Today's schedule has changed; it was because Kapranov didn't come. The morning session was by David Kazhdan and Iain Gordon, instead. Kazhdan was on "Iwahori-Hecke algebras for reductive groups over higher dimensional fields" -- which started from "what is integration? -- it is Grothendieck's K_0 group of differential invariant functionals" then some examples of dependable sets, definable but not differentiable maps, and almost representability. Then he looked at his small memo from his pocket and explained the Iwahori subgroup of SL_2(F = Q_p ((t))), H: its definable functions with bounded support, and its convolution. He defined the rational section of H on M and its relation to representability. Ian Gordon was on the rational Cherednik algebras by OHP. PBW by Cherednik and Etingof-Ginzburg and the Theorem by Ginzburg-Guay-Opdam-Rouquier on the existence of KZ functor which is fully faithful on projections and surjective on objects, utilizing the Ariki-Koike algebra. And also the theorem of Rouquier for the decomposition of chambers, and the theorem

(Gordon-Stafford, Musson, Boyarchenko, Vale) the Gordon-Stafford paper on the Hilbert scheme.

Afternoon session was done by Ian Grojnowski on "some semi-infinite geometry" which is "work in progress, many details to check... attempt to understand work of Feigin-Frenkel [FF] and Beilinson-Tate linear algebra". The first problem was to define a category of D-modules on semi-infinity flag variety by loop space LG / (LB)^0 [well-defined homotopy type, tangent space: Beilinson-Tate space] and an equivalence of categories D^{\le 0} (D-modules) to D (\hat{g}-enhanced). The main problem was to define the category. Examples started from sl_2 / P^1 case and its Iwahori subalgebra. [FF] defined \hat{g}-modules, which they called Wakimoto modules, which look like they should be D-modules attached to Iwahori orbits. When we wish to "localize LG / (LB)^0" around the middle dimension with finite codimension intersections, it is like to defining by a manifold y gluing net along open embeddings but closed also. Then he defined lci of mappings for the Quillen model category of dg algebra if it is locally finitely presented (lfp) and L_{B/A} has Tor dim \le 1. In the commutative algebra case, it coincides with usual notion. However, by gluing the \Triangle^{op} Presh(Aff)_{\tau}-stacks, we had to deal with some of the Grothendieck topology and the surjectivity seems not enough for the definition of Cech covering for the half-infinite geometry.

## Friday, June 23, 2006

## Thursday, June 22, 2006

### 3rd day of Affine Hecke Algebras

I felt very bad at the morning; in addition, the bus neglected me while I was sitting to wait. So that it was just on time 9:00 when I arrived at the auditorium.

Etingof was on the rational Cherednik algebras and Hecke algebras attached to complex orbifolds, which started from the past work of Hochshild cohomology, the definition of rational Cherednik algebras, PBW theorem, Etingof-Ginzburg theorem on H_{1,c} (with formal c), and polynomial representations. Then it turned to the generalization to the case of any irreducible smooth algebraic variety X over complex numbers with a faithful G-action, which is also devided into the case |G|=1 and X affine. Some definitions of reflection hypersurface, Dunkl-Opdam operators, and filtration by order. The main theorem for the Hecke algebras was on the flat deformation when X: complex manifolds has 2nd cohomology 0, which includes the lower degree del Pezzo surfaces. The idea of proof of theorem was by the KZ functor. Next, Rouqier was on the representations of rational Cherednik algebras, which started from the classical classification of finite Hecke, affine Hecke, double affine Hecke (by the order of affinization), -, deg. affine Hecke, trigonometric DAHA, -, -. rational DAHA (by the order of degeneration). Type A was done by Suzuki and also we used the Hilbert scheme of points of C^2 and PBW then a correspondence towards the Cartan subalgebras and nilpotent radical of Borel subalgebras is obtained. The problem is 1) the functor CKZ (Cherednik-Knizknik-Zamolodchikov) / CKZ with regular singularitiy connection, 2) compute local monodromy, 3) CKZs fully faithful on projective objects.

The lunch was a fried rice with cheese and cafe. After that, I turned back to the hotel to take a rest because of the sickness. Then I went to a supermarket and bought food and the meal including sandwiches.

Etingof was on the rational Cherednik algebras and Hecke algebras attached to complex orbifolds, which started from the past work of Hochshild cohomology, the definition of rational Cherednik algebras, PBW theorem, Etingof-Ginzburg theorem on H_{1,c} (with formal c), and polynomial representations. Then it turned to the generalization to the case of any irreducible smooth algebraic variety X over complex numbers with a faithful G-action, which is also devided into the case |G|=1 and X affine. Some definitions of reflection hypersurface, Dunkl-Opdam operators, and filtration by order. The main theorem for the Hecke algebras was on the flat deformation when X: complex manifolds has 2nd cohomology 0, which includes the lower degree del Pezzo surfaces. The idea of proof of theorem was by the KZ functor. Next, Rouqier was on the representations of rational Cherednik algebras, which started from the classical classification of finite Hecke, affine Hecke, double affine Hecke (by the order of affinization), -, deg. affine Hecke, trigonometric DAHA, -, -. rational DAHA (by the order of degeneration). Type A was done by Suzuki and also we used the Hilbert scheme of points of C^2 and PBW then a correspondence towards the Cartan subalgebras and nilpotent radical of Borel subalgebras is obtained. The problem is 1) the functor CKZ (Cherednik-Knizknik-Zamolodchikov) / CKZ with regular singularitiy connection, 2) compute local monodromy, 3) CKZs fully faithful on projective objects.

