# 41st ARTIN meeting

### University of Edinburgh, Wednesday 11th through Friday 13th of June 2014.

The 41st ARTIN meeting will be hosted at the School of Mathematics of University of Edinburgh on Wednesday 11th through Friday 13th of June 2014. It will be funded by the London Mathematical Society, Glasgow Mathematical Journal Trust, and the Edinburgh Hodge Institute.

ARTIN funding: Thanks to generous funding from the Edinburgh Hodge Institute, we have an unusual amount of funding available to support non-speaking participants coming from within the ARTIN network. If you would like to attend, and are interested in applying for funding, please send an email to David Jordan.

If you would like to participate, please REGISTER HERE.

#### Schedule:

##### Wednesday 11th June James Clerk Maxwell Building, Room 5215
• 14:00-14:50 Claudia Scheimbauer (ETH Zurich):Factrorization homology as a fully extended TFT (Homotopy) algebras and bimodules over them can be viewed as factorization algebras on the real line R which are locally constant with respect to a certain stratification. Moreover, Lurie proved that E_n-algebras are equivalent to locally constant factorization algebras on $$R^n$$. Starting from these two facts I will explain how to model the Morita category of E_n-algebras as an $$(\infty, n)$$-category. Every object in this category, i.e. any E_n-algebra A, is "fully dualizable" and thus gives rise to a fully extended TFT by the cobordism hypothesis of Baez-Dolan-Lurie. I will explain how this TFT can be explicitly constructed by (essentially) taking factorization homology with coefficients in the E_n-algebra A.

• 15:00-15:50 Adrien Brochier (Edinburgh): Quantum D-modules and topological field theories. Given a braided tensor category C and a punctured surface S, the formalism of factorization homology produces a category equipped with a canonical action of the braid group of S. The main goal of the talk is to explain how this category can be identified with the category of modules over a very explicit algebra. In the case C is the category of modules over a quantum group, this provides a uniform quantization of character varieties of surfaces. Notably, if S is a punctured torus, one get a certain deformation of the category of D-modules over the underlying algebraic group. I'll try to explain how this construction fits into a certain 4d topological field theory closely related to Reshetikhin-Turaev 3d TFT. I'll also emphazise the analogy between factorization homology and conformal field theory. This is a joint project with David Ben-Zvi and David Jordan.

• 16:00-16:50 Vanessa Miemietz (East Anglia): Morita theory for finitary 2-categories. Motivated by the success methods of categorification have had in representation theory, I will give an introduction to finitary 2-categories (which cover most known examples of 2-categories arising in categorification) and their 2-representation theory. The main goal of the talk is to state a Morita theorem for these and to point out analogies and differences to classical Morita theory.

##### Thursday 12th June James Clerk Maxwell Building, Room 5215
• 10:00-10:50 Stefan Kolb (Newcastle): Radial part calculations for affine $$\mathfrak{sl}_2$$, revisited. Radial part calculations for affine $$\mathfrak{sl}_2$$ lead to a generalisation of the $$BC_1$$ elliptic Calogero-Moser Hamiltonian. In this talk I will review this construction by first looking at the corresponding finite case in some detail. The Hamiltonian comes with a natural set of formal solutions of the corresponding heat equation. I will discuss the convergence of these solutions following arguments by Etingof and Kirillov Jr in the group case.

• 11:00-11:50 Alexander Premet (Manchester): Regular derivations of truncated polynomial rings. Let K be an algebraically closed field of characteristic p>2 and let O be the quotient of the polynomial ring in n variables over K by the p-th power of its augmentation ideal. Let L be the derivation algebra of O, a simple Lie algebra of type W. It is known that the ring of all regular functions on L invariant under the action of the automorphism group is freely generated by n elements and a version of Chevalley's restriction theorem holds for L. Furthermore, the related quotient morphism is faithfully flat and all its fibres are irreducible complete intersections.

An element x of L is called regular if the centraliser of x in L has the smallest possible dimension. In my talk I will give a description of regular elements of L and show that an analogue of Kostant's differential criterion for regularity holds in L. Normality of fibres of the above mentioned quotient morphism will also be discussed.

• 12:00-12:50 Alexey Sevastyanov (Aberdeen): A proof of De Concini-Kac-Procesi conjecture. In 1992 De Concini, Kac and Procesi observed that isomorphism classes of irreducible representations of a quantum group at odd primitive root of unity m are parameterized by conjugacy classes in the corresponding algebraic group. They also conjectured that the dimensions of irreducible representations corresponding to a given conjugacy class O are divisible by m^{1/2 dim O}. In this talk I shall prove an improved version of this conjecture and derive some important consequences of it related to q-W algebras.

• 15:00-15:50 Eric Opdam (Amsterdam): Unipotent formal degrees and adjoint gamma factors. A conjecture of Hiraga, Ichino and Ikeda (2008) expresses the formal degree of a discrete series representation in terms of the adjoint gamma factor of its conjectural local Langlands parameter. We will present a proof of this conjecture for the unipotent discrete series representations of an unramified reductive group over a nonarchimedian local field. We will also discuss a recent conjecture expressing formal degrees in terms of elliptic fake degrees and Lusztig's nonabelian Fourier transform (joint work with Dan Ciubotaru).

