Complex reﬂection groups as Weyl groups

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Complex reflection groups as Weyl groups

Michel Brou´e

Institut Henri–Poincar´e

July 2007

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FINITE COMPLEX REFLECTION GROUPS

Afinite reflection group onK is a finite subgroup of GL_K(V) (V a finite dimensional K–vector space) generated by pseudo–reflections, i.e., linear maps represented by







ζ 0 · · · 0 0 1 · · · 0 ... ... . .. ... 0 0 · · · 1







A finite reflection group on Ris called a Coxeter group. A finite reflection group on Qis called a Weyl group.

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FINITE COMPLEX REFLECTION GROUPS Let K be a characteristic zero field.







ζ 0 · · · 0 0 1 · · · 0 ... ... . .. ... 0 0 · · · 1







A finite reflection group on Ris called a Coxeter group. A finite reflection group on Qis called a Weyl group.

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Let K be a characteristic zero field.







ζ 0 · · · 0 0 1 · · · 0 ... ... . .. ...

0 0 · · · 1







R

A finite reflection group on Qis called a Weyl group.

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





ζ 0 · · · 0 0 1 · · · 0 ... ... . .. ...

0 0 · · · 1







A finite reflection group on Ris called a Coxeter group.

A finite reflection group on Qis called a Weyl group.

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





ζ 0 · · · 0 0 1 · · · 0 ... ... . .. ...

0 0 · · · 1







A finite reflection group on is called a Coxeter group.

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

There is one infinite seriesG(de,e,r) (d ,eandr integers), ...and 34 exceptional groupsG4,G5, . . . ,G37.

2 The group G(de,e,r) (d ,e andr integers) consists of all r×r monomial matrices with entries inµ_de such that the product of entries belongs toµd.

3 We have

G(d,1,r)'C_doS_r

G(e,e,2) =D2e (dihedral group of order 2e) G(2,2,r) =W(Dr).

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There is one infinite seriesG(de,e,r) (d ,eandr integers),

3 We have

G(d,1,r)'C_doS_r

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3 We have

G(d,1,r)'C_doS_r

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G(d,1,r)'C_doS_r

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3 We have

G(d,1,r)'C_doS_r

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FINITE REDUCTIVE GROUPS : POLYNOMIAL ORDER

G is a connected reductive algebraic group over ¯Fq, with Weyl group W, endowed with a Frobenius–like endomorphism F. The group G :=G^F is a finite reductive group.

G= GLn(¯Fq) , F : (a_i,j)7→(a^q_i,j), G = GLn(q)

Type of G— The typeG= (X,Y,R,R^∨;Wφ) of G consists of the root datum of Gendowed with the outer automorphismWφdefined byF.

Example

GL_n= (X =Y =Zⁿ,R =R^∨ =A_n;φ= 1)

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Example

GL_n= (X =Y =Zⁿ,R =R^∨ =A_n;φ= 1)

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Example

GL_n= (X =Y =Zⁿ,R =R^∨ =A_n;φ= 1)

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Example

GL_n = (X =Y =Zⁿ,R =R^∨ =A_n;φ= 1)

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Polynomial order —There is a polynomial in Z[x]

|G|(x) =x^NY

d

Φ_d(x)^a(d)

such that |G|(q) =|G|.

|GL_n|(x) =x(ⁿ₂)^d=nY

d=1

(x^d−1) =x(ⁿ₂)^d=nY

d=1

Φ_d(x)^[n/d]

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Polynomial order —There is a polynomial in Z[x]

|G|(x) =x^NY

d

Φ_d(x)^a(d)

such that |G|(q) =|G|. Example

|GL_n|(x) =x(ⁿ₂)^d=nY

d=1

(x^d−1) =x(ⁿ₂)^d=nY

d=1

Φ_d(x)^[n/d]

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Admissible subgroups — The toriofG are the subgroups of the shape T^F whereT is anF–stable torus (i.e., isomorphic to some

F¯^×× · · · ×F¯^× in G).

The Levi subgroupsof G are the subgroups of the shape L^F where L is a centralizer of anF–stable torus inG.

The split maximal torusT₁ = (F^×q)ⁿ of order (q−1)ⁿ The Coxeter maximal torusTc = GL1(Fqⁿ) of order qⁿ−1

Levi subgroups have shape GLn1(q^a¹)× · · · ×GLns(q^a^s) Cauchy theorem — The (polynomial) order of an admissible subgroup divides the (polynomial) order of the group.

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Admissible subgroups — The toriofG are the subgroups of the shape T^F whereT is anF–stable torus (i.e., isomorphic to some

F¯^×× · · · ×F¯^× in G).

Example

The split maximal torusT₁ = (F^×_q)ⁿ of order (q−1)ⁿ The Coxeter maximal torusTc = GL1(Fqⁿ) of order qⁿ−1

Levi subgroups have shape GL_n₁(q^a¹)× · · · ×GL_n_s(q^a^s)

Cauchy theorem — The (polynomial) order of an admissible subgroup divides the (polynomial) order of the group.

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T^F whereT is anF–stable torus (i.e., isomorphic to some F¯^×× · · · ×F¯^× in G).

Example

The split maximal torusT₁ = (F^×_q)ⁿ of order (q−1)ⁿ The Coxeter maximal torusTc = GL1(Fqⁿ) of order qⁿ−1

Levi subgroups have shape GL_n₁(q^a¹)× · · · ×GL_n_s(q^a^s)

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Levi subgroups and type — For

G= (X,Y,R,R^∨;Wφ) a type, a Levi subtype ofG is a type of the shape

L= (X,Y,R⁰,R^0∨;W⁰wφ)

whereR⁰ is a parabolic system ofR, with Weyl groupW⁰, and where w ∈W is such thatwφstablizesR⁰ andR^0∨.

