ϕ stabilizes a set of simple roots of W, then W ϕ contains a 1-regular element by Springer [18, 6.6]. This applies to the cases of G(2, 2, 4) = W (D_{4}) and G_{28} = W (F_{4}).

Finally, for G(3, 3, 3) one verifies by direct computation that the 1-dimensional eigenspace of ϕ in (2.9)(2) for the eigenvalue 1 does not lie in any of the reflecting hyperplanes, so that ϕ is 1-regular.

It remains to consider the cases listed in (a)–(c). For W = G(4, 2, 2) with ϕ as in (2.9)(a), we have that ⟨W, ζ_{3}ϕ⟩ = G_{6}. But the reflecting hyperplane of the reflection ζ_{3}ϕ of G_{6} of order 3 is different from those of reflections (of order 2) in W, so ζ_{3}ϕ is regular with respect to W.

For W = G_{7} we have ⟨W, ϕ⟩ = G_{10}. But d = 8 is a regular number for G_{10}, so ⟨W, ζ_{8}ϕ⟩ contains 1-regular elements. Since W does not contain elements of order 8, these elements must lie in the coset W ϕζ_{8}. (The group ⟨W, ϕζ_{8}⟩ is the complex reflection group G_{15}.)

Finally, consider the imprimitive groups. The group G(d, 1, n) has a presentation given by the Coxeter type diagram

◯_{t}_{d}  ◯_{s_{1}}_{2}  ◯_{s_{2}}_{2} ⋯ ◯_{s_{n−1}}_{2}

where t, s_{1}, …, s_{n−1} are the generating reflections, of order d, 2, …, 2 respectively. (Thus, generators are represented by nodes, a single respectively double edge between r_{1}, r_{2} stands for the relation r_{1}r_{2}r_{1} = r_{2}r_{1}r_{2}, respectively r_{1}r_{2}r_{1}r_{2} = r_{2}r_{1}r_{2}r_{1}, and non-connected generators commute.)

The reflections t^{−1}s_{1}t, s_{1}, …, s_{n−1} generate the normal reflection subgroup G(d, d, n), so G(d, 1, n) is generated by t and also by w := s_{n−1} … s_{2}s_{1}t over G(d, d, n). It is easy to see that w is a dn-regular element of G(d, 1, n), with a regular eigenvector with eigenvalue exp(2πi/dn). Thus ψ := exp(−2πi/dn)w is 1-regular in G(d, 1, n), of order n, and modulo scalars generates G(d, 1, n) over G(d, d, n). A fortiori for every e_{1}|e_{2}|d we have that G(d, e_{1}, n) is generated over G(d, e_{2}, n) modulo scalars by a suitable power of the 1-regular element ψ. This is the required element.

We can now resolve the case of arbitrary W:

Theorem 3.4. Let W ≤ GL(V) be a complex reflection group, ϕ ∈ N_{GL(V)}(W). Then there exist w ∈ W, z ∈ C_{GL(V)}(W) such that zwϕ is 1-regular with respect to W, and thus ⟨W, zϕ⟩ is split.

Proof. If W is irreducible, this is shown in the previous proposition. So now assume that W = W_{1}×…×W_{r} is a direct product of isomorphic reflection subgroups W_{i} ≅ W_{1}, permuted cyclically by ϕ. Let V = V_{1} ⊕ … ⊕ V_{r} be the corresponding decomposition of V such that W_{i} acts trivially on V_{j} for i ≠ j. Then ϕ^{r} normalizes each W_{i} and each V_{i}. Let ϕ_{1} := ϕ^{r}|_{V_{1}}. By the irreducible case there exist w ∈ W_{1}, z ∈ C_{GL(V_{1})}(W_{1}) such that zwϕ_{1}|_{V_{1}} is 1-regular with respect to W_{1}, with regular eigenvector v_{1} ∈ V_{1} say. Then v := v_{1} ⊕ ϕ(v_{1}) ⊕ … ⊕ ϕ^{r−1}(v_{1}) ∈ V is 1-regular for W ϕ.

Finally, if V = V_{1} ⊕V_{2} is a ⟨W, ϕ⟩-invariant direct decomposition, with corresponding ϕ-stable decomposition W = W_{1} × W_{2} into reflection subgroups such that W_{i} is trivial on V_{3−i}, then by induction we may choose z_{i} ∈ C_{GL(V_{i})}(W_{i}) and w_{i} ∈ W_{i} such that z_{i}w_{i}ϕ|_{V_{i}} has a 1-regular eigenvector v_{i} ∈ V_{i} with respect to W_{i} for i = 1, 2. Then v_{1} ⊕v_{2} is 1-regular for W ϕ.

