## Titles and abstracts

**Noriyuki Abe: **Soergel bimodules and homomorphism between Bott-Samelson bimodules

**Abstract:** The Hecke categories by which we mean categorifications of the Hecke algebra play important roles in modular representation theory. I will introduce the category of Soergel bimodules as the category generated by Bott-Samelson bimodules. This gives a reincarnation of the Hecke category under an assumption of an existence of certain morphism between Bott-Samelson bimodules. I will also explain certain sufficient condition of the existence.

**Alexis Bouthier: **Affine Springer theory and affine character sheaves

**Abstract:** We will survey results on affine Springer theory, how one can extend classical results to the affine setting where everything is infinite dimensional and use it to geometrize certain characters of p-adic groups.

**Kęstutis Česnavičius: **The affine Grassmannian as a presheaf quotient

**Abstract:** The affine Grassmannian of a reductive group G is usually defined as the étale sheafification of the quotient of the loop group LG by the positive loop subgroup. I will discuss various triviality results for G-torsors which imply that this sheafification is often not necessary.

**Stéphane Gaussent: **Hecke algebras associated to loop groups over local fields

**Abstract: **The local fields we will focus on are the non-archimedean ones, like p-adic numbers. The loop groups that we are interested in are the loops on the reductive groups, that we see as affine Kac-Moody groups. If one consider them over a non-archimedean local field, the classical decompositions such as Cartan or Bruhat are no longer covering the whole group. However, one can still multiply double cosets and define some Hecke algebras. If we think of the "Cartan decomposition" we get the spherical Hecke algebra, whereas the "Bruhat decomposition" give the Iwahori-Hecke algebra. I will explain how these algebras are defined and present some of their properties.

**Naoki Genra:** Reduction by stages on W-algebras

**Abstract:** Let X be a Poisson variety with a Hamiltonian G-action and H be a normal subgroup of G. Then X//G is obtained by a (Hamiltonian) reduction of X//H by the induced G/H-action under suitable assumptions, called reduction by stages. We apply for the Slodowy slices and show that the Slodowy slice associated to (g, O) is obtained by a reduction of the Slodowy slice associated to (g, O’) for a simple Lie algebra g and nilpotent orbits O, O’ such that O > O’ with some conditions. The quantum cases imply that the finite/affine W-algebras associated to (g, O) are obtained by W-algebras associated to (g, O’), which proves a conjecture of Morgan in finite cases and gives a conjectural generalization of results of Madsen and Ragoucy in affine cases. This is a joint work with Thibault Juillard.

**Tatsuyuki Hikita: **Elliptic bar involutions for quiver varieties

**Abstract: **K-theoretic canonical bases for symplectic resolutions which admit good Hamiltonian torus actions are characterized by K-theoretic bar involutions defined using (abelian) stable envelopes. In this talk, after reviewing expected properties of K-theoretic canonical bases, I will explain another construction of the bar involutions for quiver varieties using nonabelian stable envelopes defined by Okounkov. If time permits, I will also explain a construction of (level 0) elliptic quantum groups and bar involutions on them.

**Syu Kato (first talk): **Catalan varieties

**Abstract:** We first explain the notion of k-Schur polynomials due to La Pointe, Lascoux, and Morse, and their relation to the geometry of loop spaces of flag manifolds. We then exhibit a(nother) geometric interpretation of k-Schur polynomials conjectured in the thesis of Chen, and outline our proof of her conjecture (and its extension offered by Blasiak-Morse-Pun) by introducing and examining certain family of smooth projective algebraic varieties.

**Syu Kato (second talk, part of the SAGA): **Loop spaces of flag manifolds, quantum geometry, and representation theory

**Abstract:** We briefly recall a description of the algebraic loop spaces of flag manifolds (a.k.a semi-infinite flag manifolds) and its connection to the quantum K-groups of flag manifolds. Then, we discuss about some difference operators acting on the equivariant quantum K-groups of flag manifolds that are controlled by some quasi-coherent sheaves on semi-infinite flag manifolds, and its (partly conjectural) implication(s) in representation theory.

**Kazuya Kawasetsu:** Representation theory of affine vertex algebras

**Abstract: **The characters of the integrable (more generally, admissible) representations over affine Kac-Moody Lie algebras are modular (Jacobi) forms of weight 0. These modules also appear as modules over affine vertex algebras, and the modularity is related to that of vertex algebras (modular invariance properties). In this talk, we will explain recent developments on modular invariant representations of affine vertex algebras focusing on non-integrable ones.

