Entrance hall of the ICMS Workshop

Cherednik Algebras

Jun 18, 2007 - Jun 22, 2007

Kings Buildings, University of Edinburgh

Organisers

Name Institution
Gordon, Iain University of Edinburgh

Scientific Organisers
Dr Karin Baur, University of Leicester
Professor Pavel Etingof, Massachusetts Institute of Technology
Professor Victor Ginzburg, University of Chicago
Professor Iain Gordon, University of Edinburgh
Dr Alison Parker, University of Leicester
Dr Catharina Stroppel, University of Glasgow

Short Report

The workshop was attended by around seventy mathematicians, a mixture of PhD students and postdocs, UK based researchers working in related topics and international leaders in the field of Cherednik Algebras. There were twenty six talks during the week, the majority of which reported on research in Cherednik algebras that had been carried out in the last six months. These talks represented many different points of view on the subject - algebraic, geometric, combinatorial and analytic - and featured several significant breakthroughs from the last couple of months. The week included a special lecture on Open Problems designed for non-experts. There was a great deal of interaction throughout the workshop, and one of the great bonuses was the chance for on-the-spot discussion with people with intellectually and geographically diverse backgrounds. An immediate outcome of the workshop will be a special issue of Selecta Math devoted to Cherednik Algebras: this will feature mostly the ICMS participants, writing either research or survey articles.

Participants list and links to available presentations are further down this page.

Download the pdf file of the full report

Original Details

Cherednik algebras were first introduced in 1994 in the guise of double affine Hecke algebras to solve the Macdonald inner product conjectures. Since then, these algebras and their offshoots have become increasingly influential in a growing number of fields. Their representation theory has significant connections with algebraic geometry, combinatorics, finite dimensional algebra, homological algebra, integrable systems, Lie theory, noncommutative algebra and q–calculus: they have been used to confirm conjectures and answer questions in all of these subjects.

The first goal of this workshop is to bring together the international figures working on Cherednik algebras. There are many different strands to the field, and it is vital that as much discussion as possible takes place between researchers. The workshop to act as a primer for future research. To achieve this the meeting will address the following general areas (as well as others that prove to be important or that arise in the meantime):

  • quantisation of symplectic algebraic varieties;
  • representation theory of cyclotomic Hecke algebras;
  • generalised Calogero–Moser systems;
  • geometric representation theory through quantum Hamiltonian reduction;
  • Lie theoretic combinatorics (Kazhdan–Lusztig theory, orthogonal polynomials).

A second aim is to widen interest amongst the UK representation theory community in Cherednik algebras and their applications, and to found new collaborations between UK mathematicians.

A third aim is to give young mathematicians access to established, international researchers in this field. This will be achieved through our objective of having a series of six short talks in the workshop, through the relatively small number of participants, and through the timetabling of long afternoon discussion breaks in our programme.

The Organisers would like to thank the Glasgow Mathematical Journal Research Support Fund, the Edinburgh Mathematical Society Research Support Fund and the London Mathematical Society for their support for this workshop.

Arrangements

Venue
The Workshop will be held in the James Clerk Maxwell Building (JCMB), King's Buildings, University of Edinburgh . The lectures will be held n Room 6301, with registration, teas, coffees and break-out space in Rooms 5326 and 5327.

Registration
There will be a combined Registration and Welcome Buffet from 18.30 to 21.00 on Sunday 17 June, in the St Trinnean's Room, St Leonard's Hall, Pollock Halls. Food will be available throughout that period and you may register at any time between 18.30 and 21.00.

Those who cannot register on Sunday evening may do so at King's Buildings from 08.30-09.00 before the talks start on Monday morning.

Accommodation
ICMS has arranged en suite rooms in university accommodation in Pollock Halls for those who require it. Participants who are being funded are also free to make their own arrangements and may claim back the cost, with receipts, up to a maximum of £35 per night for bed and breakfast. A list of Edinburgh accommodation of various sorts and prices can be found on the Accommodation Page on the ICMS web site. Section 4 is particularly relevant. June is very busy in Edinburgh and early booking is recommended.

Meals and Refreshments
There will be a Registration buffet in the St Trinneans Room in St Leonards, Pollock Halls, (near the Reception Centre – see map) on Sunday evening and a Workshop dinner on Thursday evening in the St Trinnean's Room. Morning and afternoon refreshments and lunch will be provided at the workshop venue. You will be given vouchers for lunch in the first floor restaurant in King’s Buildings House (No 10 on the map). There will be a reception with wine and nibbles at 17.30 on Wednesday 20 June at ICMS, 14 India Street.