The last assertion is Lemma 3.2.

Clearly, in general there will exist several choices for z ∈ C_{GL(V)}(W) as in the assertion of Theorem 3.4. For example, if ϕ = 1, then z = ζ_{d} will work for all regular numbers d of W. The different possibilities for irreducible W are determined in [3].

4. Cyclic extensions of cyclotomic Hecke algebras

In this section we change our objects of study: from complex reflection groups and their representations we turn to associated braid groups and Hecke algebras.

4.1. Action on braid groups. Cyclotomic Hecke algebras of a complex reflection group W are defined as certain quotients of the group algebra of the braid group attached to W. It thus seems natural to study automorphisms of cyclotomic Hecke algebras via this construction.

Let W be an irreducible complex reflection group on the complex vector space V. For a reflection s ∈ W let H_{s} := ker_{V} (s−1) be the reflecting hyperplane in V. Then the centralizer C_{W} (H_{s}) of H_{s} in W is cyclic, of order e say, and a generator r of C_{W} (H_{s}) with determinant det_{V} (r) = exp(2πi/e) is called a distinguished reflection in W. We write S for the set of all distinguished reflections of W.

We let M := V \ ⋃_{s∈S} H_{s} be the hyperplane complement of W and M̄ := M/W the space of regular orbits of W on V. By definition, the braid group of W is the fundamental group

B(W) := π_{1}(M̄, x̄_{0}),

for a choice of base point x̄_{0} ∈ M̄, see [5]. For each distinguished reflection s ∈ S there is a conjugacy class of so-called generators of the monodromy around H_{s}, or braidreflections s ∈ B(W), see for example [5, 2.A].

Now let ϕ ∈ N_{GL(V)}(W). Then ϕ acts continuously on M, hence on M̄. In particular, it induces an automorphism of π_{1}(M̄, x̄_{0}) if x̄_{0} is ϕ-invariant. Now x̄_{0} is ϕ-invariant if and only if there exists a preimage x_{0} ∈ M and w ∈ W such that x_{0} is wϕ-invariant, that is, if wϕ is 1-regular wrt. W with 1-regular eigenvector x_{0}.

Thus ϕ naturally defines an automorphism σ_{ϕ} of B(W) (which by construction permutes the generators of monodromy) if and only if W ϕ contains a 1-regular element wrt. W. In this case we write B(W, ϕ) for the semidirect product of B(W) by the cyclic group generated by ϕ (acting via σ_{ϕ}).

Let u = (u_{s,j} | s ∈ S, 0 ≤ j ≤ |s| − 1) be a system of indeterminates over ℤ with

u_{s,j} = u_{t,j} if s, t are W -conjugate,

and A := ℤ[u^{±1}]. The cyclotomic Hecke algebra H(W, u) is the quotient

H(W, u) := ℤ[u^{±1}]B(W)/( ∏_{j=0}^{|s|−1}(s − u_{s,j})|s ∈ S)

of the group algebra of B(W) over A modulo the ideal generated by certain monic polynomials in the braid reflections s. More generally, for any commutative ring R and any set of parameters v_{s,j} ∈ R^{×} satisfying v_{s,j} = v_{t,j} if s, t are conjugate in W, there is a natural map A → R, u_{s,j} ↦ v_{s,j}, and we define H_{R}(W, v) := R ⊗ H(W, u).

An automorphism σ_{ϕ} of B(W) induces an automorphism of the cyclotomic Hecke algebra H(W, u) if the parameters u satisfy the condition

∏_{j}(X − u_{s,j}) = ∏_{j}(X − u_{t,j}) whenever s, t are ⟨W, ϕ⟩-conjugate.

From now on, given W and ϕ, we will always assume that the indeterminates u satisfy this (possibly stronger) condition. We’ll then also say that the parameters u are ϕadapted. In this case we obtain an extended cyclotomic Hecke algebra

H(W ϕ, u) := ℤ[u^{±1}]B(W, ϕ)/( ∏_{j=0}^{|s|−1}(s − u_{s,j})|s ∈ S).

