Reflection groups and q -reflection groups Yuri Bazlov Geometry seminar 24 November 2009
Reflections ❱ ❂ vector space over k , ❞✐♠ ❱ is finite. s ✷ ●▲ ✭ ❱ ✮ is a (pseudo) reflection if s is of finite order, ❝♦❞✐♠❱ s ❂ ✶ . • real reflections ( k ❂ R ): s ✘ ❞✐❛❣✭✶ ❀ ✶ ❀ ✿ ✿ ✿ ❀ ✶ ❀ � ✶✮ X reflecting hyperplane ❂ ❦❡r ✭ ■❞ � s ✮ O s(X) • complex reflections ( k ❂ C ): s ✘ ❞✐❛❣✭✶ ❀ ✶ ❀ ✿ ✿ ✿ ❀ ✶ ❀ ✧ ✮ ✧ ✻ ❂ ✶ a root of ✶ • char k ❃ ✵ : s may not be diagonalisable
Finite reflection groups ( subgps of ●▲ ✭ ❱ ✮ generated by reflections ) NB: Finiteness is a very strong condition! Only very special arrangements of reflecting hyperplanes (“mirrors”) lead to finite reflection groups. Reflection groups over Q ❂ Weyl groups (extremely important in the theory of semisimple Lie algebras) ❭ Real reflection groups ❂ Coxeter groups ❭ Complex reflection groups
Finite reflection groups: classification over Q and R A reflection group can be characterised by the set of ✝ normals to mirrors ( roots ) For example: R ♥ ✰✶ ✸ ❢ ❡ ✐ � ❡ ❥ ✿ ✶ ✔ ✐ ✻ ❂ ❥ ✔ ♥ ✰ ✶ ❣ • reflections s ✐❥ ✿ ❡ ✐ ✩ ❡ ❥ generate symmetric group ❙ ♥ ✰✶ (Weyl group of type ❆ ♥ , ♥ ✕ ✶ ) Weyl group of type ❇ ♥ , ♥ ✕ ✷ : • R ♥ ✸ ❢✝ ❡ ✐ ✝ ❡ ❥ ✿ ✶ ✔ ✐ ✻ ❂ ❥ ✔ ♥ ❣ ❬ ❢✝ ❡ ✐ ✿ ✶ ✔ ✐ ✔ ♥ ❣ reflection-generators s ✐❥ ✿ ❡ ✐ ✩ ❡ ❥ , t ✐ ✿ ❡ ✐ ✩ � ❡ ✐ (hyperoctahedral group, order ✷ ♥ ♥ ✦ ) Also, ❉ ♥ ( ♥ ✕ ✹ ), ❊ ✻ , ❊ ✼ , ❊ ✽ , ❋ ✹ , ● ✷ are Weyl groups • ■ ✷ ✭ ♠ ✮ , ❍ ✸ , ❍ ✹ are “extra” Coxeter groups
Root systems of ❉ ✹ and ❊ ✽ (planar projection of the polytope which is the convex hull of the root system)
Complex reflection groups The Shephard – Todd classification of finite complex reflection groups (1954) They all are direct products of the following groups: • ● ❂ ● ✭ ♠ ❀ ♣ ❀ ♥ ✮ ✔ ●▲ ♥ ✭ C ✮ , ♣ ❥ ♠ (invertible ♥ ✂ ♥ matrices with exactly ♥ non-zero entries which are ♠ th roots of ✶ , their product is an ✭ ♠ ❂ ♣ ✮ th root of ✶ ) • ● ❂ one of the exceptional groups ● ✹ ❀ ✿ ✿ ✿ ❀ ● ✸✼ . Notation: ❙ ✭ ❱ ✮ ● ❂ ❢ ♣ in ❙ ✭ ❱ ✮ ✿ ❣ ✭ ♣ ✮ ❂ ♣ ✽ ❣ ✷ ● ❣ The Chevalley – Shephard – Todd theorem (1955) Assume that ❝❤❛r k ❂ ✵ . A finite ● ❁ ●▲ ✭ ❱ ✮ is a complex reflection group, if and only if ❙ ✭ ❱ ✮ ● is a polynomial algebra.
