Complex Reflection Groups and Their Associated Braid Groups and Hecke

Complex reﬂection groups and their associated braid groups and Hecke algebras

Michel Brou´e

Institut Henri–Poincar´e

January 2009

Michel Broué Reflection groups, braids, Hecke algebras Each choice of a basis (v1, v2,..., vr ) of V determines an identification of S with a graded polynomial algebra

S ' K[v1, v2,..., vr ] with deg vi = 1 .

Let G be a ﬁnite subgroup of GL(V ). The group G acts on the algebra S, and we let R := SG denote the subalgebra of G–ﬁxed polynomials.

Let K be a characteristic zero ﬁeld and let V be an r–dimensional K–vector space. Let S be the symmetric algebra of V .

Polynomial invariants of ﬁnite linear groups

Michel Broué Reflection groups, braids, Hecke algebras Let G be a finite subgroup of GL(V ). The group G acts on the algebra S, and we let R := SG denote the subalgebra of G–fixed polynomials.

Each choice of a basis (v1, v2,..., vr ) of V determines an identiﬁcation of S with a graded polynomial algebra

S ' K[v1, v2,..., vr ] with deg vi = 1 .

Polynomial invariants of ﬁnite linear groups

Let K be a characteristic zero ﬁeld and let V be an r–dimensional K–vector space. Let S be the symmetric algebra of V .

Michel Broué Reflection groups, braids, Hecke algebras Let G be a finite subgroup of GL(V ). The group G acts on the algebra S, and we let R := SG denote the subalgebra of G–fixed polynomials.

Polynomial invariants of ﬁnite linear groups

Let K be a characteristic zero ﬁeld and let V be an r–dimensional K–vector space. Let S be the symmetric algebra of V . Each choice of a basis (v1, v2,..., vr ) of V determines an identiﬁcation of S with a graded polynomial algebra

S ' K[v1, v2,..., vr ] with deg vi = 1 .

Michel Broué Reflection groups, braids, Hecke algebras Polynomial invariants of finite linear groups

S ' K[v1, v2,..., vr ] with deg vi = 1 .

Let G be a ﬁnite subgroup of GL(V ). The group G acts on the algebra S, and we let R := SG denote the subalgebra of G–ﬁxed polynomials.

Michel Brou´e Reﬂection groups, braids, Hecke algebras S = K[v1, v2,..., vr ] MMM MMMnot free unless... MMM MMM free of rank m|G| R = SG o not a polynomial algebra unless... qqq qqq qqqfree of rank m qqq P = K[u1, u2,..., ur ]

but there exists a graded polynomial algebra

P := K[u1, u2,..., ur ] with deg ui = di

and an integer m, such that

In general R is NOT a polynomial algebra,

and an integer m, such that

In general R is NOT a polynomial algebra, but there exists a graded polynomial algebra

P := K[u1, u2,..., ur ] with deg ui = di

In general R is NOT a polynomial algebra, but there exists a graded polynomial algebra

P := K[u1, u2,..., ur ] with deg ui = di and an integer m, such that

In general R is NOT a polynomial algebra, but there exists a graded polynomial algebra

P := K[u1, u2,..., ur ] with deg ui = di and an integer m, such that

S = K[v1, v2,..., vr ] MMM MMM MMM MMM free of rank m|G| R = SG qqq qqq qqqfree of rank m qqq P = K[u1, u2,..., ur ]

Michel Brou´e Reﬂection groups, braids, Hecke algebras In general R is NOT a polynomial algebra, but there exists a graded polynomial algebra

P := K[u1, u2,..., ur ] with deg ui = di and an integer m, such that

S = K[v1, v2,..., vr ] MMM MMMnot free unless... MMM MMM free of rank m|G| R = SG o not a polynomial algebra unless... qqq qqq qqqfree of rank m qqq P = K[u1, u2,..., ur ]

Michel Brou´e Reﬂection groups, braids, Hecke algebras Then

S = K[x, y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2, y 2] ⊕ K[x2, y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2, y 2]

1 m|G| = d1d2 ··· dr m 2 As a PG–module, we have S ' (PG) .

Example. 1 0 −1 0 Consider G = , ⊂ GL (K) . 0 1 0 −1 2 Denote by (x, y) the canonical basis of V = K 2.

Moreover,

Michel Brou´e Reﬂection groups, braids, Hecke algebras Then

S = K[x, y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2, y 2] ⊕ K[x2, y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2, y 2]

m 2 As a PG–module, we have S ' (PG) .

Example. 1 0 −1 0 Consider G = , ⊂ GL (K) . 0 1 0 −1 2 Denote by (x, y) the canonical basis of V = K 2.

Moreover,

1 m|G| = d1d2 ··· dr

Michel Brou´e Reﬂection groups, braids, Hecke algebras Then

S = K[x, y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2, y 2] ⊕ K[x2, y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2, y 2]

Example. 1 0 −1 0 Consider G = , ⊂ GL (K) . 0 1 0 −1 2 Denote by (x, y) the canonical basis of V = K 2.

Moreover,

1 m|G| = d1d2 ··· dr m 2 As a PG–module, we have S ' (PG) .

Michel Brou´e Reﬂection groups, braids, Hecke algebras Then

S = K[x, y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2, y 2] ⊕ K[x2, y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2, y 2]

1 0 −1 0 Consider G = , ⊂ GL (K) . 0 1 0 −1 2 Denote by (x, y) the canonical basis of V = K 2.

Moreover,

1 m|G| = d1d2 ··· dr m 2 As a PG–module, we have S ' (PG) .

Example.

Michel Brou´e Reﬂection groups, braids, Hecke algebras Then

S = K[x, y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2, y 2] ⊕ K[x2, y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2, y 2]

Denote by (x, y) the canonical basis of V = K 2.

Moreover,

1 m|G| = d1d2 ··· dr m 2 As a PG–module, we have S ' (PG) .

Example. 1 0 −1 0 Consider G = , ⊂ GL (K) . 0 1 0 −1 2

Michel Brou´e Reﬂection groups, braids, Hecke algebras Then

S = K[x, y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2, y 2] ⊕ K[x2, y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2, y 2]

Moreover,

1 m|G| = d1d2 ··· dr m 2 As a PG–module, we have S ' (PG) .

Example. 1 0 −1 0 Consider G = , ⊂ GL (K) . 0 1 0 −1 2 Denote by (x, y) the canonical basis of V = K 2.

Michel Brou´e Reﬂection groups, braids, Hecke algebras S = K[x, y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2, y 2] ⊕ K[x2, y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2, y 2]

Moreover,

1 m|G| = d1d2 ··· dr m 2 As a PG–module, we have S ' (PG) .

Example. 1 0 −1 0 Consider G = , ⊂ GL (K) . 0 1 0 −1 2 Denote by (x, y) the canonical basis of V = K 2. Then

Michel Brou´e Reﬂection groups, braids, Hecke algebras Moreover,

1 m|G| = d1d2 ··· dr m 2 As a PG–module, we have S ' (PG) .

Example. 1 0 −1 0 Consider G = , ⊂ GL (K) . 0 1 0 −1 2 Denote by (x, y) the canonical basis of V = K 2. Then

S = K[x, y] UU UUUU not free UUUU UUUU free of rank 4 R = SG = K[x2, y 2] ⊕ K[x2, y 2]xy ii iiii iiii iiii free of rank 2 P = K[x2, y 2]

Michel Broué Reflection groups, braids, Hecke algebras A finite reflection group (abbreviated frg) on K is a finite subgroup of GLK (V )(V a finite dimensional K–vector space) generated by reflections, i.e., linear maps represented by

ζ 0 ··· 0 0 1 ··· 0   . . .. . . . . . 0 0 ··· 1

A ﬁnite reﬂection group on R is called a Coxeter group.

A ﬁnite reﬂection group on Q is called a Weyl group.

Unless...

Michel Broué Reflection groups, braids, Hecke algebras A finite reflection group on R is called a Coxeter group.

A ﬁnite reﬂection group on Q is called a Weyl group.

Unless...

A finite reflection group (abbreviated frg) on K is a finite subgroup of GLK (V )(V a finite dimensional K–vector space) generated by reflections, i.e., linear maps represented by

ζ 0 ··· 0 0 1 ··· 0   . . .. . . . . . 0 0 ··· 1

Michel Broué Reflection groups, braids, Hecke algebras A finite reflection group on Q is called a Weyl group.

Unless...

A finite reflection group (abbreviated frg) on K is a finite subgroup of GLK (V )(V a finite dimensional K–vector space) generated by reflections, i.e., linear maps represented by

ζ 0 ··· 0 0 1 ··· 0   . . .. . . . . . 0 0 ··· 1

A ﬁnite reﬂection group on R is called a Coxeter group.

Michel Brou´e Reﬂection groups, braids, Hecke algebras Unless...

A finite reflection group (abbreviated frg) on K is a finite subgroup of GLK (V )(V a finite dimensional K–vector space) generated by reflections, i.e., linear maps represented by

ζ 0 ··· 0 0 1 ··· 0   . . .. . . . . . 0 0 ··· 1

A ﬁnite reﬂection group on R is called a Coxeter group.

A ﬁnite reﬂection group on Q is called a Weyl group.

Michel Broué Reflection groups, braids, Hecke algebras 1 G is generated by reflections. G 2 The ring R := S of G–fixed polynomials is a polynomial ring K[u1, u2,..., ur ] in r homogeneous algebraically independant elements. 3 S is a free R–module.

In other words, unless... m = 1, i.e., R = P.

Main characterisation

Michel Brou´e Reﬂection groups, braids, Hecke algebras m = 1, i.e., R = P.

1 G is generated by reﬂections. G 2 The ring R := S of G–ﬁxed polynomials is a polynomial ring K[u1, u2,..., ur ] in r homogeneous algebraically independant elements. 3 S is a free R–module.

In other words, unless...

Main characterisation

Michel Brou´e Reﬂection groups, braids, Hecke algebras m = 1, i.e., R = P.

G 2 The ring R := S of G–ﬁxed polynomials is a polynomial ring K[u1, u2,..., ur ] in r homogeneous algebraically independant elements. 3 S is a free R–module.

In other words, unless...

Main characterisation

Michel Brou´e Reﬂection groups, braids, Hecke algebras m = 1, i.e., R = P.

3 S is a free R–module.

In other words, unless...

Main characterisation

Theorem (Shephard–Todd, Chevalley–Serre) Let G be a finite subgroup of GL(V )(V an r–dimensional vector space over a characteristic zero field K). Let S denote the symmetric algebra of V , isomorphic to the polynomial ring K[v1, v2,..., vr ]. The following assertions are equivalent. 1 G is generated by reflections. G 2 The ring R := S of G–fixed polynomials is a polynomial ring K[u1, u2,..., ur ] in r homogeneous algebraically independant elements.

Michel Brou´e Reﬂection groups, braids, Hecke algebras m = 1, i.e., R = P.

In other words, unless...

Main characterisation

Michel Brou´e Reﬂection groups, braids, Hecke algebras m = 1, i.e., R = P.

Main characterisation

In other words, unless...

Michel Brou´e Reﬂection groups, braids, Hecke algebras Main characterisation

In other words, unless... m = 1, i.e., R = P.

Michel Brou´e Reﬂection groups, braids, Hecke algebras S = K[v1, v2,..., vr ]

free of rank |G|

G R = S = P = K[u1, u2,..., ur ]

S = K[v1, v2,..., vr ] KKK KKK KKK KKK free of rank m|G| R = SG sss sss ssfrees of rank m sss P = K[u1, u2,..., ur ] becomes

Michel Brou´e Reﬂection groups, braids, Hecke algebras S = K[v1, v2,..., vr ] KKK KKK KKK KKK free of rank m|G| R = SG sss sss ssfrees of rank m sss P = K[u1, u2,..., ur ] becomes

S = K[v1, v2,..., vr ]

free of rank |G|

G R = S = P = K[u1, u2,..., ur ]

Michel Broué Reflection groups, braids, Hecke algebras Fix d ≥ 2. For each coordinate consider the reflection vi 7→ ζd vi . We obtain the wreath product Cd o Sr , generated by reflections. This group is called G(d, 1, r). For each divisor e of d, there is a normal reflection subgroup G(d, e, r) of G(d, 1, r) of index e.

Examples

For G = he2πi/d i, cyclic group of order d acting by multiplication on V = C, we have S = K[x] and R = K[xd ] . r L Sr acts naturally on V = C = Cvi .

Examples

For G = Sr , one may choose  u = v + ··· + v  1 1 r   u2 = v1v2 + v1v3 + ··· + vr−1vr . .  . .   ur = v1v2 ··· vr

r L Sr acts naturally on V = C = Cvi .

Examples

Michel Broué Reflection groups, braids, Hecke algebras For each coordinate consider the reflection vi 7→ ζd vi . We obtain the wreath product Cd o Sr , generated by reflections. This group is called G(d, 1, r). For each divisor e of d, there is a normal reflection subgroup G(d, e, r) of G(d, 1, r) of index e.

Examples

Michel Broué Reflection groups, braids, Hecke algebras We obtain the wreath product Cd o Sr , generated by reflections. This group is called G(d, 1, r). For each divisor e of d, there is a normal reflection subgroup G(d, e, r) of G(d, 1, r) of index e.

Examples

Michel Broué Reflection groups, braids, Hecke algebras This group is called G(d, 1, r). For each divisor e of d, there is a normal reflection subgroup G(d, e, r) of G(d, 1, r) of index e.

Examples

For G = Sr , one may choose  u = v + ··· + v  1 1 r   u2 = v1v2 + v1v3 + ··· + vr−1vr . .  . .   ur = v1v2 ··· vr For G = he2πi/d i, cyclic group of order d acting by multiplication on V = C, we have S = K[x] and R = K[xd ] . r L Sr acts naturally on V = C = Cvi . Fix d ≥ 2. For each coordinate consider the reﬂection vi 7→ ζd vi . We obtain the wreath product Cd o Sr , generated by reﬂections.

Michel Broué Reflection groups, braids, Hecke algebras For each divisor e of d, there is a normal reflection subgroup G(d, e, r) of G(d, 1, r) of index e.

Examples

Michel Brou´e Reﬂection groups, braids, Hecke algebras Examples

Michel Brou´e Reﬂection groups, braids, Hecke algebras −1 −1 So, if G = hg1,..., gr i, the group hζ1 g1, . . . , ζr gr i is an frg. Note that for G irreducible, we have G/Z(G) ∈ {Dn, A4, S4, A5}. For example, the group √ 2 1 0 −3 −ζ3 ζ3 G := , 2 ≤ GL2(Q(ζ3)) , 0 ζ3 3 2ζ3 1

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

Let ζ be an eigenvalue of g. Then ζ−1g is a reﬂection.