Clearly, H(W ϕ, u) is again an associative unital algebra over A. If H(W, u) is free over A of rank m, then so is H(W ϕ, u) with rank m|σ_{ϕ}|. It is the semidirect product of H(W, u) with the cyclic group ⟨σ_{ϕ}⟩ (see [8, II.1.c], for example). The following was put forward in [5]:

Conjecture 4.2. Let W be a finite complex reflection group. Then H(W, u) is free over A of rank |W |.

This conjecture is known to hold for all but finitely many irreducible types and thus also for all W whose irreducible components are not in a certain finite list, see for example [5, Th. 4.24]. By [5] the specialization

(∗)  u_{s,j} ↦ v_{s,j} := exp(2πij/|s|) ∈ ℂ  (s ∈ S, 0 ≤ j ≤ |s| − 1)

gives H_{ℂ}(W, v) = ℂW, the group algebra of W over ℂ. Thus, by Tits’ deformation theorem [7, §64], if W satisfies Conjecture 4.2 then H(W, u) ⊗ L is isomorphic to LW for any splitting field L ≥ A. By construction, H(W ϕ, u) specializes to the group algebra of W ⟨ϕ⟩ under the specialization (*), so if W satisfies Conjecture 4.2 then by the above remark H(W ϕ, u) is isomorphic to LW ⟨ϕ⟩ over any splitting field L ≥ A. In particular, the specialization homomorphism induces a bijection

Irr (H(W ϕ, u)) ⟷^{1−1} Irr (W ⟨ϕ⟩).

We’ll say that a finite complex reflection group W satisfies the freeness property, if the assertion of Conjecture 4.2 holds for W, and thus the above consequences are true for H(W, u) and H(W ϕ, u).

4.3. Choice of 1-regular element. As we have seen in Theorem 3.4 for any ϕ ∈ N_{GL(V)}(W) there exists w ∈ W and z ∈ C_{GL(V)}(W) such that zwϕ is 1-regular, so defines an automorphism of the Hecke algebra H(W, u) (with the ϕ-adapted choice of parameters u). In general, though, there may be various choices of zw (corresponding to the various regular elements in the coset W ϕ). If W is a Weyl group, there is one natural choice for the 1-regular element zwϕ: since all simple systems of reflections are conjugate under W, we may choose w such that wϕ stabilizes such a simple system. Moreover, z may be chosen such that zwϕ also fixes the associated system of simple roots. This yields a canonical 1-regular element up to conjugation.

For general complex reflection groups, there isn’t (yet) a notion of simple system, and correspondingly we are not aware of a canonical choice of 1-regular elements in this case. Instead, in what follows we’ll present a choice of such an element for each irreducible type. With this, we’ll define and investigate the corresponding extended cyclotomic Hecke algebra.

By Corollary 2.8 the non-trivial choices for ϕ can all be realized inside some bigger reflection group of the same dimension, with the sole exception of W (F_{4}). We’ll see that the corresponding extended cyclotomic Hecke algebra can similarly be realized as a subalgebra of the cyclotomic Hecke algebra of that bigger reflection group. This simplifies very much the study of these extended algebras.

4.4. The infinite series. We now look at the non-trivial extensions of irreducible reflection groups in Proposition 2.7 in some more detail. We first consider the imprimitive groups.

(a) The complex reflection group W = G(e, e, 2), e ≥ 4 even, is the Coxeter group of type I_{2}(e). Any reflection ϕ (of order 2) from G(2e, 2e, 2) not lying in G(e, e, 2) induces the non-trivial graph automorphism of W. The extended Hecke algebra in this case was considered in [10, Sect. 7].

(b) Let now W = G(e, e, n), n ≥ 3, e ≥ 2, ϕ a reflection of order e from G(e, 1, n). (For n = 2 this would correspond to a choice of 1-regular element wrt. G(e, e, 2) different from the one in (a).) The braid group B(W) has a presentation given by the Coxeter type diagram

e

s^′_1◯ ⧹

s_1◯ ⧸ ◯_{s2} ◯_{s3} ⋯ ◯ s_{n−1}

with the usual Coxeter relations plus the further relation

(s_{2}s_{1}s′_{1})^{2} = (s_{1}s′_{1}s_{2})^{2},

(see [5, Table 5]). Hence the cyclotomic Hecke algebra H(W, u) is generated by elements T_{s_{1}}, T_{s′_{1}}, T_{s_{2}}, …, T_{s_{n−1}} subject to the braid relations above and the further relations

(T_{r} − u_{1})(T_{r} − u_{2}) = 0 for r = s_{1}, s′_{1}, s_{2}, …, s_{n−1}.