• 16:00-16:50 Sam Gunningham (UT Austin): Springer Theory on a reductive group. Lusztig's generalized Springer correspondence gives a description of the category of equivariant perverse sheaves on the unipotent cone of a complex reductive group in terms of cuspidal local systems on Levi subgroups L of G and representations of the relative Weyl group associated to L. In this talk I will provide a new perspective on Lusztig's results and show how they can be extended to a description of the equivariant derived category of D-modules on a reductive group (or its Lie algebra). This result brings together ideas and results from the theory of orbital and character sheaves, Springer theory, quantum Hamiltonian reduction, and Cherednik algebras.

• 17:00-17:50 Gwyn Bellamy (Glasgow): Counting resolutions of symplectic quotient singularities. If V is a symplectic vector space and G a finite subgroup of Sp(V), then the quotient singularity V/G is a very interesting object to study, both from the geometric and representation-theoretic point of view. One of the motivational problems in trying to understand the singularities of V/G is that of deciding whether V/G admits a symplectic resolution or not. More generally, one can ask how many symplectic resolutions it admits. The goal of this talk is to explain how one can count the number of symplectic resolutions of V/G. We’ll present an explicit formula for this number in terms of the dimension of a certain Orlik-Solomon algebra. The key to deriving this formula is to relate the resolutions of V/G to the Calogero-Moser deformations, where one can use the representation theory of symplectic reflection algebras. This is a report on work in progress.

##### Friday 13th June James Clerk Maxwell Building, Room 5215
• 9:00-9:50 Martina Balagovic (Newcastle): Irreducible modules for degenerate DAHA of type $$A$$ Double affine Hecke algebras and their degenerate versions have been an active area of study since their original definition in 1993, when they were used by Ivan Cherednik to solve Macdonald conjectures. When studying their representation theory, one usually focuses on Category $$\mathcal{O}$$. Here one defines Verma modules $$M$$ as certain induced modules, and shows that with appropriate assumptions these modules always have an irreducible quotient $$L=M/K$$, and that all irreducible Category $$\mathcal{O}$$ modules can be realized in this way. However, the description of this maximal submodule $$K$$, and consequently the characters of the irreducible module $$L$$, are not known in general. This is why alternative characterizations of irreducible modules are interesting.

In this talk, I will focus on the question "Can irreducible Category $$\mathcal{O}$$ modules for degenerate double affine Hecke algebras of type $$A$$ be realized as submodules of Verma modules?". The answer for affine Hecke algebras of type $$A$$, as discovered by Guzzi, Nazarov and Papi, is that all irreducible modules admit such an embedding. The answer for double affine Hecke algebras is more involved. I will describe both the modules which do not allow such a realization and the embedding of the modules which do allow it in terms of the combinatorics of the affine symmetric group and periodic skew Young diagrams.

• 10:00-10:50 Sachin Gautam (Columbia): Tensor isomorphism between Yangians and quantum loop algebras. The Yangian and the quantum loop algebra of a simple Lie algebra g arise naturally in the study of the rational and trigonometric solutions of the Yang–Baxter equation, respectively. These algebras are deformations of the current algebra $$\mathfrak{g}[s]$$ and the loop algebra $$\mathfrak{g}[z, z^{−1} ]$$ respectively.

The aim of this talk is to establish an explicit relation between the finite–dimensional representation categories of these algebras, as meromorphic braided tensor categories. The notion of meromorphic tensor categories was introduced by Y. Soibelman and finite–dimensional representations of Yangians and quantum loop algebras are among the first non–trivial examples of these.

The isomorphism between these two categories is governed by the monodromy of an abelian difference equation. Moreover, the twist relating the tensor products is a solution of an abelian version of the qKZ equations of Frenkel and Reshetikhin. The main result of this talk is an analog of the Kohno–Drinfeld theorem for abelian qKZ equations. These results are part of an ongoing project, joint with V. Toledano Laredo.

• 11:00-11:50 Giorgia Fortuna (ETH Zurich): The factorization KL category at critical level and the closure of opers. Let X be a smooth curve and g a simple Lie algebra. To every point x on X, we can consider the formal disc D of x, and attach to it the Kazhdan-Lusztig category at the critical level. This category it is knows to be related to the space of opers on the punctured disc of D. In this talk we will consider the factorization version of these two objects. In particular we will focus on the case of two moving discs and study the closure of certain subschemes as the two discs collide. We will explain how this closure can be used to describe the factorization KL category over two copies of the curve.

#### Travel details:

Talks will be held in the James Clerk Maxwell Building at the University of Edinburgh. Walking from the Edinburgh train station to the University takes approximately 30 minutes, and there is also a bus that runs directly from the train station with a stop at the University very near the James Clerk Maxwell Building. The University of Edinburgh travel advice webpage can be found here.

#### Accommodation and conference dinner:

Most participants are staying at the Masson House hotel. Walking directions to James Clerk Maxwell Building are here: http://goo.gl/maps/x6PL8. Note that it's a 30 minute walk to JCMB!

The conference dinner will be at Voujon Indian restaurant, http://voujonedinburgh.co.uk‎, located here: http://goo.gl/maps/BgiA6.