• Peleg Bar-Sever (Weizmann Institute of Science)

Title: On the conjugacy of Cartan subalgebras of affine Kac-Moody  superalgebras

Abstract: Given two isomorphic Kac-Moody superalgebras, it is interesting to describe how connected are their Cartan matrices. This question was reduced to proving the conjugacy of any pair of Cartan subalgebras of a Kac-Moody superalgebra. For the symmetrizable non-super case,

this was resolved by Kac and Peterson in 1983. In this talk, I will present a new result on the conjugacy of Cartan subalgebras of affine Kac-Moody superalgebras.

• Joseph Bernstein (Tel Aviv University)

Title: Groups, Groupoids, Stacks and Representation Theory

Abstract: see here.

Slides

• Galyna Dobrovolska (Ariel University)

Title: Geometric representation theory, canonical bases, and explicit character formulas for affine Lie algebras

Abstract: I will begin with a tour of classical ideas in geometric representation theory including Kazhdan-Lusztig polynomials and Beilinson-Bernstein localization. Then I will delve into a logical continuation of these ideas in the work of Bezrukavnikov on exotic t-structures. Finally I will explain an application of these ideas to finding explicit character formulas for affine Lie algebras in our joint work with Andrei Ionov and Vasily Krylov.

Slides

• Boris Feigin (Hebrew University of Jerusalem)

Title: Extensions  of deformed vertex algebras and " shifted toroidal algebras"

Abstract: I will present some generalization of toroidal algebras and discuss several applications ( in geometry, topology, integrable systems).

• Evgeny Feigin (Tel Aviv University)

Title:  Positroids, Grassmannians and affine Schubert varieties.

Abstract: The classical Grassmann varieties admit a positroid stratification with a rich algebraic, geometric and combinatorial structure. The stratification plays an important role in the description of totally nonnegative  Grassmannians and can be defined by projecting Richardson varieties from the full flag  variety. The positroid strata are labelled by juggling patterns which also enter the description of the degeneration of Grassmannians to the unions of affine Schubert varieties. 

We will describe the objects showing up in this picture and  explain the role of quiver Grassmannians for the equioriented cyclic quivers. 

Partially based on joint works with Martina Lanini and Alexander Puetz.

• Michael Finkelberg (Hebrew University of Jerusalem)

Title: Lagrangian subvarieties of hyperspherical varieties

Abstract: A few years ago, in the framework of relative Langlands duality, D. Ben-Zvi, Y. Sakellaridis and A. Venkatesh proposed the notion of S-duality of certain coisotropic symplectic varieties with hamiltonian action of Langlands dual groups. 

We consider certain Lagrangian subvarieties of coisotropic symplectic varieties and conjecture that the Lagrangian subvarieties of S-dual varieties have the same number of irreducible components. We check this conjecture for a family of examples related to basic classical Lie superalgebras and equivariant slices. This is our joint work with Victor Ginzburg and Roman Travkin.

• Yasmine Fittouhi (Weizmann Institute of Science)

Title: Description of the Nilfibre $\mathscr N$

Abstract: Let  $P$ a parabolic subgroup of $SL(n)$, $P’$ its derived subgroup, and $\mathfrak{m}$ the nilradical of its Lie algebra $\mathfrak{p}$. This presentation delves into the geometry of the  nilfibre  $\mathscr N$, the zero locus of the augmented invariant ideal $\mathbb{C}[\mathfrak{m}]^{P'}$ , through the concept of component tableau associated with Weierstrass sections. We show that for component tableaux  define distinct irreducible subvarieties in $\mathscr N$.  The results highlight the intricate interplay between combinatorial data, tableau construction,  and the underlying geometry of the nilfibre, offering new insights.

Slides

• Dmitry Gourevitch (Weizmann Institute of Science)

Title: A Stone-von Neumann equivalence of categories for smooth representations of the Heisenberg group

Abstract: The classical Stone-von Neuman theorem relates the irreducible unitary representations of the Heisenberg group Hn to non-trivial unitary characters of its center Z, and plays a crucial role in the construction of the oscillator representation for the metaplectic group.

In a joint work with R. Gomez and S. Sahi we extended these ideas to non-unitary and non-irreducible representations, thereby obtaining an equivalence of categories between certain representations of Z and those of Hn. Our main result is a smooth equivalence, which involves the fundamental ideas of du Cloux on differentiable representations and smooth imprimitivity systems for Nash groups. I will explain the idea behind this equivalence, show how to extend the oscillator representation to the smooth setting and give some applications.

Slides

• Max Gurevich (Technion)

Title: Deforming Springer into Arthur, and other tales of small representations of p-adic groups

Abstract: This talk reports on recent and ongoing joint work with Emile Okada.

When can an infinite-dimensional representation be considered small?

For p-adic reductive groups, one approach involves minimizing the size of the associated wavefront set.

Another automorphic-flavored approach considers the class of spherical/unramified representations.

For split symplectic or odd orthogonal groups, we show that Arthur's endoscopic theory (in its local aspects) effectively sharpens the links between the two genres. In the process, we resolve a heuristic posed by Ciubotaru- Mason-Brown- Okada regarding p-adic variants of ‘weak Arthur packets’.