The induced automorphism σ := σ_{ϕ} fixes s_{2}, …, s_{n−1} and satisfies

σ(s_{1}) = (s_{2}s_{1}s′_{1}s_{2})s_{1}(s_{2}s_{1}s′_{1}s_{2})^{−1}, σ(s′_{1}) = s_{1}.

It follows that the extended cyclotomic Hecke algebra H(W ϕ, u) is generated by elements T_{s_{1}}, T_{s′_{1}}, T_{s_{2}}, …, T_{s_{n−1}}, T_{σ} subject to

(T_{r} − u_{1})(T_{r} − u_{2}) = T_{σ}^{e} − 1 = 0.

This is just a specialization of the cyclotomic Hecke algebra of the complex reflection group G(e, 1, n),

(c) Let now W = G(d, e, n), n ≥ 2, 1 < e < d, and ϕ the 1-regular element constructed in the proof of Proposition 3.3. Then ϕ induces an automorphism σ_{ϕ} (of order n) of B(G(d, e, n)), but indeed on B(G(d, e′, n)) for all e′|d. To describe σ_{ϕ} it suffices to give its action on the largest group, on B(G(d, 1, n)). The latter has a presentation given by the Coxeter type diagram

◯_{t}  ◯_{s_{1}}  ◯_{s_{2}} ⋯ ◯_{s_{n−1}}

with the usual Coxeter relations (see [5, Table 5]). The induced automorphism σ := σ_{ϕ} acts by conjugation with (s_{n−1} ⋯ s_{2}s_{1}t)^{−1} (the nth power of which acts indeed trivially by [5, Table 5]). So for example

σ(s_{j}) = s_{j−1} for 2 ≤ j ≤ n − 1.

It induces an automorphism of the Hecke algebra H(G(d, 1, n), u) generated by elements T, S_{1}, …, S_{n−1} subject to the braid relations implied by the diagram above and the further relations

(T − u_{1,0}) ⋯ (T − u_{1,d−1}) = (S_{j} − u_{2,0})(S_{j} − u_{2,1}) = 0 (1 ≤ j ≤ n − 1).

It follows that the extended cyclotomic Hecke algebra H(G(d, 1, n)ϕ, u) is generated by elements T, S_{1}, …, S_{n−1}, T_{σ}, where T_{σ} acts by conjugation with (S_{n−1} ⋯ S_{2}S_{1}T)^{−1} and

T_{σ}^{n} = 1.

4.5. The primitive groups. We now consider the non-trivial automorphisms of primitive groups.

(a) Let first W = G(4, 2, 2), ϕ as in (2.9)(a). The braid group B(W) has a presentation as

B(W) = ⟨r, s, t | rst = str = trs⟩

(see [5, Table 2]). It is easily verified from the defining relations that ϕ lifts to an automorphism σ_{ϕ} of B(W) of order 3 cyclically permuting r, s, t. This induces an automorphism of the Hecke algebra H(W, u) if the parameter sets at the three braid reflections r, s, t are identical, thus on the free associative ℤ[u_{1}^{±1}, u_{2}^{±1}]-algebra, generated by elements T_{r}, T_{s}, T_{t} subject to the relations

\begin{matrix}T_{r}T_{s}T_{t} = T_{s}T_{t}T_{r} = T_{t}T_{r}T_{s}, \\ (T_{r} − u_{1})(T_{r} − u_{2}) = (T_{s} − u_{1})(T_{s} − u_{2}) = (T_{t} − u_{1})(T_{t} − u_{2}) = 0. \end{matrix}

The induced action of σ_{ϕ} permutes the three generators T_{r}, T_{s}, T_{t} cyclically. It follows that the extended cyclotomic Hecke algebra H(W ϕ, u) is generated by T_{r}, T_{σ} satisfying

(T_{r}T_{σ})^{3} = (T_{σ}T_{r})^{3}, (T_{r} − u_{1})(T_{r} − u_{2}) = T_{σ}^{3} − 1 = 0.

This is just a specialization of the cyclotomic Hecke algebra of the complex reflection group G_{6}.