If g ∈ SL3(C) is an involution, then −g is a reﬂection. Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

Let G ≤ SL2(C) be ﬁnite and g ∈ G.

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

If g ∈ SL3(C) is an involution, then −g is a reﬂection. Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

Let ζ be an eigenvalue of g. Then ζ−1g is a reﬂection.

Let G ≤ SL2(C) be ﬁnite and g ∈ G.

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

If g ∈ SL3(C) is an involution, then −g is a reﬂection. Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

Then ζ−1g is a reﬂection.

Let G ≤ SL2(C) be ﬁnite and g ∈ G. Let ζ be an eigenvalue of g.

Michel Brou´e Reﬂection groups, braids, Hecke algebras with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

If g ∈ SL3(C) is an involution, then −g is a reﬂection. Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

−1 −1 So, if G = hg1,..., gr i, the group hζ1 g1, . . . , ζr gr i is an frg. Note that for G irreducible, we have G/Z(G) ∈ {Dn, A4, S4, A5}. For example, the group √ 2 1 0 −3 −ζ3 ζ3 G := , 2 ≤ GL2(Q(ζ3)) , 0 ζ3 3 2ζ3 1

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

Let G ≤ SL2(C) be ﬁnite and g ∈ G. Let ζ be an eigenvalue of g. Then ζ−1g is a reﬂection.

Michel Brou´e Reﬂection groups, braids, Hecke algebras with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

If g ∈ SL3(C) is an involution, then −g is a reﬂection. Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

Note that for G irreducible, we have G/Z(G) ∈ {Dn, A4, S4, A5}. For example, the group √ 2 1 0 −3 −ζ3 ζ3 G := , 2 ≤ GL2(Q(ζ3)) , 0 ζ3 3 2ζ3 1

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

Let G ≤ SL2(C) be ﬁnite and g ∈ G. Let ζ be an eigenvalue of g. Then ζ−1g is a reﬂection.

−1 −1 So, if G = hg1,..., gr i, the group hζ1 g1, . . . , ζr gr i is an frg.

Michel Brou´e Reﬂection groups, braids, Hecke algebras with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

If g ∈ SL3(C) is an involution, then −g is a reﬂection. Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

For example, the group √ 2 1 0 −3 −ζ3 ζ3 G := , 2 ≤ GL2(Q(ζ3)) , 0 ζ3 3 2ζ3 1

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

Let G ≤ SL2(C) be ﬁnite and g ∈ G. Let ζ be an eigenvalue of g. Then ζ−1g is a reﬂection.

−1 −1 So, if G = hg1,..., gr i, the group hζ1 g1, . . . , ζr gr i is an frg. Note that for G irreducible, we have G/Z(G) ∈ {Dn, A4, S4, A5}.

Michel Brou´e Reﬂection groups, braids, Hecke algebras with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

If g ∈ SL3(C) is an involution, then −g is a reﬂection. Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

Let G ≤ SL2(C) be ﬁnite and g ∈ G. Let ζ be an eigenvalue of g. Then ζ−1g is a reﬂection.

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3.

Michel Broué Reflection groups, braids, Hecke algebras If g ∈ SL3(C) is an involution, then −g is a reflection. Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

Let G ≤ SL2(C) be ﬁnite and g ∈ G. Let ζ be an eigenvalue of g. Then ζ−1g is a reﬂection.

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

Michel Brou´e Reﬂection groups, braids, Hecke algebras Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

If g ∈ SL3(C) is an involution, then −g is a reﬂection.

Let G ≤ SL2(C) be ﬁnite and g ∈ G. Let ζ be an eigenvalue of g. Then ζ−1g is a reﬂection.

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

Michel Brou´e Reﬂection groups, braids, Hecke algebras Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

Let G ≤ SL2(C) be ﬁnite and g ∈ G. Let ζ be an eigenvalue of g. Then ζ−1g is a reﬂection.

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

If g ∈ SL3(C) is an involution, then −g is a reﬂection.

Michel Broué Reflection groups, braids, Hecke algebras Let G ≤ SL2(C) be finite and g ∈ G. Let ζ be an eigenvalue of g. Then ζ−1g is a reflection.

with ζ3 := exp(2πi/3), is a frg of order 72, denoted G5, isomorphic to SL2(3) × C3. We may choose

6 3 3 6 3 9 6 6 9 3 12 u1 := v1 + 20v1 v2 − 8v2 , u2 := 3v1 v2 + 3v1 v2 + v1 v2 + v2 ,

with degrees d1 = 6, d2 = 12 (note that d1d2 = 72 = |G|).

If g ∈ SL3(C) is an involution, then −g is a reﬂection. Note that A5, PSL2(7) and 3.A6 have faithful 3-dimensional representations and are generated by involutions.

Michel Brou´e Reﬂection groups, braids, Hecke algebras G4 , G5 ,..., G37.

There is one inﬁnite series G(de, e, r)(d ,e and r integers), ...and 34 exceptional groups

2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

3 We have

G(d, 1, r) ' Cd o Sr

G(e, e, 2) = D2e (dihedral group of order 2e)

G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

1 The finite reflection groups on C have been classified by Coxeter, Shephard and Todd.

Classiﬁcation

Michel Brou´e Reﬂection groups, braids, Hecke algebras G4 , G5 ,..., G37.

2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

3 We have

G(d, 1, r) ' Cd o Sr

G(e, e, 2) = D2e (dihedral group of order 2e)

G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

There is one inﬁnite series G(de, e, r)(d ,e and r integers), ...and 34 exceptional groups

Classiﬁcation

1 The finite reflection groups on C have been classified by Coxeter, Shephard and Todd.

Michel Brou´e Reﬂection groups, braids, Hecke algebras G4 , G5 ,..., G37.

2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

3 We have

G(d, 1, r) ' Cd o Sr

G(e, e, 2) = D2e (dihedral group of order 2e)

G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

...and 34 exceptional groups

Classiﬁcation

1 The finite reflection groups on C have been classified by Coxeter, Shephard and Todd. There is one infinite series G(de, e, r)(d ,e and r integers),

Michel Brou´e Reﬂection groups, braids, Hecke algebras 2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

3 We have

G(d, 1, r) ' Cd o Sr

G(e, e, 2) = D2e (dihedral group of order 2e)

G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

G4 , G5 ,..., G37.

Classiﬁcation

1 The finite reflection groups on C have been classified by Coxeter, Shephard and Todd. There is one infinite series G(de, e, r)(d ,e and r integers), ...and 34 exceptional groups

Michel Brou´e Reﬂection groups, braids, Hecke algebras G(d, 1, r) ' Cd o Sr

G(e, e, 2) = D2e (dihedral group of order 2e)

G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

3 We have

Classiﬁcation

1 The finite reflection groups on C have been classified by Coxeter, Shephard and Todd. There is one infinite series G(de, e, r)(d ,e and r integers), ...and 34 exceptional groups G4 , G5 ,..., G37.

Michel Brou´e Reﬂection groups, braids, Hecke algebras G(d, 1, r) ' Cd o Sr

G(e, e, 2) = D2e (dihedral group of order 2e)

G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

3 We have

Classiﬁcation

2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

Michel Brou´e Reﬂection groups, braids, Hecke algebras G(d, 1, r) ' Cd o Sr

G(e, e, 2) = D2e (dihedral group of order 2e)

G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

Classiﬁcation

2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

3 We have

Michel Brou´e Reﬂection groups, braids, Hecke algebras G(e, e, 2) = D2e (dihedral group of order 2e)

G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

Classiﬁcation

2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

3 We have

G(d, 1, r) ' Cd o Sr

Michel Brou´e Reﬂection groups, braids, Hecke algebras G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

Classiﬁcation

2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

3 We have

G(d, 1, r) ' Cd o Sr

G(e, e, 2) = D2e (dihedral group of order 2e)

Michel Broué Reflection groups, braids, Hecke algebras Classification

2 The group G(de, e, r)(d ,e and r integers) consists of all r × r monomial matrices with entries in µde such that the product of entries belongs to µd .

3 We have

G(d, 1, r) ' Cd o Sr

G(e, e, 2) = D2e (dihedral group of order 2e)

G(2, 2, r) = W (Dr )

G23 = H3 , G28 = F4 , G30 = H4

G35,36,37 = E6,7,8 .

Michel Brou´e Reﬂection groups, braids, Hecke algebras hence, in terms of normalizers,

NG (H) = NG (L) = NG (H, L) .

The fixator GH (pointwise stabilizer) of H is a cyclic group consisting of reflections with reflecting hyperplane H and reflecting line L.

a reflecting hyperplane H := ker(s − 1), a reflecting line L := im (s − 1), a reflecting pair( H, L). Properties : H ⊕ L = V , H determines L, and L determines H,

Reﬂecting hyperplanes, lines, pairs

Let G be a ﬁnite subgroup of GL(V ). A reﬂection s is associated with

Michel Brou´e Reﬂection groups, braids, Hecke algebras hence, in terms of normalizers,

NG (H) = NG (L) = NG (H, L) .

The fixator GH (pointwise stabilizer) of H is a cyclic group consisting of reflections with reflecting hyperplane H and reflecting line L.

a reﬂecting line L := im (s − 1), a reﬂecting pair( H, L). Properties : H ⊕ L = V , H determines L, and L determines H,

Reﬂecting hyperplanes, lines, pairs

Let G be a finite subgroup of GL(V ). A reflection s is associated with a reflecting hyperplane H := ker(s − 1),

Michel Brou´e Reﬂection groups, braids, Hecke algebras hence, in terms of normalizers,

NG (H) = NG (L) = NG (H, L) .

The fixator GH (pointwise stabilizer) of H is a cyclic group consisting of reflections with reflecting hyperplane H and reflecting line L.

a reﬂecting pair( H, L). Properties : H ⊕ L = V , H determines L, and L determines H,

Reﬂecting hyperplanes, lines, pairs

Let G be a finite subgroup of GL(V ). A reflection s is associated with a reflecting hyperplane H := ker(s − 1), a reflecting line L := im (s − 1),

Michel Brou´e Reﬂection groups, braids, Hecke algebras hence, in terms of normalizers,

NG (H) = NG (L) = NG (H, L) .

The fixator GH (pointwise stabilizer) of H is a cyclic group consisting of reflections with reflecting hyperplane H and reflecting line L.

Properties : H ⊕ L = V , H determines L, and L determines H,

Reﬂecting hyperplanes, lines, pairs

Let G be a finite subgroup of GL(V ). A reflection s is associated with a reflecting hyperplane H := ker(s − 1), a reflecting line L := im (s − 1), a reflecting pair( H, L).

Michel Brou´e Reﬂection groups, braids, Hecke algebras hence, in terms of normalizers,

NG (H) = NG (L) = NG (H, L) .

The fixator GH (pointwise stabilizer) of H is a cyclic group consisting of reflections with reflecting hyperplane H and reflecting line L.

H ⊕ L = V , H determines L, and L determines H,

Reﬂecting hyperplanes, lines, pairs

Let G be a finite subgroup of GL(V ). A reflection s is associated with a reflecting hyperplane H := ker(s − 1), a reflecting line L := im (s − 1), a reflecting pair( H, L). Properties :

Michel Brou´e Reﬂection groups, braids, Hecke algebras hence, in terms of normalizers,

NG (H) = NG (L) = NG (H, L) .

The fixator GH (pointwise stabilizer) of H is a cyclic group consisting of reflections with reflecting hyperplane H and reflecting line L.

H determines L, and L determines H,

Reﬂecting hyperplanes, lines, pairs

Let G be a finite subgroup of GL(V ). A reflection s is associated with a reflecting hyperplane H := ker(s − 1), a reflecting line L := im (s − 1), a reflecting pair( H, L). Properties : H ⊕ L = V ,

Michel Broué Reflection groups, braids, Hecke algebras The fixator GH (pointwise stabilizer) of H is a cyclic group consisting of reflections with reflecting hyperplane H and reflecting line L.

hence, in terms of normalizers,

NG (H) = NG (L) = NG (H, L) .

Reﬂecting hyperplanes, lines, pairs

Let G be a finite subgroup of GL(V ). A reflection s is associated with a reflecting hyperplane H := ker(s − 1), a reflecting line L := im (s − 1), a reflecting pair( H, L). Properties : H ⊕ L = V , H determines L, and L determines H,

Reﬂecting hyperplanes, lines, pairs

NG (H) = NG (L) = NG (H, L) .

Michel Broué Reflection groups, braids, Hecke algebras Reflecting hyperplanes, lines, pairs

NG (H) = NG (L) = NG (H, L) .

The fixator GH (pointwise stabilizer) of H is a cyclic group consisting of reflections with reflecting hyperplane H and reflecting line L.

Michel Broué Reflection groups, braids, Hecke algebras called a distinguished reflection.

is ramiﬁed at q = SL if and only if L is a reﬂecting line

A := {H | H reﬂecting hyperplane of some reﬂection in G}

For H ∈ A, eH := |GH | 2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H ,

For L a line in V , the ideal q := SL of S is a height one prime ideal. In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

S V O Now the extension (corresponding to the covering ) ? R = SG V /G .

Notation

Michel Broué Reflection groups, braids, Hecke algebras called a distinguished reflection.

is ramiﬁed at q = SL if and only if L is a reﬂecting line

For H ∈ A, eH := |GH | 2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H ,

For L a line in V , the ideal q := SL of S is a height one prime ideal. In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

S V O Now the extension (corresponding to the covering ) ? R = SG V /G .

Notation

A := {H | H reﬂecting hyperplane of some reﬂection in G}

Michel Broué Reflection groups, braids, Hecke algebras called a distinguished reflection.

is ramiﬁed at q = SL if and only if L is a reﬂecting line

2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H ,

For L a line in V , the ideal q := SL of S is a height one prime ideal. In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

S V O Now the extension (corresponding to the covering ) ? R = SG V /G .

Notation

A := {H | H reﬂecting hyperplane of some reﬂection in G}

For H ∈ A, eH := |GH |

Michel Broué Reflection groups, braids, Hecke algebras is ramified at q = SL if and only if L is a reflecting line

called a distinguished reﬂection.

For L a line in V , the ideal q := SL of S is a height one prime ideal. In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

S V O Now the extension (corresponding to the covering ) ? R = SG V /G .

