ICMS

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
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Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. They are a class of algebras with triangular decomposition, arising from a deformation problem whose solutions are called quasi-Yetter-Drinfeld modules. One-dimensional quasi-YD modules lead to classical U(g), Smith's algebras, and their generalisations. Nice quasi-YD modules are Yetter-Drinfeld modules, related to quantum Yang-Baxter equation, which give rise to quantum versions of the Weyl algebra where the role of polynomial algebras is played by Nichols-Woronowicz-Majid algebras. The latter enjoy an increasing number of uses in Hopf algebra theory (as an illustration, we give a very easy counterexample to the third Kaplansky's conjecture). Rational Cherednik algebras for t=0 embed in doubles of NWM algebras. For nonzero t, the NWM algebra is replaced with an algebra associated to the classical Yang-Baxter equation (discovered in type A by Bartholdi, Enriquez, Etingof and Rains), which turns out to be Koszul, becomes a module for the rational Cherednik algebra, and acts on polynomials by deformed Demazure operators. This is a joint work with Arkady Berenstein.
Berenstein, Arkady
Quasiharmonic polynomials for Coxeter groups and rational Cherednik algebras
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In my talk (based on a joint paper with Yurii Burman) I will introduce a certain generalization of usual harmonic polynomials for each Coxeter group - the so-called quasiharmonic polynomials. Using them I will construct a deformation of each finite-dimensional module over rational Cherednik algebras. A surprising by-product of this "quasiharmonic'' approach is the construction of canonical elementary invariants for all Coxeter groups. In the case of the symmetric group these invariants turn out to be certain 1-parameter deformations of the elementary symmetric functions.
Etingof, Pavel
Geometry of rational Cherednik algebras
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In the first lecture I will explain how to attach a Cherednik algebra and a Hecke algebra to any smooth algebraic variety X with an action of a finite group G. If X is a vector space and G is a reflection group, we get the usual rational Cherednik algebra and Hecke algebra, but in general this construction also yields affine and double affine Hecke algebras and many new algebras. In the second lecture I will talk about my recent work with R. Bezrukavnikov, where the geometric ideology from the first lecture is used to construct parabolic induction and restriction functors for rational Cherednik algebras, and will explain some applications of these functors.
Ginzburg, Victor
Character sheaves and Cherednik algebras
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I will discuss various connections between Cherednik algebras, D-modules, and quantum Hamiltonian reduction.
Hernandez, David
Double affine quantum algebras and their representations
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Double affine (or toroïdal) quantum algebras are the quantum group analogs of double affine Hecke algebras. We present several results on their representation theory, including the construction of a tensor category structure which can be defined by using a renormalization of the Drinfeld "coproduct". We also expect to present additional recent results directly related to the representation theory of Cherednik algebras.
Martino, Maurizio
Restricted rational Cherednik algebras and two-sided cells
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We shall discuss an intriguing connection between restricted rational Cherednik algebras and Hecke algebras at unequal parameters. The former are finite dimensional quotient algebras of rational Cherednik algebras (at t=0) for any complex reflection group. They are in some sense the 'zero fibre' of these algebras and carry interesting information about the underlying geometry. The latter algebras are generalisations of the usual Hecke algebra. We shall explain how, in type B_n, the combinatorics governing the blocks of rational restricted Cherednik algebras are the same as those (conjecturally) describing two-sided cells for the corresponding Hecke algebras. We also formulate some conjectures extending this connection. This is joint work with Iain Gordon.
Montarani, Silvia
Finite dimensional representations of symplectic reflection algebras associated to wreath products
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Symplectic reflection algebras were introduced by Etingof and Ginzburg. They arise from the action of a finite group $Gsubset mathrm{Sp}(V)$ of automorphisms on a symplectic vector space $V$, and are a multi-parameter deformation of the algebra $S(V)sharp G$, smash product of $G$ with the polynomial algebra on $V$. While rational Cherednik algebras represent a very useful special case of this construction, another series of interesting examples is provided by the wreath product symplectic reflection algebras, when $G=S_N ltimes G^{times N },$ for some finite subgroup $Gsubset SL(2, C)$. In this talk we will explain how to produce finite dimensional representations for the wreath product case using deformation theory, and a Morita equivalence with Crawley-Boevey and Holland's deformed preprojective algebras. We will then illustrate how Wee Liang Gan was able to recover the same representations by defining "reflection functors" for the wreath product symplectic reflection algebras, and how his approach can be extended to the "continuous case" (when $G$ is an infinite reductive subgroup of $SL(2,C)$).