(b) Let now W = G(3, 3, 3), ϕ as in (2.9)(b). The braid group B(W) has a presentation as

B(W) = ⟨r, s, t | rsr = srs, rtr = trt, sts = tst, (rst)^{2} = (str)^{2}⟩

(see [5, Table 5]). It can be checked from the defining relations that ϕ lifts to an automorphism σ := σ_{ϕ} of B(W) of order 4 with orbits

σ : r ↦ s ↦ r^{st} ↦ r^{t}, σ : t ↦ r^{s} ↦ s^{tr^{−1}} ↦ s^{t}

on the generating reflections. It induces an automorphism of the Hecke algebra H(W, u) generated by elements T_{r}, T_{s}, T_{t} subject to the braid relations from B(W) above plus

(T_{r} − u_{1})(T_{r} − u_{2}) = (T_{s} − u_{1})(T_{s} − u_{2}) = (T_{t} − u_{1})(T_{t} − u_{2}) = 0.

It follows that the extended cyclotomic Hecke algebra H(W ϕ, u) is generated by elements T_{r}, T_{s}, T_{t}, T_{σ} subject to

T_{r}T_{s}T_{r} = T_{s}T_{r}T_{s}, T_{r}T_{t}T_{r} = T_{t}T_{r}T_{t}, T_{s}T_{t}T_{s} = T_{t}T_{s}T_{t}, (T_{r}T_{s}T_{t})^{2} = (T_{s}T_{t}T_{r})^{2}

T_{σ}T_{r}T_{σ}^{−1} = T_{s}, T_{σ}T_{s}T_{σ}^{−1} = T_{t}^{−1}T_{s}^{−1}T_{r}T_{s}T_{t}, T_{σ}T_{t}T_{σ}^{−1} = T_{s}^{−1}T_{r}T_{s},

(T_{r} − u_{1})(T_{r} − u_{2}) = T_{σ}^{4} − 1 = 0.

It can be checked that H(W ϕ, u) is isomorphic to the free subalgebra of the Hecke algebra H(W̃ , v) of the complex reflection group W̃ = G_{26}, with presentation given by the Coxeter type diagram

◯_{T_{r}}  ◯_{T_{2}}  ◯_{T_{3}}

and the relations

(T_{r} − u_{1})(T_{r} − u_{2}) = T_{2}^{3} − 1 = T_{3}^{3} − 1 = 0,

generated by T_{r}, T_{s} = T_{2}T_{r}T_{2}^{−1}, T_{t} = T_{3}T_{2}T_{r}T_{2}^{−1}T_{3}^{−1}, and T_{σ} = T_{2}T_{3}^{−1}.

(c) Let now W = G_{5}, ϕ a reflection of order 2 from G_{14}. The braid group B(W) has a presentation as

B(W) = ⟨r, s | rsrs = srsr⟩

(see [5, Table 1]). It can be checked that ϕ lifts to an automorphism σ_{ϕ} of B(W) of order 2 interchanging r, s. This induces an automorphism of the Hecke algebra H(W, u) if the parameter sets at the two braid reflections r, s are identical, thus on the free associative ℤ[u^{±1}]-algebra, generated by elements T_{r}, T_{s} subject to the relations

\begin{matrix}T_{r}T_{s}T_{r}T_{s} = T_{s}T_{r}T_{s}T_{r}, \\ (T_{r} − u_{1})(T_{r} − u_{2})(T_{r} − u_{3}) = (T_{s} − u_{1})(T_{s} − u_{2})(T_{s} − u_{3}) = 0. \end{matrix}

The induced action of σ_{ϕ} permutes the two generators T_{r}, T_{s}. It follows that the extended cyclotomic Hecke algebra H(W ϕ, u) is generated by T_{r}, T_{σ} subject to

(T_{r}T_{σ})^{4} = (T_{σ}T_{r})^{4}, (T_{r} − u_{1})(T_{r} − u_{2})(T_{r} − u_{3}) = T_{σ}^{2} − 1 = 0.

This is just a specialization of the cyclotomic Hecke algebra of the complex reflection group G_{14}.

(d) Let now W = G_{7}, ϕ a reflection of order 4 from G_{10} (or, what amounts to the same modulo inner elements and scalars, a reflection of order 2 from G_{15}). The braid group B(W) has a presentation as

B(W) = ⟨r, s, t | rst = str = trs⟩

where s, t come from reflections of order 3 (see [5, Table 2]). Then ϕ lifts to an automorphism σ := σ_{ϕ} of B(W) of order 2 acting by

σ(r) = trt^{−1}, σ(s) = t, σ(t) = s.