Notation

A := {H | H reﬂecting hyperplane of some reﬂection in G}

For H ∈ A, eH := |GH | 2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H ,

Michel Broué Reflection groups, braids, Hecke algebras is ramified at q = SL if and only if L is a reflecting line

For L a line in V , the ideal q := SL of S is a height one prime ideal. In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

S V O Now the extension (corresponding to the covering ) ? R = SG V /G .

Notation

A := {H | H reﬂecting hyperplane of some reﬂection in G}

For H ∈ A, eH := |GH | 2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H , called a distinguished reﬂection.

Michel Broué Reflection groups, braids, Hecke algebras is ramified at q = SL if and only if L is a reflecting line

In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

S V O Now the extension (corresponding to the covering ) ? R = SG V /G .

Notation

A := {H | H reﬂecting hyperplane of some reﬂection in G}

For H ∈ A, eH := |GH | 2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H , called a distinguished reﬂection.

For L a line in V , the ideal q := SL of S is a height one prime ideal.

Michel Broué Reflection groups, braids, Hecke algebras is ramified at q = SL if and only if L is a reflecting line

S V O Now the extension (corresponding to the covering ) ? R = SG V /G .

Notation

A := {H | H reﬂecting hyperplane of some reﬂection in G}

For H ∈ A, eH := |GH | 2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H , called a distinguished reﬂection.

For L a line in V , the ideal q := SL of S is a height one prime ideal. In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

Michel Broué Reflection groups, braids, Hecke algebras is ramified at q = SL if and only if L is a reflecting line

V (corresponding to the covering ) V /G .

Notation

A := {H | H reﬂecting hyperplane of some reﬂection in G}

For H ∈ A, eH := |GH | 2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H , called a distinguished reﬂection.

For L a line in V , the ideal q := SL of S is a height one prime ideal. In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

S O Now the extension ? R = SG

Michel Broué Reflection groups, braids, Hecke algebras is ramified at q = SL if and only if L is a reflecting line

Notation

A := {H | H reﬂecting hyperplane of some reﬂection in G}

For H ∈ A, eH := |GH | 2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H , called a distinguished reﬂection.

For L a line in V , the ideal q := SL of S is a height one prime ideal. In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

S V O Now the extension (corresponding to the covering ) ? R = SG V /G .

Michel Brou´e Reﬂection groups, braids, Hecke algebras Notation

A := {H | H reﬂecting hyperplane of some reﬂection in G}

For H ∈ A, eH := |GH | 2iπ/e sH is the generator of GH whose nontrivial eigenvalue is e H , called a distinguished reﬂection.

For L a line in V , the ideal q := SL of S is a height one prime ideal. In other words, the hypersurface of V deﬁned by q is a codimension one irreducible variety.

S V O Now the extension (corresponding to the covering ) ? R = SG V /G is ramiﬁed at q = SL if and only if L is a reﬂecting line.

Michel Broué Reflection groups, braids, Hecke algebras 1 S The ramification locus of V / / V /G is H∈A H. 2 Let X be a subset of V . Then the fixator of X in G is generated by the reflections which fix X . 3 The set Par(G) of fixators (“parabolic subgroups” of G) is in (reverse–order) bijection with the set I(A) of intersections of elements of A : ∼ I(A) −→ Par(G) , X 7→ GX .

Thus there are G–equivariant bijections

A o / {reﬂecting lines} o / {ramiﬁed height one prime ideals of S}

Ramiﬁcation and parabolic subgroups

Steinberg Theorem Assume G generated by reﬂections.

Ramiﬁcation and parabolic subgroups

Steinberg Theorem Assume G generated by reﬂections.

Thus there are G–equivariant bijections

A o / {reﬂecting lines} o / {ramiﬁed height one prime ideals of S}

Steinberg Theorem Assume G generated by reﬂections.

Thus there are G–equivariant bijections

A o / {reﬂecting lines} o / {ramiﬁed height one prime ideals of S}

Ramiﬁcation and parabolic subgroups

Thus there are G–equivariant bijections

A o / {reﬂecting lines} o / {ramiﬁed height one prime ideals of S}

Ramiﬁcation and parabolic subgroups

Steinberg Theorem Assume G generated by reﬂections.

Michel Broué Reflection groups, braids, Hecke algebras 2 Let X be a subset of V . Then the fixator of X in G is generated by the reflections which fix X . 3 The set Par(G) of fixators (“parabolic subgroups” of G) is in (reverse–order) bijection with the set I(A) of intersections of elements of A : ∼ I(A) −→ Par(G) , X 7→ GX .

Thus there are G–equivariant bijections

A o / {reﬂecting lines} o / {ramiﬁed height one prime ideals of S}

Ramiﬁcation and parabolic subgroups

Steinberg Theorem Assume G generated by reﬂections.

1 S The ramiﬁcation locus of V / / V /G is H∈A H.

Michel Broué Reflection groups, braids, Hecke algebras 3 The set Par(G) of fixators (“parabolic subgroups” of G) is in (reverse–order) bijection with the set I(A) of intersections of elements of A : ∼ I(A) −→ Par(G) , X 7→ GX .

Thus there are G–equivariant bijections

A o / {reﬂecting lines} o / {ramiﬁed height one prime ideals of S}

Ramiﬁcation and parabolic subgroups

Steinberg Theorem Assume G generated by reﬂections.

1 S The ramification locus of V / / V /G is H∈A H. 2 Let X be a subset of V . Then the fixator of X in G is generated by the reflections which fix X .

Michel Brou´e Reﬂection groups, braids, Hecke algebras Thus there are G–equivariant bijections

A o / {reﬂecting lines} o / {ramiﬁed height one prime ideals of S}

Ramiﬁcation and parabolic subgroups

Steinberg Theorem Assume G generated by reﬂections.

1 S The ramification locus of V / / V /G is H∈A H. 2 Let X be a subset of V . Then the fixator of X in G is generated by the reflections which fix X . 3 The set Par(G) of fixators (“parabolic subgroups” of G) is in (reverse–order) bijection with the set I(A) of intersections of elements of A : ∼ I(A) −→ Par(G) , X 7→ GX .

Michel Brou´e Reﬂection groups, braids, Hecke algebras Set reg S V := V − H∈A H .

Since the covering V reg / / V reg/G is Galois, it induces a short exact sequence

- reg - reg - - 1 Π1(V , x0)Π1(V /G, x0) G 1

PG BG (Pure braid group) (Braid group)

Braid groups

Michel Brou´e Reﬂection groups, braids, Hecke algebras Since the covering V reg / / V reg/G is Galois, it induces a short exact sequence

- reg - reg - - 1 Π1(V , x0)Π1(V /G, x0) G 1

PG BG (Pure braid group) (Braid group)

Braid groups

Set reg S V := V − H∈A H .

Michel Brou´e Reﬂection groups, braids, Hecke algebras - reg - reg - - 1 Π1(V , x0)Π1(V /G, x0) G 1

PG BG (Pure braid group) (Braid group)

Braid groups

Set reg S V := V − H∈A H .

Since the covering V reg / / V reg/G is Galois, it induces a short exact sequence

Michel Brou´e Reﬂection groups, braids, Hecke algebras PG BG (Pure braid group) (Braid group)

Braid groups

Set reg S V := V − H∈A H .

Since the covering V reg / / V reg/G is Galois, it induces a short exact sequence

- reg - reg - - 1 Π1(V , x0)Π1(V /G, x0) G 1

Michel Brou´e Reﬂection groups, braids, Hecke algebras (Pure braid group) (Braid group)

Braid groups

Set reg S V := V − H∈A H .

Since the covering V reg / / V reg/G is Galois, it induces a short exact sequence

- reg - reg - - 1 Π1(V , x0)Π1(V /G, x0) G 1

PG BG

Michel Brou´e Reﬂection groups, braids, Hecke algebras Braid groups

Set reg S V := V − H∈A H .

Since the covering V reg / / V reg/G is Galois, it induces a short exact sequence

- reg - reg - - 1 Π1(V , x0)Π1(V /G, x0) G 1

PG BG (Pure braid group) (Braid group)

Michel Brou´e Reﬂection groups, braids, Hecke algebras For x ∈ V , we set

x = xL + xH (with xL ∈ L and xH ∈ H) .

2iπ/e = Thus, we have sH (x) = e H xL + xH .

If t ∈ R, we set :

t 2iπt/eH sH (x) = e xL + xH deﬁning a path sH,x from x to sH (x)

We have

teH 2πit sH (x) = e xL + xH deﬁning a loop πH,x with origin x In other words, eH πH,x = sH,x ∈ PG

Let H ∈ A, with associated line L.

Notation around H

Michel Brou´e Reﬂection groups, braids, Hecke algebras If t ∈ R, we set :

t 2iπt/eH sH (x) = e xL + xH deﬁning a path sH,x from x to sH (x)

We have

teH 2πit sH (x) = e xL + xH deﬁning a loop πH,x with origin x In other words, eH πH,x = sH,x ∈ PG

For x ∈ V , we set

x = xL + xH (with xL ∈ L and xH ∈ H) .

2iπ/e = Thus, we have sH (x) = e H xL + xH .

Notation around H

Let H ∈ A, with associated line L.

Michel Brou´e Reﬂection groups, braids, Hecke algebras If t ∈ R, we set :

t 2iπt/eH sH (x) = e xL + xH deﬁning a path sH,x from x to sH (x)

We have

teH 2πit sH (x) = e xL + xH deﬁning a loop πH,x with origin x In other words, eH πH,x = sH,x ∈ PG

2iπ/e = Thus, we have sH (x) = e H xL + xH .

Notation around H

Let H ∈ A, with associated line L. For x ∈ V , we set

x = xL + xH (with xL ∈ L and xH ∈ H) .

Michel Brou´e Reﬂection groups, braids, Hecke algebras We have

teH 2πit sH (x) = e xL + xH deﬁning a loop πH,x with origin x In other words, eH πH,x = sH,x ∈ PG

If t ∈ R, we set :

t 2iπt/eH sH (x) = e xL + xH deﬁning a path sH,x from x to sH (x)

Notation around H

Let H ∈ A, with associated line L. For x ∈ V , we set

x = xL + xH (with xL ∈ L and xH ∈ H) .

2iπ/e = Thus, we have sH (x) = e H xL + xH .

Michel Brou´e Reﬂection groups, braids, Hecke algebras We have

teH 2πit sH (x) = e xL + xH deﬁning a loop πH,x with origin x In other words, eH πH,x = sH,x ∈ PG

Notation around H

Let H ∈ A, with associated line L. For x ∈ V , we set

x = xL + xH (with xL ∈ L and xH ∈ H) .

2iπ/e = Thus, we have sH (x) = e H xL + xH .

If t ∈ R, we set :

t 2iπt/eH sH (x) = e xL + xH deﬁning a path sH,x from x to sH (x)

Michel Brou´e Reﬂection groups, braids, Hecke algebras In other words, eH πH,x = sH,x ∈ PG

Notation around H

Let H ∈ A, with associated line L. For x ∈ V , we set

x = xL + xH (with xL ∈ L and xH ∈ H) .

2iπ/e = Thus, we have sH (x) = e H xL + xH .

If t ∈ R, we set :

t 2iπt/eH sH (x) = e xL + xH deﬁning a path sH,x from x to sH (x)

We have

teH 2πit sH (x) = e xL + xH deﬁning a loop πH,x with origin x

Michel Brou´e Reﬂection groups, braids, Hecke algebras Notation around H

Let H ∈ A, with associated line L. For x ∈ V , we set

x = xL + xH (with xL ∈ L and xH ∈ H) .

2iπ/e = Thus, we have sH (x) = e H xL + xH .

If t ∈ R, we set :

t 2iπt/eH sH (x) = e xL + xH deﬁning a path sH,x from x to sH (x)

We have

teH 2πit sH (x) = e xL + xH deﬁning a loop πH,x with origin x In other words, eH πH,x = sH,x ∈ PG

Michel Brou´e Reﬂection groups, braids, Hecke algebras −1 sH (γ )

Braid reﬂections reg Let γ be a path in V from x0 to xH . −1 We deﬁne : σH,γ := sH (γ ) · sH,x · γ

• sH (x0)

H • PP• sH (xH ) γ B •· · xH • < b x0

• x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Michel Brou´e Reﬂection groups, braids, Hecke algebras −1 sH (γ )

reg Let γ be a path in V from x0 to xH . −1 We deﬁne : σH,γ := sH (γ ) · sH,x · γ

• sH (x0)

H • PP• sH (xH ) γ B •· · xH • < b x0

• x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections

Michel Brou´e Reﬂection groups, braids, Hecke algebras −1 sH (γ )

reg Let γ be a path in V from x0 to xH . −1 We deﬁne : σH,γ := sH (γ ) · sH,x · γ

• sH (x0)

PP• sH (xH ) γ B •· · xH • < b x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections

H •

• x0

Michel Brou´e Reﬂection groups, braids, Hecke algebras −1 sH (γ )

reg Let γ be a path in V from x0 to xH . −1 We deﬁne : σH,γ := sH (γ ) · sH,x · γ

• sH (x0)

PP• sH (xH ) γ •·< b ·x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections

H • B xH •

• x0

Michel Brou´e Reﬂection groups, braids, Hecke algebras −1 sH (γ )

reg Let γ be a path in V from x0 to xH . −1 We deﬁne : σH,γ := sH (γ ) · sH,x · γ

• sH (x0)

γ •·< b ·x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections

H •PP sH (xH ) B • xH •

• x0

Michel Broué Reflection groups, braids, Hecke algebras reg Let γ be a path in V from x0 to xH . −1 We define : σH,γ := sH (γ ) · sH,x · γ

−1 sH (γ )

γ •·< b ·x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections

• sH (x0)

H •PP sH (xH ) B • xH •

• x0

Michel Broué Reflection groups, braids, Hecke algebras −1 We define : σH,γ := sH (γ ) · sH,x ·

−1 sH (γ )

γ •·< b ·x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections reg Let γ be a path in V from x0 to xH . γ

• sH (x0)

H •PP sH (xH ) B • xH •

• x0

Michel Broué Reflection groups, braids, Hecke algebras −1 We define : σH,γ := sH (γ ) ·

−1 sH (γ )

γ •·< b ·x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections reg Let γ be a path in V from x0 to xH .

sH,x · γ

• sH (x0)

H •PP sH (xH ) B • xH •

• x0

Michel Broué Reflection groups, braids, Hecke algebras We define : σH,γ :=

γ •·< b ·x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections reg Let γ be a path in V from x0 to xH . −1 sH (γ ) · sH,x · γ

−1 • sH (x0) sH (γ )

H •PP sH (xH ) B • xH •

• x0

Michel Brou´e Reﬂection groups, braids, Hecke algebras γ •·< b ·x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections reg Let γ be a path in V from x0 to xH . −1 We deﬁne : σH,γ := sH (γ ) · sH,x · γ

−1 • sH (x0) sH (γ )

H •PP sH (xH ) B • xH •

• x0

Michel Broué Reflection groups, braids, Hecke algebras Definition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Braid reﬂections reg Let γ be a path in V from x0 to xH . −1 We deﬁne : σH,γ := sH (γ ) · sH,x · γ

−1 • sH (x0) sH (γ )

H • PP• sH (xH ) γ B •· · xH • < b x0

• x0

Michel Broué Reflection groups, braids, Hecke algebras Braid reflections reg Let γ be a path in V from x0 to xH . −1 We define : σH,γ := sH (γ ) · sH,x · γ

−1 • sH (x0) sH (γ )

H • PP• sH (xH ) γ B •· · xH • < b x0

• x0

Deﬁnition

We call braid reﬂections the elements sH,γ ∈ B deﬁned by the paths σH,γ.