Remark on generators of ❙ ✭ ❱ ✮ ● ❙ ✭ ❱ ✮ is an algebra of polynomials in ♥ ❂ ❞✐♠ ❱ variables. If ● ❁ ●▲ ✭ ❱ ✮ is a finite complex reflection group, ❙ ✭ ❱ ✮ ● has ♥ algebraically independent generators ♣ ✶ ❀ ✿ ✿ ✿ ❀ ♣ ♥ . Moreover, ♣ ✶ ❀ ✿ ✿ ✿ ❀ ♣ ♥ may be chosen to be homogeneous. ♣ ✶ ❀ ✿ ✿ ✿ ❀ ♣ ♥ are not unique, but ❢ ❞ ✶ ❀ ✿ ✿ ✿ ❀ ❞ ♥ ❣ ❂ ❢ ❞❡❣ ♣ ✶ ❀ ✿ ✿ ✿ ❀ ❞❡❣ ♣ ♥ ❣ is uniquely determined by ● ( degrees of ● ). One has ❞ ✶ ❞ ✷ ✿ ✿ ✿ ❞ ♥ ❂ ❥ ● ❥ . Example ● ❂ S ♥ symmetric group ✔ ●▲ ♥ ✭ C ✮ ♣ ✶ ❀ ✿ ✿ ✿ ❀ ♣ ♥ are, e.g., elementary symmetric polynomials in ♥ variables Degrees: ❞ ✶ ❂ ✶ ❀ ❞ ✷ ❂ ✷ ❀ ✿ ✿ ✿ ❀ ❞ ♥ ❂ ♥
Generalisations of the C-S-T theorem (1) ❝❤❛r k ❃ ✵ . Serre (1970s) proved that if ❙ ✭ ❱ ✮ ● is polynomial, then ● is a reflection group, and for any proper subspace ❲ ✚ ❱ , ❍ =the stabiliser of ❲ has polynomial ❙ ✭ ❲ ✮ ❍ . Kemper, Malle (1997) proved “if and only if” (using a classification of pseudoreflection groups due to Kantor, Wagner, Zalesskii, Serezhin). (2) Replace ❙ ✭ ❱ ✮ with some noncommutative algebra, on which the group ● acts. (In other words, consider a “ noncommutative space ” with an action of ● .) Below is a particular case of this: ❱ ❂ C -span of ① ✶ ❀ ✿ ✿ ✿ ❀ ① ♥ ; q ❂ ❢ q ✐❥ ❣ ♥ ✐ ❀ ❥ ❂✶ , q ✐✐ ❂ ✶ , q ✐❥ q ❥✐ ❂ ✶ ✽ ✐ ❀ ❥ ❙ q ✭ ❱ ✮ ❂ ❤ ① ✶ ❀ ✿ ✿ ✿ ❀ ① ♥ ❥ ① ✐ ① ❥ ❂ q ✐❥ ① ❥ ① ✐ ✐ “the algebra of q -polynomials”
Problem 1: Find finite ● such that ● acts on ❙ q ✭ ❱ ✮ and ❙ q ✭ ❱ ✮ ● is also a q ✵ -polynomial algebra. (“ q -reflection groups”?) B.-Berenstein, 2009: instead of solving Problem 1, solved a different problem (Problem 2 below) such that: • if q ✐❥ ❂ ✶ ✽ ✐ ❀ ❥ (the commutative case ), the solution to Problem 1 AND to Problem 2 are reflection groups.
The semidirect product ❙ ✭ ❱ ✮ ⋊ ● To see what Problem 2 is about, condider the following. Definition: The semidirect product ❙ ✭ ❱ ✮ ⋊ ● is the algebra generated by ❱ and by the algebra C ● subject to relations ❣ ✁ ✈ ❂ ❣ ✭ ✈ ✮ ✁ ❣ for ❣ ✷ ● , ✈ ✷ ❱ ; ❬ ✈ ✶ ❀ ✈ ✷ ❪ ❂ ✵ ✽ ✈ ✶ ❀ ✈ ✷ ✷ ❱ . Important property: if ① ✶ ❀ ✿ ✿ ✿ ❀ ① ♥ are a basis of ❱ , ❢ ① ❦ ✶ ✶ ✿ ✿ ✿ ① ❦ ♥ ♥ ❣ ❥ ❦ ✐ ✷ Z ✕ ✵ ❀ ❣ ✷ ● ❣ is a basis of ❙ ✭ ❱ ✮ ⋊ ● . In other words, ❙ ✭ ❱ ✮ ⋊ ● is ❙ ✭ ❱ ✮ ✡ C ● as a vector space.