Programme

Sunday 17 June
18.30-21.00 Registration & buffet, St Trinnean's Room, St Leonard's, Pollock Halls

Monday 18 June
08.30-09.00 Registration - Room 5326, JCMB
09.00-10.00 Iain Gordon (University of Edinburgh)
What is a Cherednik algebra
10.00-11.00 Raphael Rouquier (University of Oxford)
Introduction to Cherednik algebras, I
11.00-11.30 Coffee break
11.30-12.30 Victor Ginzburg (University of Chicago)
Character sheaves and Cherednik algebras I
12.30-14.30 Lunch, KB House
14.30-15.30 Raphael Rouquier (University of Oxford)
Introduction to Cherednik algebras, II
15.30-16.30 Will Turner (University of Oxford)
Cubist algebras
16.30-17.00 Coffee break
17.00-18.00 Monica Vazirani (University of California at Davis)
The combinatorics of tableaux for the double affine Hecke algebra

Tuesday 19 June
09.00-10.00 Yuri Bazlov (University of Warwick)
Braided doubles and rational Cherednik algebras
10.00-11.00 Victor Ginzburg (University of Chicago)
Character sheaves and Cherednik algebras II
11.00-11.30 Coffee break
11.30-12.30 Arkady Berenstein (University of Oregon)
Quasiharmonic polynomials for Coxeter groups and rational Cherednik algebras
12.30-14.30 Lunch, KB House
14.30-15.30 Takeshi Suzuki (Okayama University)
Cylindrical combinatorics and the representation theory of Cherednik algebras
15.30-16.30 Maurizio Martino (Universität Bonn)
Restricted rational Cherednik algebras and two-sided cells
16.30-17.00 Coffee break
17.00-18.00 Raphael Rouquier (University of Oxford)
Microlocalization of rational Cherednik algebras

Wednesday 20 June
09.00-10.00 Alexander Premet (University of Manchester)
Primitive ideals and finite W-algebras
10.00-11.00 Pavel Etingof (Massachusetts Institute of Technology)
Geometry of rational Cherednik algebras, I
11.00-11.30 Coffee break
11.30-12.30 Eric Opdam (University of Amsterdam)
Classification and structure of discrete representations of affine Hecke algebras
12.30-13.00 Pavel Etingof (Massachusetts Institute of Technology)
Open problems
13.00-14.30 Lunch, KB House
14.30 onwards Free afternoon
17.30-19.00
Reception with wine & nibbles, ICMS, 14 India Street

Thursday 21 June
09.00-10.00 Nicholas Guay (Université de Versailles)
Quantum algebras associated to quivers and symplectic reflection algebras
10.00-11.00 David Hernandez (Université de Versailles)
Double affine quantum algebras and their representations
11.00-11.30 Coffee break
11.30-12.30 Pavel Etingof (Massachusetts Institute of Technology)
Geometry of rational Cherednik algebras, II
12.30-14.30 Lunch, KB House
14.30-15.30 Eric Rains (University of California at Davis)
New deformations of Coxeter groups
15.30-16.30 Michela Varagnolo (Université de Cergy-Pontoise)
Representation theory of double affine Hecke algebras, I
16.30-17.00 Coffee break
17.00-18.00 Eric Vasserot (Université Paris 7)
Representation theory of double affine Hecke algebras, II
19.30 Workshop Dinner, St Trinnean's Room, St Leonard's, Pollock Halls

Friday 22 June
09.30-10.30 Maxim Nazarov (University of York)
Extremal cocycles on Weyl groups and the reflection equation
10.30-11.00 Coffee break
11.00-12.00 Silvia Montarani (Massachusetts Institute of Technology)
Finite dimensional representations of symplectic reflection algebras associated to wreath products
12.00-13.00 Pavel Etingof (Massachusetts Institute of Technology)
Surprise
13.00-14.00 Lunch, KB House
14.00-15.00 Yuri Berest (Cornell University)
The generalised Calogero-Moser spaces

Background Lecture Notes:

Here are some notes from a graduate school on Cherednik Algebras in Luminy from 2/4-6/4 2007 . They are from 4 series of introductory lectures.

  1. Category O and finite dimensional representations of rational Cherednik algebras by Nicolas Guay (1, 2, 3, 4)
  2. Rational Cherednik algebras, category O and shift operators by Eric Opdam
  3. Representations of double affine Hecke algebras by Eric Vasserot
  4. Rational Cherednik algebras and Hilbert schemes by Iain Gordon

A further paper has also been received: Open problems by Pavel Etingof and Victor Ginzburg.