Michel Broué Reflection groups, braids, Hecke algebras 1 The braid group BG is generated by the braid reflections (sH,γ) (for all H and all γ). e 2 H The pure braid group PG is generated by the elements (sH,γ)

sH,γ and sH,γ0 are conjugate in P. γ eH reg sH,γ is a loop in V : •·< b ·x0

The variety V (resp. V /G) is connected, the hyperplanes are irreducible divisors (irreducible closed subvarieties of codimension one), and the braid reﬂections are “generators of the monodromy” around the irreducible divisors. Then Theorem

The following properties are immediate.

γ eH reg sH,γ is a loop in V : •·< b ·x0

The following properties are immediate.

sH,γ and sH,γ0 are conjugate in P.

The following properties are immediate.

sH,γ and sH,γ0 are conjugate in P. γ eH reg sH,γ is a loop in V : •·< b ·x0

Theorem

The following properties are immediate.

sH,γ and sH,γ0 are conjugate in P. γ eH reg sH,γ is a loop in V : •·< b ·x0

The following properties are immediate.

sH,γ and sH,γ0 are conjugate in P. γ eH reg sH,γ is a loop in V : •·< b ·x0

Michel Brou´e Reﬂection groups, braids, Hecke algebras e 2 H The pure braid group PG is generated by the elements (sH,γ)

The following properties are immediate.

sH,γ and sH,γ0 are conjugate in P. γ eH reg sH,γ is a loop in V : •·< b ·x0

1 The braid group BG is generated by the braid reﬂections (sH,γ) (for all H and all γ).

Michel Brou´e Reﬂection groups, braids, Hecke algebras The following properties are immediate.

sH,γ and sH,γ0 are conjugate in P. γ eH reg sH,γ is a loop in V : •·< b ·x0

1 The braid group BG is generated by the braid reﬂections (sH,γ) (for all H and all γ). e 2 H The pure braid group PG is generated by the elements (sH,γ)

Michel Broué Reflection groups, braids, Hecke algebras × 1 The linear character detH : G → C is defined by g(jH ) = detH (g)jH ( det(s) if Hs =G H 2 det (s) = H 1 if not

∼ G 3 × Q × Q × Hom(G, C ) −→ H∈A Hom(GH , C ) ' H∈A/G Hom(GH , C )

Proposition

jH denotes a nontrivial element of L, Q j := 0 0 j 0 (depends only on the orbit of H under G) H {H |(H =G H)} H

Linear characters of the reﬂection groups

For H ∈ A,

Michel Broué Reflection groups, braids, Hecke algebras × 1 The linear character detH : G → C is defined by g(jH ) = detH (g)jH ( det(s) if Hs =G H 2 det (s) = H 1 if not

∼ G 3 × Q × Q × Hom(G, C ) −→ H∈A Hom(GH , C ) ' H∈A/G Hom(GH , C )

Proposition

Q j := 0 0 j 0 (depends only on the orbit of H under G) H {H |(H =G H)} H

Linear characters of the reﬂection groups

For H ∈ A,

jH denotes a nontrivial element of L,

Michel Broué Reflection groups, braids, Hecke algebras × 1 The linear character detH : G → C is defined by g(jH ) = detH (g)jH ( det(s) if Hs =G H 2 det (s) = H 1 if not

∼ G 3 × Q × Q × Hom(G, C ) −→ H∈A Hom(GH , C ) ' H∈A/G Hom(GH , C )

Proposition

Linear characters of the reﬂection groups

For H ∈ A,

jH denotes a nontrivial element of L, Q j := 0 0 j 0 (depends only on the orbit of H under G) H {H |(H =G H)} H

Michel Brou´e Reﬂection groups, braids, Hecke algebras Q × ' H∈A/G Hom(GH , C )

× 1 The linear character detH : G → C is deﬁned by g(jH ) = detH (g)jH ( det(s) if Hs =G H 2 det (s) = H 1 if not

∼ G 3 × Q × Hom(G, C ) −→ H∈A Hom(GH , C )

Linear characters of the reﬂection groups

For H ∈ A,

jH denotes a nontrivial element of L, Q j := 0 0 j 0 (depends only on the orbit of H under G) H {H |(H =G H)} H

Proposition

Michel Brou´e Reﬂection groups, braids, Hecke algebras Q × ' H∈A/G Hom(GH , C )

( det(s) if Hs =G H 2 det (s) = H 1 if not

∼ G 3 × Q × Hom(G, C ) −→ H∈A Hom(GH , C )

Linear characters of the reﬂection groups

For H ∈ A,

jH denotes a nontrivial element of L, Q j := 0 0 j 0 (depends only on the orbit of H under G) H {H |(H =G H)} H

Proposition

× 1 The linear character detH : G → C is deﬁned by g(jH ) = detH (g)jH

Michel Brou´e Reﬂection groups, braids, Hecke algebras Q × ' H∈A/G Hom(GH , C )

∼ G 3 × Q × Hom(G, C ) −→ H∈A Hom(GH , C )

Linear characters of the reﬂection groups

For H ∈ A,

jH denotes a nontrivial element of L, Q j := 0 0 j 0 (depends only on the orbit of H under G) H {H |(H =G H)} H

Proposition

× 1 The linear character detH : G → C is deﬁned by g(jH ) = detH (g)jH ( det(s) if Hs =G H 2 det (s) = H 1 if not

Michel Brou´e Reﬂection groups, braids, Hecke algebras Q × ' H∈A/G Hom(GH , C )

Linear characters of the reﬂection groups

For H ∈ A,

jH denotes a nontrivial element of L, Q j := 0 0 j 0 (depends only on the orbit of H under G) H {H |(H =G H)} H

Proposition

× 1 The linear character detH : G → C is deﬁned by g(jH ) = detH (g)jH ( det(s) if Hs =G H 2 det (s) = H 1 if not

∼ G 3 × Q × Hom(G, C ) −→ H∈A Hom(GH , C )

Michel Broué Reflection groups, braids, Hecke algebras Linear characters of the reflection groups

For H ∈ A,

jH denotes a nontrivial element of L, Q j := 0 0 j 0 (depends only on the orbit of H under G) H {H |(H =G H)} H

Proposition

× 1 The linear character detH : G → C is deﬁned by g(jH ) = detH (g)jH ( det(s) if Hs =G H 2 det (s) = H 1 if not

∼ G 3 × Q × Q × Hom(G, C ) −→ H∈A Hom(GH , C ) ' H∈A/G Hom(GH , C )

Michel Brou´e Reﬂection groups, braids, Hecke algebras `H : BG −→ Z

∗ ∗ ∗G hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C hence deﬁnes a morphism O u: OOO uu OO' uu V reg/G

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e.,

For H ∈ A, BGH ' Z

L ↓ Ó Ø ÓÓ ØØ G ' /e ÓÓ • ØØ H Z H Z ÓÓ ØØ ÓÓ Ø H ØØ

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH G ∆H ∈ R = S

Linear characters of the braid groups

Michel Brou´e Reﬂection groups, braids, Hecke algebras `H : BG −→ Z

∗ ∗ ∗G hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C hence deﬁnes a morphism O u: OOO uu OO' uu V reg/G

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e.,

For H ∈ A, BGH ' Z

L ↓ Ó Ø ÓÓ ØØ G ' /e ÓÓ • ØØ H Z H Z ÓÓ ØØ ÓÓ Ø H ØØ

G ∆H ∈ R = S

Linear characters of the braid groups

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH

Michel Brou´e Reﬂection groups, braids, Hecke algebras `H : BG −→ Z

For H ∈ A, BGH ' Z

L ↓ Ó Ø ÓÓ ØØ G ' /e ÓÓ • ØØ H Z H Z ÓÓ ØØ ÓÓ Ø H ØØ

∗ ∗ ∗G hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C hence deﬁnes a morphism O u: OOO uu OO' uu V reg/G

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e.,

Linear characters of the braid groups

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH G ∆H ∈ R = S

Michel Brou´e Reﬂection groups, braids, Hecke algebras `H : BG −→ Z

For H ∈ A, BGH ' Z

L ↓ Ó Ø ÓÓ ØØ G ' /e ÓÓ • ØØ H Z H Z ÓÓ ØØ ÓÓ Ø H ØØ

hence deﬁnes a morphism

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e.,

Linear characters of the braid groups

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH G ∗ ∗ ∗G ∆H ∈ R = S hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C O u: OOO uu OO' uu V reg/G

Michel Brou´e Reﬂection groups, braids, Hecke algebras `H : BG −→ Z

For H ∈ A, BGH ' Z

L ↓ Ó Ø ÓÓ ØØ G ' /e ÓÓ • ØØ H Z H Z ÓÓ ØØ ÓÓ Ø H ØØ

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e.,

Linear characters of the braid groups

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH G ∗ ∗ ∗G ∆H ∈ R = S hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C hence deﬁnes a morphism O u: OOO uu OO' uu V reg/G

Michel Brou´e Reﬂection groups, braids, Hecke algebras BGH ' Z ↓

GH ' Z/eH Z

`H : BG −→ Z

For H ∈ A,

L Ó Ø ÓÓ ØØ ÓÓ • ØØ ÓÓ ØØ ÓÓ Ø H ØØ

Linear characters of the braid groups

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH G ∗ ∗ ∗G ∆H ∈ R = S hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C hence deﬁnes a morphism O u: OOO uu OO' uu V reg/G

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e.,

Michel Brou´e Reﬂection groups, braids, Hecke algebras BGH ' Z ↓

GH ' Z/eH Z

For H ∈ A,

L Ó Ø ÓÓ ØØ ÓÓ • ØØ ÓÓ ØØ ÓÓ Ø H ØØ

Linear characters of the braid groups

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH G ∗ ∗ ∗G ∆H ∈ R = S hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C hence deﬁnes a morphism O u: OOO uu OO' uu V reg/G

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e., `H : BG −→ Z

Michel Brou´e Reﬂection groups, braids, Hecke algebras BGH ' Z ↓

GH ' Z/eH Z

Linear characters of the braid groups

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH G ∗ ∗ ∗G ∆H ∈ R = S hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C hence deﬁnes a morphism O u: OOO uu OO' uu V reg/G

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e., `H : BG −→ Z

For H ∈ A,

L Ó Ø ÓÓ ØØ ÓÓ • ØØ ÓÓ ØØ ÓÓ Ø H ØØ

Michel Brou´e Reﬂection groups, braids, Hecke algebras ↓

GH ' Z/eH Z

Linear characters of the braid groups

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH G ∗ ∗ ∗G ∆H ∈ R = S hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C hence deﬁnes a morphism O u: OOO uu OO' uu V reg/G

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e., `H : BG −→ Z

For H ∈ A, BGH ' Z

L Ó Ø ÓÓ ØØ ÓÓ • ØØ ÓÓ ØØ ÓÓ Ø H ØØ

Michel Brou´e Reﬂection groups, braids, Hecke algebras Linear characters of the braid groups

eH The discriminant at H ∈ A (or rather A/G) is ∆H := jH G ∗ ∗ ∗G ∆H ∈ R = S hence its dual ∆H ∈ R = S deﬁnes a (continuous) map

∗ reg × ∆H : V / C hence deﬁnes a morphism O u: OOO uu OO' uu V reg/G

∗ reg × Π1(∆H ):Π1(V /G) → Π1(C ) i.e., `H : BG −→ Z

For H ∈ A, BGH ' Z

L ↓ Ó Ø ÓÓ ØØ G ' /e ÓÓ • ØØ H Z H Z ÓÓ ØØ ÓÓ Ø H ØØ

Michel Brou´e Reﬂection groups, braids, Hecke algebras ∼ G 1 × Q × Hom(G, C ) −→ H∈A Hom(GH , C ) ∼ Q G Hom(BG , Z) −→ H∈A Hom(BGH , Z)

2 `H is a length : X ` (sn1 · sn2 ··· snk ) = n H H1,γ1 H2,γ2 Hk ,γk i {i |(Hi =G H)}

Proposition

`H BG / Z

detH G /Z/eH Z

Michel Brou´e Reﬂection groups, braids, Hecke algebras ∼ Q G Hom(BG , Z) −→ H∈A Hom(BGH , Z)

2 `H is a length : X ` (sn1 · sn2 ··· snk ) = n H H1,γ1 H2,γ2 Hk ,γk i {i |(Hi =G H)}

∼ G 1 × Q × Hom(G, C ) −→ H∈A Hom(GH , C )

`H BG / Z

detH G /Z/eH Z

Proposition

Michel Brou´e Reﬂection groups, braids, Hecke algebras 2 `H is a length : X ` (sn1 · sn2 ··· snk ) = n H H1,γ1 H2,γ2 Hk ,γk i {i |(Hi =G H)}

∼ Q G Hom(BG , Z) −→ H∈A Hom(BGH , Z)

`H BG / Z

detH G /Z/eH Z

Proposition ∼ G 1 × Q × Hom(G, C ) −→ H∈A Hom(GH , C )

Michel Brou´e Reﬂection groups, braids, Hecke algebras X ` (sn1 · sn2 ··· snk ) = n H H1,γ1 H2,γ2 Hk ,γk i {i |(Hi =G H)}

2 `H is a length :

`H BG / Z

detH G /Z/eH Z

Proposition ∼ G 1 × Q × Hom(G, C ) −→ H∈A Hom(GH , C ) ∼ Q G Hom(BG , Z) −→ H∈A Hom(BGH , Z)

Michel Brou´e Reﬂection groups, braids, Hecke algebras X ` (sn1 · sn2 ··· snk ) = n H H1,γ1 H2,γ2 Hk ,γk i {i |(Hi =G H)}

`H BG / Z

detH G /Z/eH Z

Proposition ∼ G 1 × Q × Hom(G, C ) −→ H∈A Hom(GH , C ) ∼ Q G Hom(BG , Z) −→ H∈A Hom(BGH , Z)

2 `H is a length :

Michel Brou´e Reﬂection groups, braids, Hecke algebras `H BG / Z

detH G /Z/eH Z

Proposition ∼ G 1 × Q × Hom(G, C ) −→ H∈A Hom(GH , C ) ∼ Q G Hom(BG , Z) −→ H∈A Hom(BGH , Z)

2 `H is a length : X ` (sn1 · sn2 ··· snk ) = n H H1,γ1 H2,γ2 Hk ,γk i {i |(Hi =G H)}

Michel Brou´e Reﬂection groups, braids, Hecke algebras Proposition ∼ G 1 × Q × Hom(G, C ) −→ H∈A Hom(GH , C ) ∼ Q G Hom(BG , Z) −→ H∈A Hom(BGH , Z)

2 `H is a length : X ` (sn1 · sn2 ··· snk ) = n H H1,γ1 H2,γ2 Hk ,γk i {i |(Hi =G H)}

`H BG / Z

detH G /Z/eH Z

Michel Brou´e Reﬂection groups, braids, Hecke algebras 1 ZPG = hπi and ZBG = hζi.