Drinfeld’s degenerate affine Hecke algebra Drinfeld (1985) suggested the following deformation of the defining relations of ❙ ✭ ❱ ✮ ⋊ ● . Let ❆ be the algebra generated by ❱ and by the algebra C ● subject to relations ❣ ✁ ✈ ❂ ❣ ✭ ✈ ✮ ✁ ❣ for ❣ ✷ ● , ✈ ✷ ❱ ; ❬ ✈ ✶ ❀ ✈ ✷ ❪ ❂ P ❣ ✷ ● ❛ ❣ ✭ ✈ ✶ ❀ ✈ ✷ ✮ ❣ . Here ❛ ❣ ✿ ❱ ✂ ❱ ✦ C are bilinear forms. Clearly, the above set ❢ ① ❦ ✶ ✶ ✿ ✿ ✿ ① ❦ ♥ ♥ ❣ ❣ ( † ) of monomials spans ❆ , but it may now be linearly dependent, and ❆ may be “strictly smaller” than ❙ ✭ ❱ ✮ ✡ C ● . The set ❢ ❛ ❣ ✿ ❣ ✷ ● ❣ ✚ ✭ ❱ ✡ ❱ ✮ ✄ is called admissible , if the monomials ( † ) are a basis of ❆ . • PBW-type basis • ❆ is a flat deformation of ❙ ✭ ❱ ✮ ⋊ ●
The following conditions are necessary for ❢ ❛ ❣ ✿ ❣ ✷ ● ❣ to be admissible: for ✈ ✐ ✷ ❱ , ❣ ✷ ● , • ❬ ✈ ✶ ❀ ✈ ✷ ❪ ❂ � ❬ ✈ ✷ ❀ ✈ ✶ ❪ , so ❛ ❣ is skew-symmetric; • ❣ ✁ ❬ ✈ ✶ ❀ ✈ ✷ ❪ ❂ ❬ ❣ ✭ ✈ ✶ ✮ ❀ ❣ ✭ ✈ ✷ ✮❪ ✁ ❣ , so ❛ ❤ ✭ ✈ ✶ ❀ ✈ ✷ ✮ ❂ ❛ ❣❤❣ � ✶ ✭ ❣ ✭ ✈ ✶ ✮ ❀ ❣ ✭ ✈ ✷ ✮✮ ; • ❬❬ ✈ ✶ ❀ ✈ ✷ ❪ ❀ ✈ ✸ ❪ ✰ ❬❬ ✈ ✷ ❀ ✈ ✸ ❪ ❀ ✈ ✶ ❪ ✰ ❬❬ ✈ ✸ ❀ ✈ ✶ ❪ ❀ ✈ ✷ ❪ ❂ ✵ (Jacobi identity), which rewrites as ❦❡r✭ ❛ ❣ ✮ ❂ ❱ ❣ and ❝♦❞✐♠ ✭ ❱ ❣ ✮ ❂ ✷ . ❣ ✻ ❂ ✶ ❀ ❛ ❣ ✻ ❂ ✵ ✮ Here ❱ ❣ ❂ ❢ ✈ ✷ ❱ ✿ ❣ ✭ ✈ ✮ ❂ ✈ ❣ . Drinfeld claimed that the above conditions are sufficient for ❢ ❛ ❣ ❣ to be admissible. This claim is true. Definition ❆ , which is a flat deformation of ❙ ✭ ❱ ✮ ⋊ ● , is called a degenerate affine Hecke algebra. Problem 2(D): Find such ❆ for a given ● ❁ ●▲✭ ❱ ✮ . ([Dr’85]: ● ❂ ❙ ♥ or Coxeter gp.)
History Q. Why study flat deformations of ❙ ✭ ❱ ✮ ⋊ ● ? A. Representation theory, geometry (orbifolds ❱ ❂ ● ), Lie theory etc. For example: • Lusztig (1989) introduced the “graded affine Hecke algebra” of a Weyl group ● , a deformation of the semidirect product relation in ❙ ✭ ❱ ✮ ⋊ ● . • Etingof, Ginzburg (2002) introduced the symplectic reflection algebras which are degenerate affine Hecke algebras for ● which preserves a symplectic form ✦ on ❱ . ( Both were done without knowing about Drinfeld’s earlier construction. )
Particular case: The split symplectic case ● ❁ ●▲✭ ❱ ✮ , the algebra to be deformed is ❙ ✭ ❱ ✟ ❱ ✄ ✮ ⋊ ● . There is always a non-trivial deformation, the Heisenberg-Weyl algebra ❆ ✭ ❱ ✮ : ✽ ① ❀ ① ✵ ✷ ❱ ✄ ❀ ✈ ❀ ✈ ✵ ✷ ❱ ❬ ① ❀ ① ✵ ❪ ❂ ✵ ❀ ❬ ✈ ❀ ✈ ✵ ❪ ❂ ✵ ❀ ❬ ✈ ❀ ① ❪ ❂ ❤ ✈ ❀ ① ✐ ✁ ✶ ❀ where ❤ ❀ ✐ is the canonical pairing between ❱ and ❱ ✄ . ❆ ✭ ❱ ✮ is the most straightforward quantisation of the phase space ❱ ✟ ❱ ✄ . If ❤ ✘❀ ① ✐ ✁ ✶ is replaced by an expression in C ● and the deformation is still flat, one has a rational Cherednik algebra of ● . These are introduced and classified in [EG, Invent. Math., ’02] and correspond to complex reflection groups . Problem 2: Find finite ● for which there is a q -analogue of the rational Cherednik algebra of ● .
More recommend