A key method compares the local systems on nilpotent orbits that arise from the geometric Springer construction, parameterizing representations of Weyl groups, with those appearing in endoscopy. This comparison reveals a curious interplay between the theory of Green functions for finite groups and p-adic phenomena.

In particular, we highlight the concept of ‘weak sphericity’ - the property of a representation having non-zero vectors invariant under a maximal compact subgroup (which may not be hyperspecial).

Slides

• Anthony Joseph (Weizmann Institute of Science)

Title: Dessins d’enfants pour noel. 

Slides

• Guy Kapon (Weizmann Institute  of Science)

Title: Distinguished representations of symmetric subgroups of GL_n

Abstract: A conjecutre of Erez Lapid suggested by relative Langlands theory states a necessary condition for an irreducible represntaion to be distinguished with respect to a symmetric subgroup. We prove the analogue of the conjecture for the case of $GL_n$ over finite fileds and any symmetric subgroup.  

The proof has two parts. The cuspidal case follows from the theory of Deligne-Lustig, and the general case then follows by Zelevinsky theory. 

• Anton Khoroshkin (University of Haifa)

Title: On Demazure modules and Cauchy identity for staircase matrices

Abstract: First, I will discuss some combinatorial aspects of Demazure modules, whose characters are known as "key polynomials" $\kappa_{\lambda}(x)$. Second, I will present a generalization of the classical Cauchy identity for staircase-shaped matrices.

The Cauchy identity expresses the product of terms $(1 - x_i y_j)^{-1}$ (where indexes $(i,j)$ are the entries of a rectangular matrix), as a sum over partitions  $\lambda$ of products of Schur polynomials: $s_{\lambda}(x) s_{\lambda}(y)$. By breaking the symmetry, we describe the decomposition of the product $(1 - x_i y_j)^{-1}$, (where $(i,j)$ ranges the cells of a staircase matrix), in terms of a sum of products of key polynomials: $\kappa_{\lambda}(x) \kappa^{\mu}(y)$.

Surprisingly, the proof of this polynomial identity relies on certain (homological) properties of Demazure modules and their tensor products discussed in the first part of the talk.

The talk is based on joint work with E. Feigin and Ie. Makedonskyi (arxiv:2411.03117).

• Leonid Makar-Limanov (Wayne University)

Title: The Newton approach to the two-dimensional Jacobian conjecture

Abstract: In my talk I’ll explain how Isaac Newton would attack the two-dimensional Jacobian conjecture if he knew about it.

• Anna Melnikov (University of Haifa)

Title: The sphericity phenomenon in the intersection of a nilpotent orbit with a maximal nilradical in classical Lie algebras.

Abstract: Let G be a Lie complex group and  g=Lie(G). Let B be a Borel subgroup of G. G acts adjointly on g.  A nilpotent orbit in g is called spherical if it admits the dense B-orbit. As it was shown by E. Vinberg and M. Brion an orbit is spherical if and only if  it decomposes into a finite number of B-orbits. Further D. Panyushev constructed a full classification of spherical orbits in all simple Lie algebras.

Let n be the maximal nilradical of Lie(B). Its intersection with a nilpotent orbit O is a Lagrangian subvariety of O and in particular it is an equidimensional union of irreducible components, called orbital varieties. They are naturally B-stable, but do not admit a dense B-orbit in general. Moreover the existence of a dense B-orbit does not provide in general that an orbital variety decomposes into a finite number of B-orbits. 

However, there is a global spherical phenomena: in classical Lie algebras  each orbital variety in O has a dense B-orbit if and only if the intersection of O with n decomposes into a finite number of B-orbits. We provide the classification of such nilpotent orbits for classical Lie algebras.

As for orbital varieties with a dense B-orbits, there is at least one such variety for any nilpotent orbit and there is an interesting duality between orbital varieties with a dense B-orbit and smooth orbital varieties.

Joint work with L. Fresse.

• Shifra Reif (Bar Ilan University)

Title: The Harish Chandra Theorem for Symmetric superspaces

Abstract:  The Harish-Chandra Theorem for the center of the universal enveloping algebra can be generalized to symmetric spaces. It states that the algebra of invariant differential operators on a symmetric space G/K is isomorphic to the algebra of polynomials on the Cartan subspace that are invariant under the shifted action of the Weyl group. We will discuss this generalization for symmetric superspaces.

Joint work with Siddhartha Sahi, Vera Serganova and Alexander Sherman.

• Guy Shtotland (Ben Gurion University)

Title: Distinction of the Steinberg representation with respect to split symmetric subgroups

Abstract: We study the distinction of the Steinberg representation of a split p adic group G with respect to a split symmetric subgroup H. 

We relate this problem to the problem of determining the existence of a non zero harmonic function on a certain hyper graph related to X=G/H. 

We show that the Steinberg representation is distinguished only if X is quasi-split and that GL_{2n+1}/GL_{n}xGL_{n+1}  is essentially the only example where X is quasi-split but the Steinberg representation isn't distinguished. As a corollary, we obtain a proof of the relative local Langlands conjecture for the Steinberg representation, we show that the Steinberg representation is distinguished if and only if its parameter factors through the dual group of X.

Slides