2 We have the short exact sequence

1 −→ ZPG −→ ZBG −→ ZG −→ 1

2iπt Let π ∈ PG deﬁned by π : t 7→ e x0 2iπt/z Let ζ ∈ BG deﬁned by ζ : t 7→ e x0

Theorem

From now on we assume that G is irreducible on V . Hence the centre of G is cyclic. Set z := |ZG| and ζ := e2iπ/z .

Center of the braid groups

Michel Brou´e Reﬂection groups, braids, Hecke algebras 1 ZPG = hπi and ZBG = hζi.

2 We have the short exact sequence

1 −→ ZPG −→ ZBG −→ ZG −→ 1

2iπt Let π ∈ PG deﬁned by π : t 7→ e x0 2iπt/z Let ζ ∈ BG deﬁned by ζ : t 7→ e x0

Theorem

Hence the centre of G is cyclic. Set z := |ZG| and ζ := e2iπ/z .

Center of the braid groups

From now on we assume that G is irreducible on V .

Michel Brou´e Reﬂection groups, braids, Hecke algebras 1 ZPG = hπi and ZBG = hζi.

2 We have the short exact sequence

1 −→ ZPG −→ ZBG −→ ZG −→ 1

Theorem

2iπt Let π ∈ PG deﬁned by π : t 7→ e x0 2iπt/z Let ζ ∈ BG deﬁned by ζ : t 7→ e x0

Center of the braid groups

From now on we assume that G is irreducible on V . Hence the centre of G is cyclic. Set z := |ZG| and ζ := e2iπ/z .

Michel Brou´e Reﬂection groups, braids, Hecke algebras 1 ZPG = hπi and ZBG = hζi.

2 We have the short exact sequence

1 −→ ZPG −→ ZBG −→ ZG −→ 1

Theorem

2iπt/z Let ζ ∈ BG deﬁned by ζ : t 7→ e x0

Center of the braid groups

From now on we assume that G is irreducible on V . Hence the centre of G is cyclic. Set z := |ZG| and ζ := e2iπ/z .

2iπt Let π ∈ PG deﬁned by π : t 7→ e x0

Michel Brou´e Reﬂection groups, braids, Hecke algebras 1 ZPG = hπi and ZBG = hζi.

2 We have the short exact sequence

1 −→ ZPG −→ ZBG −→ ZG −→ 1

Theorem

Center of the braid groups

From now on we assume that G is irreducible on V . Hence the centre of G is cyclic. Set z := |ZG| and ζ := e2iπ/z .

2iπt Let π ∈ PG deﬁned by π : t 7→ e x0 2iπt/z Let ζ ∈ BG deﬁned by ζ : t 7→ e x0

Michel Brou´e Reﬂection groups, braids, Hecke algebras 1 ZPG = hπi and ZBG = hζi.

2 We have the short exact sequence

1 −→ ZPG −→ ZBG −→ ZG −→ 1

Center of the braid groups

From now on we assume that G is irreducible on V . Hence the centre of G is cyclic. Set z := |ZG| and ζ := e2iπ/z .

2iπt Let π ∈ PG deﬁned by π : t 7→ e x0 2iπt/z Let ζ ∈ BG deﬁned by ζ : t 7→ e x0

Theorem

Michel Brou´e Reﬂection groups, braids, Hecke algebras 2 We have the short exact sequence

1 −→ ZPG −→ ZBG −→ ZG −→ 1

Center of the braid groups

From now on we assume that G is irreducible on V . Hence the centre of G is cyclic. Set z := |ZG| and ζ := e2iπ/z .

2iπt Let π ∈ PG deﬁned by π : t 7→ e x0 2iπt/z Let ζ ∈ BG deﬁned by ζ : t 7→ e x0

Theorem

1 ZPG = hπi and ZBG = hζi.

Michel Brou´e Reﬂection groups, braids, Hecke algebras Center of the braid groups

From now on we assume that G is irreducible on V . Hence the centre of G is cyclic. Set z := |ZG| and ζ := e2iπ/z .

2iπt Let π ∈ PG deﬁned by π : t 7→ e x0 2iπt/z Let ζ ∈ BG deﬁned by ζ : t 7→ e x0

Theorem

1 ZPG = hπi and ZBG = hζi.

2 We have the short exact sequence

1 −→ ZPG −→ ZBG −→ ZG −→ 1

Michel Broué Reflection groups, braids, Hecke algebras The choice of a Coxeter generating set for G defines a presentation of BG

Example : ··· s t1 t2 tr−1

2 2 2 ··· 2 s t1 t2 tr−1

and a “section” (not a group morphism !) of the map BG G using reduced decompositions.

Let w0 be the longest element of G, and let g0 be its lift in BG .

2 π = g0

2r Example : π = (st1t2 ··· tr−1)

Case of Coxeter groups

Michel Brou´e Reﬂection groups, braids, Hecke algebras Example : ··· s t1 t2 tr−1

2 2 2 ··· 2 s t1 t2 tr−1

and a “section” (not a group morphism !) of the map BG G using reduced decompositions.

Let w0 be the longest element of G, and let g0 be its lift in BG .

2 π = g0

2r Example : π = (st1t2 ··· tr−1)

Case of Coxeter groups

The choice of a Coxeter generating set for G deﬁnes a presentation of BG

Michel Brou´e Reﬂection groups, braids, Hecke algebras and a “section” (not a group morphism !) of the map BG G using reduced decompositions.

Let w0 be the longest element of G, and let g0 be its lift in BG .

2 π = g0

2r Example : π = (st1t2 ··· tr−1)

Case of Coxeter groups

The choice of a Coxeter generating set for G deﬁnes a presentation of BG

Example : ··· s t1 t2 tr−1

2 2 2 ··· 2 s t1 t2 tr−1

Michel Brou´e Reﬂection groups, braids, Hecke algebras Let w0 be the longest element of G, and let g0 be its lift in BG .

2 π = g0

2r Example : π = (st1t2 ··· tr−1)

Case of Coxeter groups

The choice of a Coxeter generating set for G deﬁnes a presentation of BG

Example : ··· s t1 t2 tr−1

2 2 2 ··· 2 s t1 t2 tr−1 and a “section” (not a group morphism !) of the map BG G using reduced decompositions.

Michel Brou´e Reﬂection groups, braids, Hecke algebras 2 π = g0

2r Example : π = (st1t2 ··· tr−1)

Case of Coxeter groups

The choice of a Coxeter generating set for G deﬁnes a presentation of BG

Example : ··· s t1 t2 tr−1

2 2 2 ··· 2 s t1 t2 tr−1 and a “section” (not a group morphism !) of the map BG G using reduced decompositions.

Let w0 be the longest element of G, and let g0 be its lift in BG .

Michel Brou´e Reﬂection groups, braids, Hecke algebras 2r Example : π = (st1t2 ··· tr−1)

Case of Coxeter groups

The choice of a Coxeter generating set for G deﬁnes a presentation of BG

Example : ··· s t1 t2 tr−1

2 2 2 ··· 2 s t1 t2 tr−1 and a “section” (not a group morphism !) of the map BG G using reduced decompositions.

Let w0 be the longest element of G, and let g0 be its lift in BG .

2 π = g0

Michel Brou´e Reﬂection groups, braids, Hecke algebras Case of Coxeter groups

The choice of a Coxeter generating set for G deﬁnes a presentation of BG

Example : ··· s t1 t2 tr−1

2 2 2 ··· 2 s t1 t2 tr−1 and a “section” (not a group morphism !) of the map BG G using reduced decompositions.

Let w0 be the longest element of G, and let g0 be its lift in BG .

2 π = g0

2r Example : π = (st1t2 ··· tr−1)

Michel Broué Reflection groups, braids, Hecke algebras (denoted by ΓG ) The minimal number of reflections needed to generate G The minimal number of braid reflections needed to generate BG d(N + Nh)/dr e

S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

1 The following integers are equal :

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees.

Artin–like presentations

1 The following integers are equal :

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where

1 The following integers are equal :

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

I is a ﬁnite set of relations which are multi–homogeneous,

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a ﬁnite set of distinguished braid reﬂections,

1 The following integers are equal :

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

1 The following integers are equal :

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

i.e., such that (for each i) vi and wi are positive words in elements of S

1 The following integers are equal :

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees.

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees. 1 The following integers are equal :

Michel Brou´e Reﬂection groups, braids, Hecke algebras (denoted by ΓG )

The minimal number of braid reﬂections needed to generate BG d(N + Nh)/dr e

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees. 1 The following integers are equal : The minimal number of reﬂections needed to generate G

Michel Brou´e Reﬂection groups, braids, Hecke algebras (denoted by ΓG )

d(N + Nh)/dr e

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

Theorem (Bessis)

Michel Brou´e Reﬂection groups, braids, Hecke algebras (denoted by ΓG )

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

Theorem (Bessis)

Michel Broué Reflection groups, braids, Hecke algebras 2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reflections.

Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees.

1 The following integers are equal (denoted by ΓG ) : The minimal number of reflections needed to generate G The minimal number of braid reflections needed to generate BG d(N + Nh)/dr e (N := number of reflections, Nh := number of hyperplanes)

Michel Brou´e Reﬂection groups, braids, Hecke algebras Artin–like presentations

An Artin–like presentation is

hs ∈ S | {vi = wi }i∈I i where S is a finite set of distinguished braid reflections, I is a finite set of relations which are multi–homogeneous,

Theorem (Bessis)

Let G ⊂ GL(V ) be a complex reﬂection group. Let d1 ≤ d2 ≤ · · · ≤ dr be the family of its invariant degrees.

1 The following integers are equal (denoted by ΓG ) : The minimal number of reﬂections needed to generate G The minimal number of braid reﬂections needed to generate BG d(N + Nh)/dr e

2 EitherΓ G = r orΓ G = r + 1, and the group BG has an Artin–like presentation by ΓG braid reﬂections.

Michel Brou´e Reﬂection groups, braids, Hecke algebras t We denote by Dbr and call braid diagram the diagram s e u n

b t Let D be a diagram like s a e D represents the relations c u

stustu ··· = tustus ··· = ustustn··· and sa = tb = uc = 1 | {z } | {z } | {z } e factors e factors e factors

which represents the relations

stustu ··· = tustus ··· = ustust ··· | {z } | {z } | {z } e factors e factors e factors Note that

3 t 3 t 3 t G7 : s 2 G11 : s 2 G19 : s 2 3 u 4 u 5 u have same braid diagram.n n n

The braid diagrams

Michel Brou´e Reﬂection groups, braids, Hecke algebras t We denote by Dbr and call braid diagram the diagram s e u n

D represents the relations

stustu ··· = tustus ··· = ustust ··· and sa = tb = uc = 1 | {z } | {z } | {z } e factors e factors e factors

which represents the relations

stustu ··· = tustus ··· = ustust ··· | {z } | {z } | {z } e factors e factors e factors Note that

3 t 3 t 3 t G7 : s 2 G11 : s 2 G19 : s 2 3 u 4 u 5 u have same braid diagram.n n n

The braid diagrams

b t Let D be a diagram like s a e c u n

Michel Brou´e Reﬂection groups, braids, Hecke algebras t We denote by Dbr and call braid diagram the diagram s e u n

and sa = tb = uc = 1

which represents the relations

stustu ··· = tustus ··· = ustust ··· | {z } | {z } | {z } e factors e factors e factors Note that

3 t 3 t 3 t G7 : s 2 G11 : s 2 G19 : s 2 3 u 4 u 5 u have same braid diagram.n n n

The braid diagrams

b t Let D be a diagram like s a e D represents the relations c u

stustu ··· = tustus ··· = ustustn··· | {z } | {z } | {z } e factors e factors e factors

Michel Brou´e Reﬂection groups, braids, Hecke algebras t We denote by Dbr and call braid diagram the diagram s e u which represents the relations n stustu ··· = tustus ··· = ustust ··· | {z } | {z } | {z } e factors e factors e factors Note that

3 t 3 t 3 t G7 : s 2 G11 : s 2 G19 : s 2 3 u 4 u 5 u have same braid diagram.n n n

The braid diagrams

b t Let D be a diagram like s a e D represents the relations c u

stustu ··· = tustus ··· = ustustn··· and sa = tb = uc = 1 | {z } | {z } | {z } e factors e factors e factors

Michel Brou´e Reﬂection groups, braids, Hecke algebras which represents the relations

stustu ··· = tustus ··· = ustust ··· | {z } | {z } | {z } e factors e factors e factors Note that

3 t 3 t 3 t G7 : s 2 G11 : s 2 G19 : s 2 3 u 4 u 5 u have same braid diagram.n n n

The braid diagrams

b t Let D be a diagram like s a e D represents the relations c u

stustu ··· = tustus ··· = ustustn··· and sa = tb = uc = 1 | {z } | {z } | {z } e factors e factors e factors

t We denote by Dbr and call braid diagram the diagram s e u n

Michel Brou´e Reﬂection groups, braids, Hecke algebras Note that

3 t 3 t 3 t G7 : s 2 G11 : s 2 G19 : s 2 3 u 4 u 5 u have same braid diagram.n n n

The braid diagrams

b t Let D be a diagram like s a e D represents the relations c u

stustu ··· = tustus ··· = ustustn··· and sa = tb = uc = 1 | {z } | {z } | {z } e factors e factors e factors

t We denote by Dbr and call braid diagram the diagram s e u which represents the relations n stustu ··· = tustus ··· = ustust ··· | {z } | {z } | {z } e factors e factors e factors

Michel Brou´e Reﬂection groups, braids, Hecke algebras have same braid diagram.