**Emmanuel Letellier: **Fourier transform from the symmetric square representation of PGL(2) and SL(2)

**Abstract: **In this talk we will first review Braverman-Kazhdan's approach of Langlands functoriality via Fourier transform (in the finite field case). We will then explain in the case of symmetric square representation of PGL(2) and SL(2) how to extend Braverman-Kazhdan's Fourier operator (which is not involutive) to an involutive Fourier transform. This is a joint work with Gérard Laumon.

**Leonardo Maltoni:** Towards a Bernstein presentation of the affine Hecke category

**Abstract:** The affine Hecke algebra has a remarkable commutative subalgebra corresponding to the coroot lattice inside the affine Weyl group. Its nature is encoded in the *Bernstein presentation* and reveals some fundamental representation theoretic properties of the Hecke algebra. If one considers *categorifications* of this algebra, for instance the diagrammatic category, this subalgebra corresponds to a class of objects (in the homotopy category) called *Wakimoto** sheaves*, that can be seen as *Rouquier complexes*. In this talk I will introduce these notions and I will present some reduction results about Rouquier complexes and the study of extension groups between Wakimoto sheaves in affine type A_1, for arbitrary coefficients.

Slides of the talk: https://www.dropbox.com/s/dotkemtxiavs87w/Maltoni-talk.pdf and tables: https://www.dropbox.com/s/jtcpmpt3ejgnifd/table_Maltoni-talk.pdf

**Ivan Mirković: **Loop Grassmannians of lattice vertex algebras

**Abstract: **Consider the lattice vertex algebra L associated to a quadratic form q. When q corresponds to the Dynkin diagram of ADE type then L coincides with the basic representation of the corresponding affine vertex algebra. We will extend this reinterpretation to arbitrary form q.

**Yuto Moriwaki: **Vertex operator algebra, braided tensor category, and colored parenthesized braid operad

**Abstract:** The configuration spaces of the Riemann sphere have a structure of an operad, and its fundamental groupoid is called a colored parenthesized braid operad. In this talk, we will explain that the monodromy of conformal blocks (multi-valued functions on the configuration space) in two-dimensional conformal field theory defines a representation of the colored parenthesized braid operad. This provides an alternative proof of the result by Huang-Lepowsky that the representation category of a vertex operator algebra inherits a braided tensor category structure.

**Kota Murakami:** Combinatorics from representation theory of generalized preprojective algebras

**Abstract:** Motivated from studies of the representation theory of quantum loop algebras, Geiss-Leclerc-Schröer introduced the notion of the generalized preprojective algebra associated with a generalized Cartan matrix and its symmetrizer. We wil survey some combinatorics arising from this algebra from the viewpoints of braid group symmetries.

Slides of the talk: https://www.dropbox.com/s/crofm97lkovvume/Murakami-talk.pdf

**Hironori Oya: **Isomorphisms among quantum Grothendieck rings and their applications

**Abstract:** A quantum Grothendieck ring of the monoidal category of finite-dimensional modules over a quantum loop algebra is a one parameter deformation of the usual Grothendieck ring. In this talk, I explain a systematic construction of algebra isomorphisms among the quantum Grothendieck rings of several Dynkin types which respect the $(q, t)$-characters of simple modules. One remarkable point is that they include the isomorphisms between the quantum Grothendieck ring of simply-laced type and that of non-simply-laced type. They lead to new positivity results for the $(q, t)$-characters of simple modules of the non-simply-laced types, and reveal non-trivial relations among the $(q, t)$-characters of simple modules of several Dynkin types. This talk is based on a joint work with Ryo Fujita, David Hernandez, and Se-jin Oh.

**Julien Sebag:** Scheme structure of arc scheme and geometry: various observations

**Abstract:** We will present properties of the arc scheme of an algebraic variety and their relations with the geometry of the involved variety.

Slides of the talk: https://www.dropbox.com/s/g98t6v82gqg1mi3/Sebag-lecture.pdf

**Susanna Zimmermann: **Cremona groups and Groethendieck rings

**Abstract: **For birational maps of the projective plane over an algebraically closed field, it is not hard to see that the number of base-points of the map and the number of base-points of the inverse are equal. This statement should intuitively be true over non-closed fields as well, but it turns out that it is not straight forward. In this talk, I will explain a way to show this for any birational self-map of a minimal smooth projective surface, which uses the Groethendieck ring of varieties of dimension 0.