It induces an automorphism of the Hecke algebra H(W, u) if the parameter sets at the two braid reflections s, t are identical, thus on the free associative ℤ[u^{±1}]-algebra, generated by elements T_{r}, T_{s}, T_{t} subject to the relations

\begin{matrix}T_{r}T_{s}T_{t} = T_{s}T_{t}T_{r} = T_{t}T_{r}T_{s}, (T_{r} − u_{1})(T_{r} − u_{2}) = 0, \\ (T_{s} − v_{1})(T_{s} − v_{2})(T_{s} − v_{3}) = (T_{t} − v_{1})(T_{t} − v_{2})(T_{t} − v_{3}) = 0. \end{matrix}

It follows that the extended cyclotomic Hecke algebra H(W ϕ, u) is generated by T_{r}, T_{t}, T_{σ} subject to

\begin{matrix}T_{r}T_{σ}T_{t} = T_{σ}T_{t}T_{r}, T_{t}T_{r}T_{σ}T_{t}T_{σ} = T_{r}T_{σ}T_{t}T_{σ}T_{t}, \\ (T_{r} − u_{1})(T_{r} − u_{2}) = (T_{t} − v_{1})(T_{t} − v_{2})(T_{t} − v_{3}) = T_{σ}^{2} − 1 = 0. \end{matrix}

This is just a specialization of the cyclotomic Hecke algebra of the complex reflection group G_{15}.

(e) In the case that W is one of the Weyl groups of type D_{4} or F_{4} and ϕ is induced by the exceptional graph automorphism, the corresponding extended Hecke algebras have been described and studied in [10, Sect. 7], for example. For W = W (D_{4}) the extended Hecke algebra can be realized as a free subalgebra of an Iwahori-Hecke algebra H(W (F_{4}), v) for the Weyl group of type F_{4}.

4.6. Automorphisms induced by roots of π. Let W ≤ W̃ ≤ GL(V) be two finite irreducible reflection groups on V having the same set of reflecting hyperplanes. Then W is normal in W̃ and the braid group B(W) is a normal subgroup of B(W̃) of index

(B(W̃) : B(W)) = (W̃ : W).

For each reflecting hyperplane H, the generator of monodromy s of B(W) around H is a power of the generator of monodromy s̃ of B(W̃) around H. We specialize to the situation that there is a single conjugacy class of hyperplanes for which the generators of monodromy differ, with s = s̃^{e}, say. Then W̃ /W is cyclic of order e. (In fact, by the classification of irreducible reflection groups, this is the only case which can arise.)

An example of this situation is given by G(d, e_{1}, n) ≤ G(d, e_{2}, n) with e_{2}|e_{1}|d, e_{1} < d. Here e = e_{1}/e_{2}. Another example is G_{7} ≤ G_{10}, where e = 2.

Let π ∈ B(W̃) = π_{1}(M̄, x̄_{0}) denote the homotopy class of the image of the path

[0, 1] ⟶ M, t ↦ x_{0} exp(2πit),

in M (see [5, 2.3]), a central element of B(W̃) with trivial image in W. Let ϕ be an lth root of π whose image ϕ ∈ W̃ is l-regular and assume that W̃ = ⟨W, ϕ⟩. Thus ϕ ∈ N_{GL(V)}(W) as before, and clearly B(W̃) = ⟨B(W), ϕ⟩.

Let u be ϕ-adapted parameters for W. Let T denote the image of s in H(W, u). Then T satisfies

(T − u_{s,0}) ⋯ (T − u_{s,|s|−1}) = 0.

Consider the Hecke algebra of W̃ with the same set of parameters u. Then the image T̃ of s̃ in H(W̃ , u) satisfies

(T̃^{e} − u_{s,0}) ⋯ (T̃^{e} − u_{s,|s|−1}) = 0.

Let T_{ϕ} denote the image of ϕ in H(W̃ , u). Then the value of χ ∈ Irr (H(W̃ , u)) on T_{ϕ} can be computed by [4, Prop. 1.23]: the quotient χ(T_{ϕ})/χ(ϕ) (where χ(ϕ) denotes the value of the specialized character of W̃ on ϕ) is a monomial in certain roots of the parameters.

Now let σ = σ_{ϕ} denote the inner automorphism of B(W̃) induced by conjugation with ϕ, and also the automorphism induced on B(W), on H(W̃ , u) and on H(W, u). The extended cyclotomic Hecke algebra H(W̃ ϕ, u) then contains H(W ϕ, u) as an A-free subalgebra.

