s X reflecting hyperplane - - PowerPoint PPT Presentation

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 O s(X)

reflecting hyperplane ❂ ❦❡r✭■❞ s✮

• 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

• f 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

• f ●.)

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 ❦♥ ♥ ❣❣

(†)

• f 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 ●.

Dunkl operators

❅ ❅✈ , ✈ ✷ ❱ , are commuting operators on ❙✭❱ ✄✮.

NB: ❅

❅✈ ♣ ❂ ❬✈❀ ♣❪ in the algebra ❆✭❱ ✮, where ♣ ✷ ❙✭❱ ✄✮.

Deformation: Replace ❆✭❱ ✮ ✘

❂ ❙✭❱ ✟ ❱ ✄✮ with a rational

Cherednik algebra ❍❈✭●✮ ✘

❂ ❙✭❱ ✟ ❱ ✄✮ ✡ C● of ● ❁ ●▲✭❱ ✮: r✈♣ ❂ ❅♣ ❅✈ ✰

❳

s

❝s ✁ ☛s✭✈✮ ✁ ♣ s✭♣✮ ☛s

, where

• s runs over complex reflections in ● ❁ ●▲✭❱ ✮
• ❝s are scalar parameters such that ❝❣s❣✶ ❂ ❝s for all ❣ ✷ ●
• ☛s ✷ ❱ ✄ is the root of s: s✭✈✮ ❂ ✈ ☛s✭✈✮☛❴

s for some ☛❴ s ✷ ❱

These operators were first introduced by Dunkl (1989) for Coxeter groups (in harmonic analysis).

Dunkl operators commute

Theorem [Du,EG]: r✈✭polynomials✮ ✒ polynomials,

r✉r✈ ❂ r✈r✉

Proof (using rational Cherednik algebras): ❍❈✭●✮ acts on ❙✭❱ ✄✮ via induced representation. The action of ✈ ✷ ❱ is via the Dunkl

• perator r✈. But ✈ ✷ ❱ commute in ❍❈✭●✮.

Example for ● ❂ S♥:

r✐ ❂ ❅ ❅①✐ ✰ ❝ P

❥ ✻❂✐

✶ ①✐ ①❥ ✭✶ s✐❥ ✮ r✶❀ ✿ ✿ ✿ ❀ r♥ act on C❬①✶❀ ✿ ✿ ✿ ❀ ①♥❪ and commute.

Braided doubles

The rational Cherenik algebra is a flat deformation of

❆✭❱ ✮ ⋊ ● ✘ ❂ ❙✭❱ ✮ ✡ C● ✡ ❙✭❱ ✄✮ (triangular decomposition).

[EG] prove this, using the Koszul deformation principle. [B.-Berenstein, Adv. Math. ’09] introduce braided doubles (a more general class of algebras defined by triangular decomposition):

❚✭❱ ✮❂■ ✡ ❍ ✡ ❚✭❲ ✮❂■ ✰ where ❱ ❀ ❲ are modules over a

Hopf algebra ❍ , ■ ✝ are two-sided ideals, ❬❱ ❀ ❲ ❪ ✚ ❍ . Example (the differential calculus on a noncommutative space):

❨ is a space with a braiding ✠ ✷ ❊♥❞✭❨ ✡ ❨ ✮,

i.e., ✭■❞ ✡ ✠✮✭✠ ✡ ■❞✮✭■❞ ✡ ✠✮ ❂ ✭✠ ✡ ■❞✮✭■❞ ✡ ✠✮✭✠ ✡ ■❞✮.

braided Weyl algebra ❆✭❨ ❀ ✠✮ ✘ ❂ ❇✭❨ ✮ ✡ ❇✭❨ ✄✮.

Here ❇✭❨ ✮, ❇✭❨ ✄✮ are Nichols algebras which have relations that depend on ✠.

Anticommuting Dunkl operators

Theorem [B.-Berenstein] If ❉ ❂ ❚✭❱ ✮❂■ ✡ k● ✡ ❚✭❲ ✮❂■ ✰ is a minimal braided double, there exist a finite-dimensional braided space ✭❨ ❀ ✠✮ so that ❉ embeds in ❆✭❨ ❀ ✠✮ ⋊ ●. Thus, one may look for algebras with triangular decomposition and with given relations among certain subalgebras of braided Weyl algebras ❆✭❨ ❀ ✠✮ ⋊ ●. For example [B.-Berenstein, Selecta Math. ’09]: Let ① ✶❀ ✿ ✿ ✿ ❀ ① ♥ be anticommuting variables, ① ✐① ❥ ❂ ① ❥ ① ✐, ✐ ✻❂ ❥ Look for algebras of the form