Nazarov, Maxim
Extremal cocycles on Weyl groups and the reflection equation
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We establish a correspondence between the "extremal cocycles" on the Weyl groups of series C,D introduced by Zhelobenko, and the canonical intertwining operators between the tensor products of fundamental representations of twisted Yangians. Twisted Yangians may be regarded as quantum affine symmetric spaces, and are defined by means of the reflection equation due to Cherednik. Our correspondence employs the Howe duality. This is a joint work with Sergey Khoroshkin.
Opdam, Eric
Classification and structure of discrete representations of affine Hecke algebras
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I will report on joint work with Erdal Emsiz and Jasper Stokman on the graded degenerate double affine Hecke algebra. This algebra has a large center, and is closely related to integrable models with delta function potentials.
Premet, Alexander
Primitive ideals and finite W-algebras
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In my talk I am going to use quantisations of Slodowy slices and reduction modulo p to construct finite generalized Gelfand-Graev models for primitive ideals in U(g).
Suzuki, Takeshi
Cylindric combinatorics and the representation theory of Cherednik algebras
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This talk is on a combinatorial aspect of the representation theory of the rational/trigonometric Cherednik algebras of type A. I introduce and investigate a cylindrical (or periodical) generalization of combinatorial theory concerning standard tableaux and plane partitions on Young diagrams, and show this "cylindrical combinatorics" provides an effective method in the study of certain class of representaions of the Cherednik algebras. In particular, concrete character formulas concerning irreducible modules are obtained by computing generating functions of plane partitions in cylindrical setting.
Turner, Will
Cubist algebras
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We associate noncommutative algebras to various tilings of Euclidean space. These algebras have even global dimension $2w$, and possess strong homological properties, representing their smooth symplectic nature. Following a general procedure, we can deform these algebras. Specialising deformation parameters in case $w=1$, we recover basic examples of rational Cherednik algebras: the deformed preprojective algebras of type $A_infty$. For all $w$, there are nontrivial derived equivalences between Cubist algebras. There are interesting relations with modular representation theory, and Calabi-Yau geometry.
Vasserot, Eric
Representation theory of double affine Hecke algebras
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We'll review the K-theoretic construction of double affine representations, and its applications to simple or finite dimensional modules.
Vazirani, Monica
The combinatorics of tableaux for the double affine Hecke algebra
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The irreducible representations (irreps) of the symmetric group $S_n$ are parameterized by combinatorial objects called ``Young diagrams,'' or shapes. A given irrep has a basis indexed by ``Young tableaux'' of that shape. In fact, this basis consists of weight vectors (simultaneous eigenvectors) for a commutative subalgebra $Y$ of the group algebra ${mathbb C} S_n$. The double affine Hecke algebra (DAHA) is a deformation of the group algebra of the {it affine /} symmetric group and it also contains a commutative subalgebra $Y$. Not every irrep of the DAHA has a basis of weight vectors (and in fact it is quite difficult to parameterize all of its irreps), but if we restrict our attention to those that do, they are parameterized by ``affine shapes'' and that basis is indexed by the ``affine tableaux'' of that shape. In this talk, we will construct these irreps. This is joint work with Takeshi Suzuki.

Participants

Name	Institution
Alev, Jacques	University of Reims
Ardakov, Konstantin	University of Nottingham
Baur, Karin	University of Leicester
Bazlov, Yuri	University of Warwick
Bellamy, Gwyn	University of Edinburgh
Berenstein, Arkady	University of Oregon
Berest, Yuri	Cornell University
Berger, Roland	Université St Étienne
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	Independent University of Moscow
Etingof, Pavel	Massachusetts Institute of Technology
Eu, Ching-Hwa	MIT
Feigin, Misha	University of Glasgow
Geck, Meinolf	University of Aberdeen
Ginzburg, Victor	University of Chicago
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	University of Edinburgh
Kraehmer, Ulrich	University of Glasgow
Kremnizer, Kobi	Massachusetts Institution of Technology
Kuwabara, Toshiro	Kyoto University
Launois, Stéphane	University of Edinburgh
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	Universität Bonn
McGerty, Kevin	Imperial College London
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	University of Chicago
Sergeev, Alexander	Loughborough University
Sevastyanov, Alexey	University of Aberdeen
Sierra, Susan	Princeton University
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