Lemma 4.7. In the above situation, let χ be a σ_{ϕ}-invariant character of H(W, u), w ∈ B(W) and T_{w} its image in H(W, u). Then χ extends to a character χ̃ of H(W̃ ϕ, u) and

χ̃(T_{w}T_{σ}) = χ̃(T_{ϕ}^{−1}T_{σ})χ̃(T_{w}T_{ϕ})/χ(1).

Proof. By construction, T_{σ}^{−1}T_{ϕ} is central in H(W̃ ϕ, u), and (T_{σ}^{−1}T_{ϕ})^{l} = T_{ϕ}^{l} = T_{π}, the image of π. The σ_{ϕ}-invariant character χ extends to H(W ϕ, u). Since T_{σ}^{−1}T_{ϕ} acts trivially on H(W ϕ, u), χ further extends to an irreducible character χ̃ of H(W̃ ϕ, u).

The central element T_{ϕ}^{−1}T_{σ} acts as the scalar c := χ̃(T_{ϕ}^{−1}T_{σ})/χ(1) in any representation affording χ̃. We obtain the desired character formula

χ̃(T_{w}T_{σ}) = χ̃(T_{w}T_{ϕ}T_{ϕ}^{−1}T_{σ}) = cχ̃(T_{w}T_{ϕ}) = χ̃(T_{ϕ}^{−1}T_{σ})χ̃(T_{w}T_{ϕ})/χ(1).

This Lemma applies in particular to the situation studied in (4.4)(c).

In order to study splitting fields for extended cyclotomic Hecke algebras we need the analogue of Proposition 2.3 in this situation. Extendability of characters will always be clear in our context due to the isomorphism to the corresponding group algebras. The second assertion of Proposition 2.3 generalizes as follows (see [7, §74], for example):

Proposition 4.8. Let H be a finite dimensional algebra over a field K_{0} and let B be a basis of H. Let K/K_{0} be an extension field such that H_{K} := H⊗_{K_{0}} K is split semisimple. Let V be a simple H_{K}-module with character χ such that χ(b) ∈ K_{0} for all b ∈ B and assume that there exists an H-module V′ such that V has multiplicity 1 in V′ ⊗_{K_{0}} K. Then V is realizable over K_{0}.

We now determine a splitting field for extended cyclotomic Hecke algebras for irreducible reflection groups.

Proposition 4.9. Let W ≤ GL(V) be an irreducible complex reflection group with the freeness property, ϕ ∈ N_{GL(V)}(W) as in (4.4) or (4.5), k the character field of W ϕ on V. Let H(W, u) be the Hecke algebra of W with ϕ-adapted parameters, H(W ϕ, u) the corresponding extended Hecke algebra. Then every σ_{ϕ}-stable character of H(W, u) has an extension to H(W ϕ, u) realizable over

K_{W ϕ} := k( (exp(−2πij/|s|)u_{s,j})^{1/|μ(k)|} | s ∈ S, 0 ≤ j ≤ |s| − 1),

where μ(k) denotes the group of roots of unity in k.

Proof. We use a case-by-case analysis. In (4.4)(a) and (b) and in (4.5)(a), (c), (d) we identified H(W ϕ, u) with a cyclotomic Hecke algebra of a larger reflection group. In this case the assertion is just [16, Th. 5.2]. Let’s now consider (4.5)(b) with W = G(3, 3, 3). We showed that H(W, u) is a free subalgebra of the cyclotomic Hecke algebra H(G_{26}, v) for G_{26}. It can easily be checked that all ϕ-stable irreducible characters of W extend to G_{26}, hence by the consequence of the freeness property all σ_{ϕ}-stable irreducible characters of H(W, u) extend to H(G_{26}, v). By [16, Th. 5.2] all irreducible characters of H(G_{26}, v) can be realized over K_{W ϕ}. Restriction to H(W ϕ, u) proves the claim in this case. The same argument works for the Weyl group W = W (D_{4}) in (4.5)(e) since again all σ_{ϕ}-stable irreducible characters of H(W, u) extend to H(W (F_{4}), v), which contains H(W ϕ, u) as a free subalgebra. For W = W (F_{4}) in (4.5)(e) it was shown in [10, Table VI] that all character values of H(W ϕ, u) lie in K_{W ϕ}. The analogue of Proposition 2.3 yields the result.