C❤✈ ✶❀ ✿ ✿ ✿ ❀ ✈ ♥✐ ✡ C● ✡ C❤① ✶❀ ✿ ✿ ✿ ❀ ① ♥✐, ① ✐✈ ❥ ✭✶✮✍✐❥ ✈ ❥ ① ✐ ✷ C●

Classification of “anticommutative Cherednik algebras”

Theorem 1 (Solution to Problem 2) The above algebras with triangular decomposition exist for, and only for, the following groups:

• ● ❂ ●✭♠❀ ♣❀ ♥✮,

✭♠❂♣✮ even

• ● ❂ ●✭♠❀ ♣❀ ♥✮✰,

✭♠❂♣✮ even, ✭♠❂✷♣✮ odd

Definition For finite ● ❁ ●▲✭❱ ✮, consider the character

❞❡t✿ ● ✦ C✂ and put ❈ ❂ ❞❡t✭●✮ (finite cyclic group). Then

• ✰ ❂ ❢❣ ✷ ● ✿ ❞❡t✭❣✮ ✷ ❈✷❣

(the subgroup of even elements of ●). (NB Either ●✰ ❂ ● or ❥● ✿ ●✰❥ ❂ ✷) Smallest group in rank ♥: ● ❂ ●✭✷❀ ✶❀ ♥✮✰ ❂ even elements in the Coxeter group of type ❇♥ (denoted ❇✰

♥ )

Anticommuting Dunkl operators for ❇✰

♥

r✐ ❂ ❅✐ ✰ ❝

❳

❥ ✻❂✐

①✐ ✰ ①❥ ① ✷

✐ ① ✷ ❥

✭✶ ✛✐❥ ✮ ✰ ①✐ ①❥ ① ✷

✐ ① ✷ ❥

✭✶ ✛❥✐✮

,

✐ ❂ ✶❀ ✿ ✿ ✿ ❀ ♥

• ❅✐ are anticommuting skew-derivations of C❤① ✶❀ ✿ ✿ ✿ ❀ ① ♥✐
• ✛✐❥ is an automorphism of C❤① ✶❀ ✿ ✿ ✿ ❀ ① ♥✐ of order ✹,

✛✐❥ ✭① ✐✮ ❂ ① ❥ ❀ ✛✐❥ ✭① ❥ ✮ ❂ ① ✐❀ ✛✐❥ ✭① ❦✮ ❂ ① ❦❀ ❦ ✻❂ ✐❀ ❥ ✿

• NB ① ✷

✐ ① ✷ ❥ is central in C❤① ✶❀ ✿ ✿ ✿ ❀ ① ♥✐, division is

well-defined; ① ✷

✐ ① ✷ ❥ ✻❂ ✭① ✐ ① ❥ ✮✭① ✐ ✰ ① ❥ ✮

Theorem 2 r✐✭skew-polynomials✮ ✒ skew-polynomials,

r✐r❥ ❂ r❥ r✐ for ✐ ✻❂ ❥

Questions • What is C❤① ✶❀ ✿ ✿ ✿ ❀ ① ♥✐●? — i.e., can the above class of groups be characterised by polynomiality of the invariants? Example C❤① ✶❀ ✿ ✿ ✿ ❀ ① ♥✐❇✰

♥ is polynomial and is generated by

• ① ✷❦

✶ ✰ ✁ ✁ ✁ ✰ ① ✷❦ ♥ ,

❦ ❂ ✶❀ ✷❀ ✿ ✿ ✿ ❀ ♥ ✶;

• ① ✶① ✷ ✿ ✿ ✿ ① ♥,

That is, ❇✰

♥ (not a reflection group in the usual sense) has

polynomial “anticommutative invariants” and has exponents

✷❀ ✹❀ ✿ ✿ ✿ ❀ ✷✭♥ ✶✮❀ ♥.

NB: the product of the exponents is precisely ❥❇✰

♥ ❥.

Kirkman, Kuzmanovich, Zhang (2009) proved [independently of B.-B.]:

❙q✭❱ ✮● is q✵-polynomial, if and only if ● is one of the above B.-B.

groups. (This settles the C-S-T theorem for ❙q✭❱ ✮● — Problem 1 is now solved.)

• The algebra of q-commuting variables ①✶❀ ✿ ✿ ✿ ❀ ①♥ (the quantum

hyperplane): if q ✻❂ ✶, need to consider finite-dimensional quotients of Manin’s quantum group ●▲q✭♥❀ C✮; “Dunkl operators” will be a deformation of the Wess-Zumino braided differential calculus. Thank you.