The braid diagrams

b t Let D be a diagram like s a e D represents the relations c u

stustu ··· = tustus ··· = ustustn··· and sa = tb = uc = 1 | {z } | {z } | {z } e factors e factors e factors

t We denote by Dbr and call braid diagram the diagram s e u which represents the relations n stustu ··· = tustus ··· = ustust ··· | {z } | {z } | {z } e factors e factors e factors Note that

3 t 3 t 3 t G7 : s 2 G11 : s 2 G19 : s 2 3 u 4 u 5 u n n n

Michel Brou´e Reﬂection groups, braids, Hecke algebras The braid diagrams

b t Let D be a diagram like s a e D represents the relations c u

stustu ··· = tustus ··· = ustustn··· and sa = tb = uc = 1 | {z } | {z } | {z } e factors e factors e factors

t We denote by Dbr and call braid diagram the diagram s e u which represents the relations n stustu ··· = tustus ··· = ustust ··· | {z } | {z } | {z } e factors e factors e factors Note that

3 t 3 t 3 t G7 : s 2 G11 : s 2 G19 : s 2 3 u 4 u 5 u have same braid diagram.n n n

Michel Brou´e Reﬂection groups, braids, Hecke algebras The groups Gn for n = 4, 5, 8, 16, 20, as well as the dihedral groups, e have diagrams of type d d , corresponding to the s t presentation

sd = td = 1 and ststs ··· = tstst ··· | {z } | {z } e factors e factors

there is a diagram D, whose set of nodes N (D) is identiﬁed with a set of distinguished reﬂections in G, such that

Theorem

For each s ∈ N (D), there exists a braid reﬂection s ∈ BG above s such that the set {s}s∈N (D), together with the braid relations of Dbr, is a presentation of BG .

For each irreducible complex irreducible group G,

Michel Brou´e Reﬂection groups, braids, Hecke algebras The groups Gn for n = 4, 5, 8, 16, 20, as well as the dihedral groups, e have diagrams of type d d , corresponding to the s t presentation

sd = td = 1 and ststs ··· = tstst ··· | {z } | {z } e factors e factors

whose set of nodes N (D) is identiﬁed with a set of distinguished reﬂections in G, such that

Theorem

For each s ∈ N (D), there exists a braid reﬂection s ∈ BG above s such that the set {s}s∈N (D), together with the braid relations of Dbr, is a presentation of BG .

For each irreducible complex irreducible group G, there is a diagram D,

Michel Brou´e Reﬂection groups, braids, Hecke algebras The groups Gn for n = 4, 5, 8, 16, 20, as well as the dihedral groups, e have diagrams of type d d , corresponding to the s t presentation

sd = td = 1 and ststs ··· = tstst ··· | {z } | {z } e factors e factors

such that

Theorem

For each s ∈ N (D), there exists a braid reﬂection s ∈ BG above s such that the set {s}s∈N (D), together with the braid relations of Dbr, is a presentation of BG .

For each irreducible complex irreducible group G, there is a diagram D, whose set of nodes N (D) is identiﬁed with a set of distinguished reﬂections in G,

Michel Brou´e Reﬂection groups, braids, Hecke algebras The groups Gn for n = 4, 5, 8, 16, 20, as well as the dihedral groups, e have diagrams of type d d , corresponding to the s t presentation

sd = td = 1 and ststs ··· = tstst ··· | {z } | {z } e factors e factors

Theorem

For each s ∈ N (D), there exists a braid reﬂection s ∈ BG above s such that the set {s}s∈N (D), together with the braid relations of Dbr, is a presentation of BG .

For each irreducible complex irreducible group G, there is a diagram D, whose set of nodes N (D) is identiﬁed with a set of distinguished reﬂections in G, such that

Michel Brou´e Reﬂection groups, braids, Hecke algebras corresponding to the presentation

sd = td = 1 and ststs ··· = tstst ··· | {z } | {z } e factors e factors

The groups Gn for n = 4, 5, 8, 16, 20, as well as the dihedral groups, e have diagrams of type d d , s t

For each irreducible complex irreducible group G, there is a diagram D, whose set of nodes N (D) is identiﬁed with a set of distinguished reﬂections in G, such that

Theorem

For each s ∈ N (D), there exists a braid reﬂection s ∈ BG above s such that the set {s}s∈N (D), together with the braid relations of Dbr, is a presentation of BG .

Michel Brou´e Reﬂection groups, braids, Hecke algebras corresponding to the presentation

sd = td = 1 and ststs ··· = tstst ··· | {z } | {z } e factors e factors

For each irreducible complex irreducible group G, there is a diagram D, whose set of nodes N (D) is identiﬁed with a set of distinguished reﬂections in G, such that

Theorem

For each s ∈ N (D), there exists a braid reﬂection s ∈ BG above s such that the set {s}s∈N (D), together with the braid relations of Dbr, is a presentation of BG .

The groups Gn for n = 4, 5, 8, 16, 20, as well as the dihedral groups, e have diagrams of type d d , s t

Michel Brou´e Reﬂection groups, braids, Hecke algebras sd = td = 1 and ststs ··· = tstst ··· | {z } | {z } e factors e factors

For each irreducible complex irreducible group G, there is a diagram D, whose set of nodes N (D) is identiﬁed with a set of distinguished reﬂections in G, such that

Theorem

For each s ∈ N (D), there exists a braid reﬂection s ∈ BG above s such that the set {s}s∈N (D), together with the braid relations of Dbr, is a presentation of BG .

The groups Gn for n = 4, 5, 8, 16, 20, as well as the dihedral groups, e have diagrams of type d d , corresponding to the s t presentation

Michel Brou´e Reﬂection groups, braids, Hecke algebras and ststs ··· = tstst ··· | {z } | {z } e factors e factors

For each irreducible complex irreducible group G, there is a diagram D, whose set of nodes N (D) is identiﬁed with a set of distinguished reﬂections in G, such that

Theorem

For each s ∈ N (D), there exists a braid reﬂection s ∈ BG above s such that the set {s}s∈N (D), together with the braid relations of Dbr, is a presentation of BG .

The groups Gn for n = 4, 5, 8, 16, 20, as well as the dihedral groups, e have diagrams of type d d , corresponding to the s t presentation

sd = td = 1

Michel Broué Reflection groups, braids, Hecke algebras For each irreducible complex irreducible group G, there is a diagram D, whose set of nodes N (D) is identified with a set of distinguished reflections in G, such that

Theorem

For each s ∈ N (D), there exists a braid reﬂection s ∈ BG above s such that the set {s}s∈N (D), together with the braid relations of Dbr, is a presentation of BG .

The groups Gn for n = 4, 5, 8, 16, 20, as well as the dihedral groups, e have diagrams of type d d , corresponding to the s t presentation

sd = td = 1 and ststs ··· = tstst ··· | {z } | {z } e factors e factors

Michel Brou´e Reﬂection groups, braids, Hecke algebras corresponding to the presentation

s2 = t2 = u2 = v 2 = w 2 = 1 , uv = vu , sw = ws , vw = wv , sut = uts = tsu , svs = vsv , tvt = vtv , twt = wtw , wuw = uwu .

The group G18 has diagram 5 3 corresponding to the s t presentation s5 = t3 = 1 and stst = tsts .

2 2 s u The group G31 has diagram 2 2 2 v tn w

Michel Brou´e Reﬂection groups, braids, Hecke algebras corresponding to the presentation

s2 = t2 = u2 = v 2 = w 2 = 1 , uv = vu , sw = ws , vw = wv , sut = uts = tsu , svs = vsv , tvt = vtv , twt = wtw , wuw = uwu .

2 2 s u The group G31 has diagram 2 2 2 v tn w

The group G18 has diagram 5 3 corresponding to the s t presentation s5 = t3 = 1 and stst = tsts .

Michel Brou´e Reﬂection groups, braids, Hecke algebras corresponding to the presentation

s2 = t2 = u2 = v 2 = w 2 = 1 , uv = vu , sw = ws , vw = wv , sut = uts = tsu , svs = vsv , tvt = vtv , twt = wtw , wuw = uwu .

The group G18 has diagram 5 3 corresponding to the s t presentation s5 = t3 = 1 and stst = tsts .

2 2 s u The group G31 has diagram 2 2 2 v tn w

Michel Brou´e Reﬂection groups, braids, Hecke algebras s2 = t2 = u2 = v 2 = w 2 = 1 , uv = vu , sw = ws , vw = wv , sut = uts = tsu , svs = vsv , tvt = vtv , twt = wtw , wuw = uwu .

The group G18 has diagram 5 3 corresponding to the s t presentation s5 = t3 = 1 and stst = tsts .

2 2 s u The group G31 has diagram 2 2 2 corresponding to the v tn w presentation

Michel Brou´e Reﬂection groups, braids, Hecke algebras uv = vu , sw = ws , vw = wv , sut = uts = tsu , svs = vsv , tvt = vtv , twt = wtw , wuw = uwu .

The group G18 has diagram 5 3 corresponding to the s t presentation s5 = t3 = 1 and stst = tsts .

2 2 s u The group G31 has diagram 2 2 2 corresponding to the v tn w presentation

s2 = t2 = u2 = v 2 = w 2 = 1 ,

Michel Brou´e Reﬂection groups, braids, Hecke algebras svs = vsv , tvt = vtv , twt = wtw , wuw = uwu .

The group G18 has diagram 5 3 corresponding to the s t presentation s5 = t3 = 1 and stst = tsts .

2 2 s u The group G31 has diagram 2 2 2 corresponding to the v tn w presentation

s2 = t2 = u2 = v 2 = w 2 = 1 , uv = vu , sw = ws , vw = wv , sut = uts = tsu ,

Michel Brou´e Reﬂection groups, braids, Hecke algebras The group G18 has diagram 5 3 corresponding to the s t presentation s5 = t3 = 1 and stst = tsts .

2 2 s u The group G31 has diagram 2 2 2 corresponding to the v tn w presentation

s2 = t2 = u2 = v 2 = w 2 = 1 , uv = vu , sw = ws , vw = wv , sut = uts = tsu , svs = vsv , tvt = vtv , twt = wtw , wuw = uwu .

Michel Brou´e Reﬂection groups, braids, Hecke algebras • Solution of an old conjecture

Theorem The space V reg is a K(π, 1).

• Springer’s theory of regular elements An element g ∈ G is ζ-regular if g has a regular eigenvector for the eigenvalue ζ. Denote by V (g, ζ) the ζ-eigenspace of g in V .

Theorem (Springer, 1974)

Let g ∈ G be d-regular. Then CG (g) is a frg on V (g, ζ), with set of degrees {di | d divides di }.

Idea of proof: show that CG (g) has polynomial invariants on V (g, ζ).

More on the work of Bessis

Michel Brou´e Reﬂection groups, braids, Hecke algebras Theorem The space V reg is a K(π, 1).

• Springer’s theory of regular elements An element g ∈ G is ζ-regular if g has a regular eigenvector for the eigenvalue ζ. Denote by V (g, ζ) the ζ-eigenspace of g in V .

Theorem (Springer, 1974)

Let g ∈ G be d-regular. Then CG (g) is a frg on V (g, ζ), with set of degrees {di | d divides di }.

Idea of proof: show that CG (g) has polynomial invariants on V (g, ζ).

More on the work of Bessis

• Solution of an old conjecture

Michel Brou´e Reﬂection groups, braids, Hecke algebras • Springer’s theory of regular elements An element g ∈ G is ζ-regular if g has a regular eigenvector for the eigenvalue ζ. Denote by V (g, ζ) the ζ-eigenspace of g in V .

Theorem (Springer, 1974)

Let g ∈ G be d-regular. Then CG (g) is a frg on V (g, ζ), with set of degrees {di | d divides di }.

Idea of proof: show that CG (g) has polynomial invariants on V (g, ζ).

More on the work of Bessis

• Solution of an old conjecture

Theorem The space V reg is a K(π, 1).

Michel Brou´e Reﬂection groups, braids, Hecke algebras An element g ∈ G is ζ-regular if g has a regular eigenvector for the eigenvalue ζ. Denote by V (g, ζ) the ζ-eigenspace of g in V .

Theorem (Springer, 1974)

Let g ∈ G be d-regular. Then CG (g) is a frg on V (g, ζ), with set of degrees {di | d divides di }.

Idea of proof: show that CG (g) has polynomial invariants on V (g, ζ).

More on the work of Bessis

• Solution of an old conjecture

Theorem The space V reg is a K(π, 1).

• Springer’s theory of regular elements

Michel Brou´e Reﬂection groups, braids, Hecke algebras Denote by V (g, ζ) the ζ-eigenspace of g in V .

Theorem (Springer, 1974)

Let g ∈ G be d-regular. Then CG (g) is a frg on V (g, ζ), with set of degrees {di | d divides di }.

Idea of proof: show that CG (g) has polynomial invariants on V (g, ζ).

More on the work of Bessis

• Solution of an old conjecture

Theorem The space V reg is a K(π, 1).

• Springer’s theory of regular elements An element g ∈ G is ζ-regular if g has a regular eigenvector for the eigenvalue ζ.

Michel Brou´e Reﬂection groups, braids, Hecke algebras Theorem (Springer, 1974)

Let g ∈ G be d-regular. Then CG (g) is a frg on V (g, ζ), with set of degrees {di | d divides di }.

Idea of proof: show that CG (g) has polynomial invariants on V (g, ζ).

More on the work of Bessis

• Solution of an old conjecture

Theorem The space V reg is a K(π, 1).

• Springer’s theory of regular elements An element g ∈ G is ζ-regular if g has a regular eigenvector for the eigenvalue ζ. Denote by V (g, ζ) the ζ-eigenspace of g in V .

Michel Brou´e Reﬂection groups, braids, Hecke algebras Idea of proof: show that CG (g) has polynomial invariants on V (g, ζ).

More on the work of Bessis

• Solution of an old conjecture

Theorem The space V reg is a K(π, 1).

• Springer’s theory of regular elements An element g ∈ G is ζ-regular if g has a regular eigenvector for the eigenvalue ζ. Denote by V (g, ζ) the ζ-eigenspace of g in V .