Let finally W = G(de, e, n) and ϕ, T_{σ} as in (4.4)(c). Then the character formula in Lemma 4.7 applies. But all factors on the right hand side are contained in K_{W ϕ}. This shows the claim. (This argument also applies to the case of G_{7} in (4.5)(d).)

4.10. Hecke algebras for reducible groups. Now let W ≤ GL(V) be an arbitrary finite complex reflection group. Let W = W_{1} × … × W_{m} be its decomposition into irreducible reflection subgroups. The braid group of W is then by definition the direct product

B(W) := B(W_{1}) × … × B(W_{m})

of the braid groups of the irreducible components.

Let ϕ ∈ N_{GL(V)}(W) be 1-regular with respect to W of finite order. Then ϕ permutes the components W_{1}, …, W_{m}. First assume that ϕ acts transitively. Clearly m is minimal with the property that ϕ^{m} stabilizes W_{1} (hence it stabilizes each component W_{j}). Then ϕ induces an automorphism σ_{ϕ} of B(W) which cyclically permutes the isomorphic factors B(W_{j}) and such that σ_{ϕ}^{m} acts as σ_{ϕ^{m}} on each B(W_{j}). (Note that ϕ^{m} is also 1-regular, so that σ_{ϕ^{m}} is defined.)

Let u_{1} be a ϕ^{m}-adapted set of parameters for W_{1}. Then u_{1} is also ϕ^{m}-adapted for all W_{j}. The Hecke algebra of W with ϕ-adapted parameters u = (u_{1}, …, u_{1}) is then the product

H(W, u) := ⨂_{j=1}^{m} H(W_{j}, u_{1}).

As before, the extended Hecke algebra H(W ϕ, u) of (W, ϕ) is defined to be the semidirect product of H(W, u) by the automorphism induced by σ_{ϕ} on H(W, u).

Now assume that W = W_{1}×W_{2} is a decomposition into ϕ-stable reflection subgroups, with corresponding W ⟨ϕ⟩-invariant decomposition V = V_{1} ⊕ V_{2}. We let ϕ_{j} := ϕ|_{V_{j}}, j = 1, 2. Then ϕ induces an automorphism σ_{ϕ} on B(W) by acting as σ_{ϕ_{j}} on B(W_{j}). Let u_{j} be a ϕ_{j}-adapted set of parameters for W_{j}, j = 1, 2. The Hecke algebra of W with ϕ-adapted parameters u = (u_{1}, u_{2}) is then the product

H(W, u) := H(W_{1}, u_{1}) ⊗ H(W_{2}, u_{2}),

and the extended Hecke algebra H(W ϕ, u) of (W, ϕ) is the semidirect product of H(W, u) with the automorphism induced by σ_{ϕ} on H(W, u).

Inductively this gives a definition of the extended Hecke algebra H(W ϕ, u) with ϕ-adapted set of parameters u for any W and any 1-regular ϕ ∈ N_{GL(V)}(W).

We can now prove that a splitting field for the extended cyclotomic Hecke algebra H(W ϕ, u) can be constructed from the character field of W ϕ and a splitting field for H(W, u):

Theorem 4.11. Let W ≤ GL(V) be a complex reflection group satisfying the freeness property and ϕ′ ∈ N_{GL(V)}(W). Then there exist w ∈ W and z ∈ C_{GL(V)}(W) such that ϕ := zwϕ′ is 1-regular and the following holds:

Let k be the character field of W ϕ on V, H(W, u) the Hecke algebra of W with ϕadapted parameters, H(W ϕ, u) the corresponding extended Hecke algebra. Then every σ_{ϕ}-stable character of H(W, u) has an extension to H(W ϕ, u) realizable over

K_{W ϕ} := k( (exp(−2πij/|s|)u_{s,j})^{1/|μ(k)|} | s ∈ S, 0 ≤ j ≤ |s| − 1),

where μ(k) denotes the group of roots of unity in k.

Proof. The case of irreducible groups was treated in Proposition 4.9. The reduction of the general case to this one is done completely analogous as in the proof of Theorem 2.16, using Proposition 4.8 instead of Proposition 2.3, and Theorem 3.4.

Added in proof: The author is grateful to the referee for drawing his attention to the recent draft preprint [3] which also contains a proof of Theorem 3.4 involving less case-by-case checking by reduction to certain minimal situations.

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