 

Presentations:

Presentation Details
Bazlov, Yuri
Braided doubles and rational Cherednik algebras
View Abstract Down
Berenstein, Arkady
Quasiharmonic polynomials for Coxeter groups and rational Cherednik algebras
View Abstract Down
Etingof, Pavel
Geometry of rational Cherednik algebras
View Abstract Down
Ginzburg, Viktor
Character sheaves and Cherednik algebras
View Abstract Down
Hernandez, David
Double affine quantum algebras and their representations
View Abstract Down
Martino, Maurizio
Restricted rational Cherednik algebras and two-sided cells
View Abstract Down
Montarani, Silvia
Finite dimensional representations of symplectic reflection algebras associated to wreath products
View Abstract Down
Nazarov, Maxim
Extremal cocycles on Weyl groups and the reflection equation
View Abstract Down
Opdam, Eric
Classification and structure of discrete representations of affine Hecke algebras
View Abstract Down
Premet, Alexander
Primitive ideals and finite W-algebras
View Abstract Down
Suzuki, Takeshi
Cylindric combinatorics and the representation theory of Cherednik algebras
View Abstract Down
Turner, Will
Cubist algebras
View Abstract Down
Vasserot, Eric
Representation theory of double affine Hecke algebras
View Abstract Down
Vazirani, Monica
The combinatorics of tableaux for the double affine Hecke algebra
View Abstract Down

Participants

Name Institution
Alev, Jacques University of Reims
Ardakov, Konstantin University of Oxford
Baur, Karin University of Graz
Bazlov, Yuri University of Manchester
Bellamy, Gwyn University of Glasgow
Berenstein, Arkady University of Oregon
Berest, Yuri Cornell University
Berger, Roland University of Saint-Etienne
Braden, Harry University of Edinburgh
Brown, Ken A University of Glasgow
Chalykh, Oleg University of Leeds
Chervov, Alexander ITEP Moscow
Cox, Anton City University London
Dotsenko, Vladimir Trinity College Dublin
Etingof, Pavel Massachusetts Institute of Technology
Eu, Ching-Hwa MIT
Feigin, Misha University of Glasgow
Geck, Meinolf University of Aberdeen
Ginzburg, Viktor University of California, Santa Cruz
Gordon, Iain University of Edinburgh
Grabowski, Jan University of Oxford
Grime, Matthew University of Bristol
Grojnowski, Ian University of Cambridge
Guay, Nicholas Université de Versailles
Gupta, Neha University of Warwick
Hernandez, David CNRS - Université de Versailles
Hinchcliffe, Victoria University of Sheffield
Iancu, Lacrimioara University of Aberdeen
Ion, Bogdan Universities of Pittsburgh & Bonn
Jacon, Nicolas Université de Franche-Comté
Kolb, Stefan Newcastle University
Krähmer, Ulrich Technische Universität Dresden
Kremnizer, Kobi Massachusetts Institution of Technology
Kuwabara, Toshiro Kyoto University
Launois, Stéphane University of Kent
Lenagan, Tom University of Edinburgh
Loktev, Sergey Institute for Theoretical and Experimental Physics
Ma, Xiaoguang MIT
Macauley, Monica University of Glasgow
Macleod, Marjory University of Glasgow
Martin, Paul University of Leeds
Martino, Maurizio University of Bonn
McGerty, Kevin University of Oxford
Montarani, Silvia Massachusetts Institute of Technology
Nazarov, Maxim University of York
Opdam, Eric Korteweg de Vries Institute/University of Amsterdam
Parashar, Deepak University of Warwick
Parker, Alison University of Leeds
Premet, Alexander University of Manchester
Rains, Eric University of California at Davis
Richard, Lionel University of Edinburgh
Rigal, Laurent Université Jean-Monnet
Rouquier, Raphaël University of Oxford
Russell, Euan University of Edinburgh
Rybnikov, Leonid Institute for Theoretical and Experimental Physics (ITEP)
Sahi, Siddhartha Rutgers University
Schedler, Travis Massachusetts Institute of Technology
Sergeev, Alexander Loughborough University
Sevastyanov, Alexey University of Aberdeen
Sierra, Susan University of Edinburgh
Smith, Paul University of Washington
Stafford, Toby University of Manchester
Stokman, Jasper University of Amsterdam
Stroppel, Catharina University of Bonn
Suzuki, Takeshi Math. Department, Faculty of Science, Okayama University
Turner, Will University of Aberdeen
Ubly, Glenn University of Southampton
Vale, Richard Universities of Glasgow & Freiburg
Varagnolo, Michela Université de Cergy-Pontoise
Vasserot, Eric Université Paris 7
Vazirani, Monica University of California at Davis
Yvonne, Xavier Université Lyon I