Theorem (Springer, 1974)

Let g ∈ G be d-regular. Then CG (g) is a frg on V (g, ζ), with set of degrees {di | d divides di }.

Michel Brou´e Reﬂection groups, braids, Hecke algebras More on the work of Bessis

• Solution of an old conjecture

Theorem The space V reg is a K(π, 1).

• Springer’s theory of regular elements An element g ∈ G is ζ-regular if g has a regular eigenvector for the eigenvalue ζ. Denote by V (g, ζ) the ζ-eigenspace of g in V .

Theorem (Springer, 1974)

Let g ∈ G be d-regular. Then CG (g) is a frg on V (g, ζ), with set of degrees {di | d divides di }.

Idea of proof: show that CG (g) has polynomial invariants on V (g, ζ).

Michel Brou´e Reﬂection groups, braids, Hecke algebras n • W = Sr in its natural permutation representation on V = C . Assume that d|r. Then the product of r/d disjoint d-cycles is d-regular, with centralizer Cd o Sr/d , with degrees

{di | d divides di } = {d, 2d,..., d · r/d = r}.

• G = W (F4), a Weyl group. There exist 3-regular elements in G. The degrees of W (F4) are 2, 6, 8, 12, so the centralizer has degrees 6, 12: It is the complex reﬂection group G5.

So, even if G is a Weyl group, CG (g) may be a truly complex reﬂection group.

Examples

Michel Brou´e Reﬂection groups, braids, Hecke algebras Then the product of r/d disjoint d-cycles is d-regular, with centralizer Cd o Sr/d , with degrees

{di | d divides di } = {d, 2d,..., d · r/d = r}.

• G = W (F4), a Weyl group. There exist 3-regular elements in G. The degrees of W (F4) are 2, 6, 8, 12, so the centralizer has degrees 6, 12: It is the complex reﬂection group G5.

So, even if G is a Weyl group, CG (g) may be a truly complex reﬂection group.

Examples n • W = Sr in its natural permutation representation on V = C . Assume that d|r.

Michel Broué Reflection groups, braids, Hecke algebras • G = W (F4), a Weyl group. There exist 3-regular elements in G. The degrees of W (F4) are 2, 6, 8, 12, so the centralizer has degrees 6, 12: It is the complex reflection group G5.

So, even if G is a Weyl group, CG (g) may be a truly complex reﬂection group.

Examples n • W = Sr in its natural permutation representation on V = C . Assume that d|r. Then the product of r/d disjoint d-cycles is d-regular, with centralizer Cd o Sr/d , with degrees

{di | d divides di } = {d, 2d,..., d · r/d = r}.

Michel Broué Reflection groups, braids, Hecke algebras The degrees of W (F4) are 2, 6, 8, 12, so the centralizer has degrees 6, 12: It is the complex reflection group G5.

So, even if G is a Weyl group, CG (g) may be a truly complex reﬂection group.

Examples n • W = Sr in its natural permutation representation on V = C . Assume that d|r. Then the product of r/d disjoint d-cycles is d-regular, with centralizer Cd o Sr/d , with degrees

{di | d divides di } = {d, 2d,..., d · r/d = r}.

• G = W (F4), a Weyl group. There exist 3-regular elements in G.

Michel Broué Reflection groups, braids, Hecke algebras so the centralizer has degrees 6, 12: It is the complex reflection group G5.

So, even if G is a Weyl group, CG (g) may be a truly complex reﬂection group.

Examples n • W = Sr in its natural permutation representation on V = C . Assume that d|r. Then the product of r/d disjoint d-cycles is d-regular, with centralizer Cd o Sr/d , with degrees

{di | d divides di } = {d, 2d,..., d · r/d = r}.

• G = W (F4), a Weyl group. There exist 3-regular elements in G. The degrees of W (F4) are 2, 6, 8, 12,

Michel Broué Reflection groups, braids, Hecke algebras So, even if G is a Weyl group, CG (g) may be a truly complex reflection group.

Examples n • W = Sr in its natural permutation representation on V = C . Assume that d|r. Then the product of r/d disjoint d-cycles is d-regular, with centralizer Cd o Sr/d , with degrees

{di | d divides di } = {d, 2d,..., d · r/d = r}.

• G = W (F4), a Weyl group. There exist 3-regular elements in G. The degrees of W (F4) are 2, 6, 8, 12, so the centralizer has degrees 6, 12: It is the complex reﬂection group G5.

Michel Brou´e Reﬂection groups, braids, Hecke algebras Examples n • W = Sr in its natural permutation representation on V = C . Assume that d|r. Then the product of r/d disjoint d-cycles is d-regular, with centralizer Cd o Sr/d , with degrees

{di | d divides di } = {d, 2d,..., d · r/d = r}.

• G = W (F4), a Weyl group. There exist 3-regular elements in G. The degrees of W (F4) are 2, 6, 8, 12, so the centralizer has degrees 6, 12: It is the complex reﬂection group G5.

So, even if G is a Weyl group, CG (g) may be a truly complex reﬂection group.

Michel Brou´e Reﬂection groups, braids, Hecke algebras 1 The ζd –regular elements in G are the images of the d-th roots of π.

2 All d-th roots of π are conjugate in BG .

3 Let g be a d-th root of π, with image g in G. Then CBG (g) is the braid group of CG (g).

Theorem (Bessis, 2007)

2iπ/d Let ζd := e .

• Springer’s theory of regular elements in complex reﬂections groups lifts to braid groups

Michel Brou´e Reﬂection groups, braids, Hecke algebras 1 The ζd –regular elements in G are the images of the d-th roots of π.

2 All d-th roots of π are conjugate in BG .

3 Let g be a d-th root of π, with image g in G. Then CBG (g) is the braid group of CG (g).

• Springer’s theory of regular elements in complex reﬂections groups lifts to braid groups

Theorem (Bessis, 2007)

2iπ/d Let ζd := e .

Michel Brou´e Reﬂection groups, braids, Hecke algebras 2 All d-th roots of π are conjugate in BG .

3 Let g be a d-th root of π, with image g in G. Then CBG (g) is the braid group of CG (g).

• Springer’s theory of regular elements in complex reﬂections groups lifts to braid groups

Theorem (Bessis, 2007)

2iπ/d Let ζd := e .

1 The ζd –regular elements in G are the images of the d-th roots of π.

Michel Brou´e Reﬂection groups, braids, Hecke algebras 3 Let g be a d-th root of π, with image g in G. Then CBG (g) is the braid group of CG (g).

• Springer’s theory of regular elements in complex reﬂections groups lifts to braid groups

Theorem (Bessis, 2007)

2iπ/d Let ζd := e .

1 The ζd –regular elements in G are the images of the d-th roots of π.

2 All d-th roots of π are conjugate in BG .

Michel Broué Reflection groups, braids, Hecke algebras • Springer’s theory of regular elements in complex reflections groups lifts to braid groups

Theorem (Bessis, 2007)

2iπ/d Let ζd := e .

1 The ζd –regular elements in G are the images of the d-th roots of π.

2 All d-th roots of π are conjugate in BG .

3 Let g be a d-th root of π, with image g in G. Then CBG (g) is the braid group of CG (g).

Michel Broué Reflection groups, braids, Hecke algebras defines a G-invariant differential form on V reg with values in CG X ω := zH ωH H∈A hence a linear differential equation df = ωf for f : V reg → CG , i.e.,

reg 1 X αH (v) ∀v ∈ V , x ∈ V , df (x)(v) = zH f (x) 2iπ αH (x) H∈A

and

1 dαH ωH := 2iπ αH Each family !G Y (zH )H∈A ∈ CGH H∈A

For H ∈ A, let αH be a linear form with kernel H,

A monodromy representation

(after Knizhnik–Zamolodchikov, Cherednik, Dunkl, Opdam, Kohno, Brou´e-Malle-Rouquier)

Michel Broué Reflection groups, braids, Hecke algebras defines a G-invariant differential form on V reg with values in CG X ω := zH ωH H∈A

hence a linear diﬀerential equation df = ωf for f : V reg → CG , i.e.,

reg 1 X αH (v) ∀v ∈ V , x ∈ V , df (x)(v) = zH f (x) 2iπ αH (x) H∈A

Each family !G Y (zH )H∈A ∈ CGH H∈A

and

1 dαH ωH := 2iπ αH

A monodromy representation

For H ∈ A, let αH be a linear form with kernel H,

Michel Brou´e Reﬂection groups, braids, Hecke algebras i.e.,

reg 1 X αH (v) ∀v ∈ V , x ∈ V , df (x)(v) = zH f (x) 2iπ αH (x) H∈A

deﬁnes a G-invariant diﬀerential form on V reg with values in CG X ω := zH ωH H∈A

hence a linear diﬀerential equation df = ωf for f : V reg → CG ,

Each family !G Y (zH )H∈A ∈ CGH H∈A

A monodromy representation

For H ∈ A, let αH be a linear form with kernel H, and

1 dαH ωH := 2iπ αH

Michel Brou´e Reﬂection groups, braids, Hecke algebras i.e.,

reg 1 X αH (v) ∀v ∈ V , x ∈ V , df (x)(v) = zH f (x) 2iπ αH (x) H∈A

deﬁnes a G-invariant diﬀerential form on V reg with values in CG X ω := zH ωH H∈A

hence a linear diﬀerential equation df = ωf for f : V reg → CG ,

A monodromy representation

For H ∈ A, let αH be a linear form with kernel H, and

1 dαH ωH := 2iπ αH Each family !G Y (zH )H∈A ∈ CGH H∈A

Michel Brou´e Reﬂection groups, braids, Hecke algebras i.e.,

reg 1 X αH (v) ∀v ∈ V , x ∈ V , df (x)(v) = zH f (x) 2iπ αH (x) H∈A

hence a linear diﬀerential equation df = ωf for f : V reg → CG ,

A monodromy representation

For H ∈ A, let αH be a linear form with kernel H, and

1 dαH ωH := 2iπ αH Each family !G Y (zH )H∈A ∈ CGH H∈A

deﬁnes a G-invariant diﬀerential form on V reg with values in CG X ω := zH ωH H∈A

Michel Brou´e Reﬂection groups, braids, Hecke algebras i.e.,

reg 1 X αH (v) ∀v ∈ V , x ∈ V , df (x)(v) = zH f (x) 2iπ αH (x) H∈A

A monodromy representation

For H ∈ A, let αH be a linear form with kernel H, and

1 dαH ωH := 2iπ αH Each family !G Y (zH )H∈A ∈ CGH H∈A

deﬁnes a G-invariant diﬀerential form on V reg with values in CG X ω := zH ωH H∈A

hence a linear diﬀerential equation df = ωf for f : V reg → CG ,

Michel Brou´e Reﬂection groups, braids, Hecke algebras A monodromy representation

For H ∈ A, let αH be a linear form with kernel H, and

1 dαH ωH := 2iπ αH Each family !G Y (zH )H∈A ∈ CGH H∈A

deﬁnes a G-invariant diﬀerential form on V reg with values in CG X ω := zH ωH H∈A

hence a linear diﬀerential equation df = ωf for f : V reg → CG , i.e.,

reg 1 X αH (v) ∀v ∈ V , x ∈ V , df (x)(v) = zH f (x) 2iπ αH (x) H∈A

Michel Broué Reflection groups, braids, Hecke algebras 1 The form ω is integrable, hence defines a group morphism

× ρ : BG −→ (CG) .

× 2 Whenever sH,γ is a braid reﬂection around H, there is uH ∈ (CG) such that −1 ρ(sH,γ) = uH (qH sH )uH In particular, we have Y (ρ(sH,γ) − qH,θθ(sH )) = 0 . ∨ θ∈GH

∨ • GH is the group of characters of GH , ∨ • for θ ∈ GH , eH,θ is the corresponding primitive idempotent in CGH X We set qH := exp ((−2iπ/eH )zH ) =: qH,θeH,θ ∨ θ∈GH

Theorem

( For H ∈ A,

Michel Broué Reflection groups, braids, Hecke algebras 1 The form ω is integrable, hence defines a group morphism

× ρ : BG −→ (CG) .

× 2 Whenever sH,γ is a braid reﬂection around H, there is uH ∈ (CG) such that −1 ρ(sH,γ) = uH (qH sH )uH In particular, we have Y (ρ(sH,γ) − qH,θθ(sH )) = 0 . ∨ θ∈GH

∨ • for θ ∈ GH , eH,θ is the corresponding primitive idempotent in CGH X We set qH := exp ((−2iπ/eH )zH ) =: qH,θeH,θ ∨ θ∈GH

Theorem

( ∨ • G is the group of characters of GH , For H ∈ A, H

Michel Broué Reflection groups, braids, Hecke algebras 1 The form ω is integrable, hence defines a group morphism

× ρ : BG −→ (CG) .

× 2 Whenever sH,γ is a braid reﬂection around H, there is uH ∈ (CG) such that −1 ρ(sH,γ) = uH (qH sH )uH In particular, we have Y (ρ(sH,γ) − qH,θθ(sH )) = 0 . ∨ θ∈GH

X We set qH := exp ((−2iπ/eH )zH ) =: qH,θeH,θ ∨ θ∈GH

Theorem

( ∨ • GH is the group of characters of GH , For H ∈ A, ∨ • for θ ∈ GH , eH,θ is the corresponding primitive idempotent in CGH

Michel Broué Reflection groups, braids, Hecke algebras 1 The form ω is integrable, hence defines a group morphism

× ρ : BG −→ (CG) .

× 2 Whenever sH,γ is a braid reﬂection around H, there is uH ∈ (CG) such that −1 ρ(sH,γ) = uH (qH sH )uH In particular, we have Y (ρ(sH,γ) − qH,θθ(sH )) = 0 . ∨ θ∈GH

Theorem

( ∨ • GH is the group of characters of GH , For H ∈ A, ∨ • for θ ∈ GH , eH,θ is the corresponding primitive idempotent in CGH X We set qH := exp ((−2iπ/eH )zH ) =: qH,θeH,θ ∨ θ∈GH

Michel Brou´e Reﬂection groups, braids, Hecke algebras In particular, we have Y (ρ(sH,γ) − qH,θθ(sH )) = 0 . ∨ θ∈GH

1 The form ω is integrable, hence deﬁnes a group morphism

× ρ : BG −→ (CG) .