The lunch was a fried rice with cheese and cafe. After that, I turned back to the hotel to take a rest because of the sickness. Then I went to a supermarket and bought food and the meal including sandwiches.

## Tuesday, June 20, 2006

### 2nd day of Affine Hecke Algebras

Today I safely took the bus 21 and used the ethernet (Gigabit) as well as the WiFi connection.

The morning talks were by three people, Delorme / Solleveld, and Stokman. Delorme was on the Schwarz algebra of affine Hecke algebra using his collabolation with Opdam. It started from the discrete series, cross relations, interwining operators, and the Fourier transforms for the Schwartz algebra. The main theorem has corollaries of Chandra completeness theorem and an analogue of Langlands disjointness theorem. Solleveld was also on the Schwartz algebra, but from topological K-theoretical language of cyclic homology HP_*, which has the properties of additivity, Morita invariance, diffeotopy invariance, and excision. The 2nd lecture was done by Stokman on the periodic quantum integrable systems with delta-interactions which consists of 1) integrability / "double degeneration" of DAHA (Dunkl operators), 2) spectral theory / Bethe ansatz equations, and 3) spin models / degenerate intertwiners. However, 3) cannot be told because of the time limit.

After the lunch, I could discuss with Mirkovic on my research project and difficulties on the construction of coherent sheaves in the non-abelian Fourier-Mukai.

Afternoon session was on the Yangian and Mickelsson algebras by Nazakov on the composition of the Cherednik functor and Drinfeld functor and its generalization for groups other than gl_n. On the Mickelsson algebras, he talked about propositions of Zhelobenko on cocycles.

After the lectures, I went to the McDonald's and did the laundries.

The morning talks were by three people, Delorme / Solleveld, and Stokman. Delorme was on the Schwarz algebra of affine Hecke algebra using his collabolation with Opdam. It started from the discrete series, cross relations, interwining operators, and the Fourier transforms for the Schwartz algebra. The main theorem has corollaries of Chandra completeness theorem and an analogue of Langlands disjointness theorem. Solleveld was also on the Schwartz algebra, but from topological K-theoretical language of cyclic homology HP_*, which has the properties of additivity, Morita invariance, diffeotopy invariance, and excision. The 2nd lecture was done by Stokman on the periodic quantum integrable systems with delta-interactions which consists of 1) integrability / "double degeneration" of DAHA (Dunkl operators), 2) spectral theory / Bethe ansatz equations, and 3) spin models / degenerate intertwiners. However, 3) cannot be told because of the time limit.

After the lunch, I could discuss with Mirkovic on my research project and difficulties on the construction of coherent sheaves in the non-abelian Fourier-Mukai.

Afternoon session was on the Yangian and Mickelsson algebras by Nazakov on the composition of the Cherednik functor and Drinfeld functor and its generalization for groups other than gl_n. On the Mickelsson algebras, he talked about propositions of Zhelobenko on cocycles.

After the lectures, I went to the McDonald's and did the laundries.

## Monday, June 19, 2006

### 1st day of Affine Hecke Algebras

I walked up at 7 am but I was waiting for the bus number 21 (towards CIRM in Luminy) at the opposite terminal so that I was late for the 9 am lecture by Cherednik -- it was on the construction of Fourier transform for Double Affine Hecke Algebras (DAHA) by the analytic continuation for the convergence. Next one was by Victor Ginzburg on the noncommutative geometry and quiver diagrams -- something like construction of the superpotential from the loop space data. (See the right photo of the auditorium.)

After the lunch, Ariki -- to whom I talked a little at the break -- talked about the tensor product of A_n type quantum groups, which demands the crystal basis of Kashiwara and is the opposite direction to the Nakajima's loop Grassmanian / quiver varieties. Last one was the seminar by Di Francesco on the combinatorics in the physics -- more precisely the statistical 6-vertex model of ice by use of the Yang-Baxter equations and unitarity.

I ate lunch at a French restaurant near the Citadine metro station.

After the lunch, Ariki -- to whom I talked a little at the break -- talked about the tensor product of A_n type quantum groups, which demands the crystal basis of Kashiwara and is the opposite direction to the Nakajima's loop Grassmanian / quiver varieties. Last one was the seminar by Di Francesco on the combinatorics in the physics -- more precisely the statistical 6-vertex model of ice by use of the Yang-Baxter equations and unitarity.

I ate lunch at a French restaurant near the Citadine metro station.

### Arrived at Marseille

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