There is a natural bijection between

the set ofG–conjugacy classes of Levi subgroups ofG, and

the set ofW–conjugacy classes of Levi subtypes ofG.

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L= (X,Y,R⁰,R^0∨;W⁰wφ)

and

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L= (X,Y,R⁰,R^0∨;W⁰wφ)

the set ofG–conjugacy classes of Levi subgroups ofG,

and

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L= (X,Y,R⁰,R^0∨;W⁰wφ)

the set ofG–conjugacy classes of Levi subgroups ofG,

G

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L= (X,Y,R⁰,R^0∨;W⁰wφ)

the set ofG–conjugacy classes of Levi subgroups ofG, and

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FINITE REDUCTIVE GROUPS : THE SYLOW THEOREMS

For Φ(x) a cyclotomic polynomial, a Φ(x)–group is a finite reductive group whose (polynomial) order is a power of Φ(x). Hence such a group is a torus.

Sylow theorem —

(polynomial) order the contribution of Φ(x) to the (polynomial) order ofG.

(2) Sylow Φ(x)–subgroups are all conjugate byG (i.e., their types are transitively permuted by the Weyl groupW).

(3) The (polynomial) index of the normalizer inG of a Sylow Φ(x)–subgroup is congruent to 1 modulo Φ(x).

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Sylow theorem —

(1) Maximal Φ(x)–subgroups (“Sylow Φ(x)–subgroups”) ofG have as (polynomial) order the contribution of Φ(x) to the (polynomial) order ofG.

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Sylow theorem —

(polynomial) order the contribution of Φ(x) to the (polynomial) order ofG.

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Sylow theorem —

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Sylow theorem —

Φ(x)–subgroup is congruent to 1 modulo Φ(x).

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Sylow theorem —

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The centralizers of Φ_d(x)–subgroups are called thed–split Levi subgroups.

Φ_d(x)–subgroups. They are all conjugate underG. Example

For eachd (1≤d ≤n), GLn(q) contains a subtorus of order Φ_d(x)^[ⁿ^d^]

Assume n=md+r withr <d. Then a minimal d–split Levi subgroup has shape GL1(q^d)^m×GLr(q).

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The minimal d–split Levi subgroups are the centralizers of Sylow Φ_d(x)–subgroups. They are all conjugate underG.

Example

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Example

For eachd (1 d n), GLn(q) contains a subtorus of order Φ_d(x)^[ⁿ^d^]

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Example

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d

subgroups.

Example

Assume n=md+r with r <d. Then a minimal d–split Levi subgroup has shape GL1(q^d)^m×GLr(q).

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GENERIC AND ORDINARY SYLOW SUBGROUPS

Let `be a prime number which does not divide|W|.

If`divides|G|=G(q), there is a unique integer d such that`divides Φ_d(q).

Then the Sylow `–subgroups ofG are nothing but the Sylow

`–subgroupsS_` of S =S^F (Sa Sylow Φ_d(x)–subgroup of G). We have

N_G(S_`) =N_G(S) and C_G(S_`) =C_G(S).

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GENERIC AND ORDINARY SYLOW SUBGROUPS Let `be a prime number which does not divide|W|.

G Φ_d(q).

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`–subgroupsS_` of S =S^F (Sa Sylow Φ_d(x)–subgroup of G).

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`–subgroupsS_` of S =S^F (Sa Sylow Φ_d(x)–subgroup of G).

We have

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CYCLOTOMIC WEYL GROUPS AND SPRINGER THEOREM

Let L(orL, orL) be a minimal d–split Levi subgroup, the centralizer of a Sylow Φd(x)–subgroupS.

G ' G G ' W L

Weyl group ofL).

Denote that group byW_G(L).

(2) Forζ a primitived-th root of the unity, we have

|W_G(L)|=G(ζ)/L(ζ).

Example

Forn =mr+d (d <r), we have W_G(L)'C_doS_r

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CYCLOTOMIC WEYL GROUPS AND SPRINGER THEOREM Let L(orL, orL) be a minimal d–split Levi subgroup, the centralizer of a Sylow Φd(x)–subgroupS.

(1) We haveN_G(L)/L'N_G(S)/C_G(S)'N_W(L)/W⁰ (where W⁰ is the Weyl group ofL).

|W_G(L)|=G(ζ)/L(ζ).

Example

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(1) We haveNG(L)/L'NG(S)/CG(S)'NW(L)/W⁰ (where W⁰ is the Weyl group ofL).

|W_G(L)|=G(ζ)/L(ζ).

Example

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|W_G(L)|=G(ζ)/L(ζ).

Example

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Let L(orL, orL) be a minimal d–split Levi subgroup, the centralizer of a Sylow Φd(x)–subgroupS.

|W_G(L)|=G(ζ)/L(ζ).

Example

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The case d = 1 — The Sylow Φ1(x)–subgroups, as well as the minimal d–split subgroups, coincide with the split maximal tori.

In case Gis split (i.e., the automorphism φinduced by F is the identity), then the groupW_G(L) coincides withW.

Springer and Springer–Lehrer theorem —The group W_G(L) is a complex reflection group (in its representation over the complex vector spaceC⊗X((ZL)Φd)).

Example

Forn =mr+d (d <r), we have W_G(L)'C_doS_r The group W_G(L) is called thed–cyclotomic Weyl group.

IfG is split, the 1–cyclotomic Weyl group is nothing but the ordinary Weyl group W.

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G L complex reflection group (in its representation over the complex vector spaceC⊗X((ZL)Φd)).

Example

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Example

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Example

G L

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Example