× 2 Whenever sH,γ is a braid reﬂection around H, there is uH ∈ (CG) such that −1 ρ(sH,γ) = uH (qH sH )uH

Theorem

Michel Brou´e Reﬂection groups, braids, Hecke algebras In particular, we have Y (ρ(sH,γ) − qH,θθ(sH )) = 0 . ∨ θ∈GH

× 2 Whenever sH,γ is a braid reﬂection around H, there is uH ∈ (CG) such that −1 ρ(sH,γ) = uH (qH sH )uH

Theorem 1 The form ω is integrable, hence deﬁnes a group morphism

× ρ : BG −→ (CG) .

Michel Brou´e Reﬂection groups, braids, Hecke algebras In particular, we have Y (ρ(sH,γ) − qH,θθ(sH )) = 0 . ∨ θ∈GH

Theorem 1 The form ω is integrable, hence deﬁnes a group morphism

× ρ : BG −→ (CG) .

× 2 Whenever sH,γ is a braid reﬂection around H, there is uH ∈ (CG) such that −1 ρ(sH,γ) = uH (qH sH )uH

Michel Brou´e Reﬂection groups, braids, Hecke algebras ( ∨ • GH is the group of characters of GH , For H ∈ A, ∨ • for θ ∈ GH , eH,θ is the corresponding primitive idempotent in CGH X We set qH := exp ((−2iπ/eH )zH ) =: qH,θeH,θ ∨ θ∈GH

Theorem 1 The form ω is integrable, hence deﬁnes a group morphism

× ρ : BG −→ (CG) .

× 2 Whenever sH,γ is a braid reﬂection around H, there is uH ∈ (CG) such that −1 ρ(sH,γ) = uH (qH sH )uH In particular, we have Y (ρ(sH,γ) − qH,θθ(sH )) = 0 . ∨ θ∈GH

Michel Broué Reflection groups, braids, Hecke algebras and a field of realization QG := Q ({trV (g) | (g ∈ G)}). The associated generic Hecke algebra is defined from such a presentation :

 STSTST = TSTSTS  H(G2) := < S, T ; (S − q0)(S − q1) = 0 >   (T − r0)(T − r1) = 0 ( STS = TST H(G4) := < S, T ; > (S − q0)(S − q1)(S − q2) = 0

Every complex reﬂection group G has an Artin-like presentation:

G2 : 2 2 , G4 : 3 3 s t s t

Hecke algebras

Michel Broué Reflection groups, braids, Hecke algebras The associated generic Hecke algebra is defined from such a presentation :

 STSTST = TSTSTS  H(G2) := < S, T ; (S − q0)(S − q1) = 0 >   (T − r0)(T − r1) = 0 ( STS = TST H(G4) := < S, T ; > (S − q0)(S − q1)(S − q2) = 0

and a ﬁeld of realization QG := Q ({trV (g) | (g ∈ G)}).

Hecke algebras

Every complex reﬂection group G has an Artin-like presentation:

G2 : 2 2 , G4 : 3 3 s t s t

Michel Broué Reflection groups, braids, Hecke algebras The associated generic Hecke algebra is defined from such a presentation :

 STSTST = TSTSTS  H(G2) := < S, T ; (S − q0)(S − q1) = 0 >   (T − r0)(T − r1) = 0 ( STS = TST H(G4) := < S, T ; > (S − q0)(S − q1)(S − q2) = 0

Hecke algebras

Every complex reﬂection group G has an Artin-like presentation:

G2 : 2 2 , G4 : 3 3 s t s t

and a ﬁeld of realization QG := Q ({trV (g) | (g ∈ G)}).

Michel Brou´e Reﬂection groups, braids, Hecke algebras Hecke algebras

Every complex reﬂection group G has an Artin-like presentation:

G2 : 2 2 , G4 : 3 3 s t s t

and a ﬁeld of realization QG := Q ({trV (g) | (g ∈ G)}). The associated generic Hecke algebra is deﬁned from such a presentation :

 STSTST = TSTSTS  H(G2) := < S, T ; (S − q0)(S − q1) = 0 >   (T − r0)(T − r1) = 0 ( STS = TST H(G4) := < S, T ; > (S − q0)(S − q1)(S − q2) = 0

Michel Brou´e Reﬂection groups, braids, Hecke algebras m if G = d e ··· , then for s t

|µ(QG )| −i |µ(QG )| −j (xi = ζd qi )i=0,1,...,d−1 , (yj = ζe rj )j=0,1,...,e−1

the algebra QG ((xi ), (yj ),... ))H(G) is split semisimple,

2 It becomes a split semisimple algebra over a ﬁeld obtained by extracting suitable roots of the indeterminates :

• Through the specialisation xi 7→ 1 yj 7→ 1,... , that algebra becomes the group algebra of G over QG . • The above specialisation deﬁnes a bijection ∼ Irr(G) −→ Irr(H(G)) , χ 7→ χH .

Theorem (G. Malle and al.)

1 The generic Hecke algebra H(G) is free of rank |G| over the ±1 ±1 corresponding Laurent polynomial ring Z[(qi ), (rj ),... ].

Michel Brou´e Reﬂection groups, braids, Hecke algebras m if G = d e ··· , then for s t

|µ(QG )| −i |µ(QG )| −j (xi = ζd qi )i=0,1,...,d−1 , (yj = ζe rj )j=0,1,...,e−1

the algebra QG ((xi ), (yj ),... ))H(G) is split semisimple,

Theorem (G. Malle and al.)

1 The generic Hecke algebra H(G) is free of rank |G| over the ±1 ±1 corresponding Laurent polynomial ring Z[(qi ), (rj ),... ]. 2 It becomes a split semisimple algebra over a ﬁeld obtained by extracting suitable roots of the indeterminates :

Michel Brou´e Reﬂection groups, braids, Hecke algebras |µ(QG )| −i |µ(QG )| −j (xi = ζd qi )i=0,1,...,d−1 , (yj = ζe rj )j=0,1,...,e−1

the algebra QG ((xi ), (yj ),... ))H(G) is split semisimple,

Theorem (G. Malle and al.)

m if G = d e ··· , then for s t

Michel Brou´e Reﬂection groups, braids, Hecke algebras the algebra QG ((xi ), (yj ),... ))H(G) is split semisimple,

Theorem (G. Malle and al.)

m if G = d e ··· , then for s t

|µ(QG )| −i |µ(QG )| −j (xi = ζd qi )i=0,1,...,d−1 , (yj = ζe rj )j=0,1,...,e−1

Michel Broué Reflection groups, braids, Hecke algebras • Through the specialisation xi 7→ 1 yj 7→ 1,... , that algebra becomes the group algebra of G over QG . • The above specialisation defines a bijection ∼ Irr(G) −→ Irr(H(G)) , χ 7→ χH .

Theorem (G. Malle and al.)

m if G = d e ··· , then for s t

|µ(QG )| −i |µ(QG )| −j (xi = ζd qi )i=0,1,...,d−1 , (yj = ζe rj )j=0,1,...,e−1

the algebra QG ((xi ), (yj ),... ))H(G) is split semisimple,

Michel Broué Reflection groups, braids, Hecke algebras • The above specialisation defines a bijection ∼ Irr(G) −→ Irr(H(G)) , χ 7→ χH .

Theorem (G. Malle and al.)

m if G = d e ··· , then for s t

|µ(QG )| −i |µ(QG )| −j (xi = ζd qi )i=0,1,...,d−1 , (yj = ζe rj )j=0,1,...,e−1

the algebra QG ((xi ), (yj ),... ))H(G) is split semisimple,

• Through the specialisation xi 7→ 1 yj 7→ 1,... , that algebra becomes the group algebra of G over QG .

Michel Brou´e Reﬂection groups, braids, Hecke algebras Theorem (G. Malle and al.)

m if G = d e ··· , then for s t

|µ(QG )| −i |µ(QG )| −j (xi = ζd qi )i=0,1,...,d−1 , (yj = ζe rj )j=0,1,...,e−1

the algebra QG ((xi ), (yj ),... ))H(G) is split semisimple,

Michel Brou´e Reﬂection groups, braids, Hecke algebras tq is a symmetrizing form on the algebra H(W , q).

tq specializes to the canonical linear form on the group algebra. For all b ∈ B, we have

−1 ∨ tq(bπ) tq(b ) = . tq(π)

1 There exists a unique linear form

−1 tq : H(W , q) → Z[q, q ] with the following properties.

Theorem–Conjecture

Michel Brou´e Reﬂection groups, braids, Hecke algebras tq is a symmetrizing form on the algebra H(W , q).

tq specializes to the canonical linear form on the group algebra. For all b ∈ B, we have

−1 ∨ tq(bπ) tq(b ) = . tq(π)

Theorem–Conjecture

1 There exists a unique linear form

−1 tq : H(W , q) → Z[q, q ] with the following properties.

Michel Brou´e Reﬂection groups, braids, Hecke algebras tq specializes to the canonical linear form on the group algebra. For all b ∈ B, we have

−1 ∨ tq(bπ) tq(b ) = . tq(π)

Theorem–Conjecture

1 There exists a unique linear form

−1 tq : H(W , q) → Z[q, q ] with the following properties.

tq is a symmetrizing form on the algebra H(W , q).

Michel Brou´e Reﬂection groups, braids, Hecke algebras For all b ∈ B, we have

−1 ∨ tq(bπ) tq(b ) = . tq(π)

Theorem–Conjecture

1 There exists a unique linear form

−1 tq : H(W , q) → Z[q, q ] with the following properties.

tq is a symmetrizing form on the algebra H(W , q).

tq specializes to the canonical linear form on the group algebra.

Michel Brou´e Reﬂection groups, braids, Hecke algebras Theorem–Conjecture

1 There exists a unique linear form

−1 tq : H(W , q) → Z[q, q ] with the following properties.

tq is a symmetrizing form on the algebra H(W , q).

tq specializes to the canonical linear form on the group algebra. For all b ∈ B, we have

−1 ∨ tq(bπ) tq(b ) = . tq(π)

Michel Brou´e Reﬂection groups, braids, Hecke algebras −1 As an element of Z[q, q ], tq(b) is multi–homogeneous with degree `H (b) in the indeterminates qH,θ. 0 If W is a parabolic subgroup of W , the restriction of tq to a parabolic 0 0 sub–algebra H(W , W , q) is the corresponding specialization of tq0 (W )

2 The form tq satisﬁes the following conditions.

The canonical forms tq are hidden behind Lusztig’s theory of characters of ﬁnite reductive groups, their generic degrees and Fourier transform matrices.

2 The form tq satisﬁes the following conditions.

Michel Brou´e Reﬂection groups, braids, Hecke algebras 0 If W is a parabolic subgroup of W , the restriction of tq to a parabolic 0 0 sub–algebra H(W , W , q) is the corresponding specialization of tq0 (W )

The canonical forms tq are hidden behind Lusztig’s theory of characters of ﬁnite reductive groups, their generic degrees and Fourier transform matrices.

2 The form tq satisﬁes the following conditions. −1 As an element of Z[q, q ], tq(b) is multi–homogeneous with degree `H (b) in the indeterminates qH,θ.

Michel Broué Reflection groups, braids, Hecke algebras The canonical forms tq are hidden behind Lusztig’s theory of characters of finite reductive groups, their generic degrees and Fourier transform matrices.

2 The form tq satisﬁes the following conditions. −1 As an element of Z[q, q ], tq(b) is multi–homogeneous with degree `H (b) in the indeterminates qH,θ. 0 If W is a parabolic subgroup of W , the restriction of tq to a parabolic 0 0 sub–algebra H(W , W , q) is the corresponding specialization of tq0 (W )

Michel Broué Reflection groups, braids, Hecke algebras 2 The form tq satisfies the following conditions. −1 As an element of Z[q, q ], tq(b) is multi–homogeneous with degree `H (b) in the indeterminates qH,θ. 0 If W is a parabolic subgroup of W , the restriction of tq to a parabolic 0 0 sub–algebra H(W , W , q) is the corresponding specialization of tq0 (W )

The canonical forms tq are hidden behind Lusztig’s theory of characters of ﬁnite reductive groups, their generic degrees and Fourier transform matrices.

Michel Brou´e Reﬂection groups, braids, Hecke algebras A 1–cyclotomic Hecke algebra for G2 = 2 2 : s t

 STSTST = TSTSTS   < S, T ; (S − q2)(S + 1) = 0 >    (T − q)(T + 1) = 0 

Cyclotomic algebras

Let ζ be a root of unity. A ζ–cyclotomic specialisation of the generic Hecke algebra is a morphism

−1 mi −1 nj ϕ : xi 7→ (ζ q) , yj 7→ (ζ q) ,... (mi , nj ∈ Z) , which gives rise to a ζ–cyclotomic Hecke algebra Hϕ(G).

Michel Brou´e Reﬂection groups, braids, Hecke algebras Cyclotomic algebras

Let ζ be a root of unity. A ζ–cyclotomic specialisation of the generic Hecke algebra is a morphism

−1 mi −1 nj ϕ : xi 7→ (ζ q) , yj 7→ (ζ q) ,... (mi , nj ∈ Z) , which gives rise to a ζ–cyclotomic Hecke algebra Hϕ(G).

A 1–cyclotomic Hecke algebra for G2 = 2 2 : s t

 STSTST = TSTSTS   < S, T ; (S − q2)(S + 1) = 0 >    (T − q)(T + 1) = 0 

Michel Broué Reflection groups, braids, Hecke algebras Relevance to character theory of finite reductive groups The unipotent characters in a given d–Harish–Chandra series F UnCh(G ;(L, λ)) are described by a suitable ζd –cyclotomic Hecke algebra HGF (L, λ) for the corresponding d–cyclotomic Weyl group WGF (L, λ):

F UnCh(G ;(L, λ)) ←→ Irr(HGF (L, λ)) .

A ζ3–cyclotomic Hecke algebra for B2(3) = 3 2 : s t

 STST = TSTS    < S, T ; (S − 1)(S − q)(S − q2) = 0 >    (T − q3)(T + 1) = 0 

Michel Brou´e Reﬂection groups, braids, Hecke algebras A ζ3–cyclotomic Hecke algebra for B2(3) = 3 2 : s t

 STST = TSTS    < S, T ; (S − 1)(S − q)(S − q2) = 0 >    (T − q3)(T + 1) = 0 

Relevance to character theory of ﬁnite reductive groups The unipotent characters in a given d–Harish–Chandra series F UnCh(G ;(L, λ)) are described by a suitable ζd –cyclotomic Hecke algebra HGF (L, λ) for the corresponding d–cyclotomic Weyl group WGF (L, λ):

F UnCh(G ;(L, λ)) ←→ Irr(HGF (L, λ)) .

Michel Brou´e Reﬂection groups, braids, Hecke algebras