Oct 17

“Poles of the Standard Lfunction and Functorial Lifts for G2” expanded, part I

Avner Segal (Bar Ilan)

This is part 1 of the speaker’s talk from last semester, expanded into a twopart series.
The functoriality conjecture is a key ingredient in the theory of automorphic forms and the Langlands program. Given two reductive groups G and H, the principle of functoriality asserts that a map r:G^>H^ between their dual complex groups should naturally give rise to a map r*:Rep(G)>Rep(H) between their automorphic representations. In this talk, I will describe the idea of functoriality, its connection to Lfunctions and recent work on weak functorial lifts to the exceptional group of type G_2.

Oct 24

Tamagawa Numbers of Linear Algebraic Groups over Function Fields

Zev Rosengarten (HUJI)

In 1981, Sansuc obtained a formula for Tamagawa numbers of reductive groups over number fields, modulo some then unknown results on the arithmetic of simply connected groups which have since been proven, particularly Weil’s conjecture on Tamagawa numbers over number fields. One easily deduces that this same formula holds for all linear algebraic groups over number fields. Sansuc’s method still works to treat reductive groups in the function field setting, thanks to the recent resolution of Weil’s conjecture in the function field setting by Lurie and Gaitsgory. However, due to the imperfection of function fields, the reductive case is very far from the general one; indeed, Sansuc’s formula does not hold for all linear algebraic groups over function fields. We give a modification of Sansuc’s formula that recaptures it in the number field case and also gives a correct answer for pseudoreductive groups over function fields. The commutative case (which is essential even for the general pseudoreductive case) is a corollary of a vast generalization of the PoitouTate nineterm exact sequence, from finite group schemes to arbitrary affine commutative group schemes of finite type. Unfortunately, there appears to be no simple formula in general for Tamagawa numbers of linear algebraic groups over function fields beyond the commutative and pseudoreductive cases. Time permitting, we may discuss some examples of noncommutative unipotent groups over function fields whose Tamagawa numbers (and relatedly, TateShafarevich sets) exhibit various types of pathological behavior.

Oct 31

Poles of the Standard Lfunction and Functorial Lifts for G2

Avner Segal (Bar Ilan)

This is part 2 of 2 of an expanded version of the speaker’s talk from last semester.
The functoriality conjecture is a key ingredient in the theory of automorphic forms and the Langlands program. Given two reductive groups G and H, the principle of functoriality asserts that a map r:G^>H^ between their dual complex groups should naturally give rise to a map r*:Rep(G)>Rep(H) between their automorphic representations. In this talk, I will describe the idea of functoriality, its connection to Lfunctions and recent work on weak functorial lifts to the exceptional group of type G_2.

Nov 7

Some SchurWeyl Dualities

Kieran Ryan (Queen Mary University of London)

SchurWeyl Duality is a remarkable theorem giving an intimate link between the representation theories of the Symmetric group S_n, and the General Linear group GL(k). Such a link also holds between other objects, in particular the Brauer Algebra and the Orthogonal group, and the Walled Brauer algebra and GL(k). I will give an introduction to these relationships.

Nov 14

The Representation Theory of the finite and infinite TemperleyLieb algebras

Stephen Moore (BGU)

The TemperleyLieb algebras are a family of finite dimensional algebras that are quotients of the symmetric groups algebras, or more generally the IwahoriHecke algebras. They appear in a number of areas of mathematics, including statistical mechanics, knot theory, quantum groups, and subfactors. We review their representation theory and give some results on an infinite dimensional generalization.

Nov 28

Structure of Degenerate Principal Series of Exceptional Groups

Avner Segal (Bar Ilan)

The reducibility and structure of parabolic inductions is a basic problem in the representation theory of padic groups. Of particular interest is its principal series and degenerate principal series representations, that is parabolic induction of 1dimensional representations of Levi subgroups. In this talk, I will start by describing the functor of normalized induction and its left adjoint the Jacquet functor and by going through several examples in the group SL_4(Q_p) will describe an algorithm which can be used to determine reducibility of such representations. This algorithm is the core of a joint project with Hezi Halawi, in which we study the structure of degenerate principal series of exceptional groups of type En (see https://arxiv.org/abs/1811.02974).

Dec 5

The Zoo of Integral Representations for Lfunctions

Nadya Gurevich (BGU)

Automorphic Lfunctions, initially defined on some right half plane, are conjectured to be have
meromorphic continuation to the whole complex plane. An effective method to prove this in some cases is by using an integral representation. Since the 1960’s, many such integrals were discovered, some of them representing the same Lfunction, but seemingly unrelated. Using recent discoveries of D.Ginzburg and D. Soudry, I will explain the relation between different integrals representing the same Lfunction.

Dec 12

Tropicalizations, tropical reductions and liftings of curves with differentials

Ilya Tyomkin (BGU)

Tropicalizations and tropical reductions provide a convenient tool to control degenerations of algebraic objects. Roughly speaking, a tropicalization is a piecewise linear object, associated to an algebraic object over a nonArchimedean field, that contains essential information about one of its integral models. The tropical reduction is then the reduction of the model over the residue field. For applications, it is often important not only to describe the tropicalization process, but also to be able to decide whether something that looks like the tropicalization and the tropical reduction comes from an algebraic object. Such statements are called lifting theorems.
Tropical techniques have been applied successfully to a number of problems in algebraic geometry, such as enumerative questions, dimension estimates, descriptions of compactifications etc. In particular, in a recent work of Bainbridge, Chen, Gendron, Grushevsky, and Moeller, a tropical approach was used to describe a new compactification of the space of smooth curves with differentials (although the authors don’t use this terminology). The proofs of BCGGM rely on transcendental techniques.
In my talk, I will present a modified version of BCGGM tropicalization, and will indicate an algebraic proof of the main result. The talk is based on a joint work with M.Temkin.

Dec 19

Tensor categories in positive characteristic

Kevin Coulembier (University of Sydney)

Tensor categories are abelian klinear monoidal categories modeled on the representation categories of affine (super)group schemes over k. Deligne gave very succinct intrinsic criteria for a tensor category to be equivalent to such a representation category, over fields k of characteristic zero. These descriptions are known to fail badly in prime characteristics. In this talk, I will present analogues in prime characteristic of these intrinsic criteria. Time permitting, I will comment on the link with a recent conjecture of V. Ostrik which aims to extend Deligne’s work in a different direction.

Dec 26

Support varieties for supergroups

Vera Serganova (UC Berkeley)

We define a functor from the category of representations of algebraic supergroups with reductive even part to the category of equivariant sheaves and show several applications of this construction to representation theory.

Jan 2

Ambidexterity in the T(n)Local Stable Homotopy Theory

Tomer Schlank (HUJI)

The monochromatic layers of the chromatic
filtration on spectra, that is the K(n)local (stable 00)categories Sp_{K(n)} enjoy many remarkable properties. One example is the vanishing of the Tate construction due to HoveyGreenleesSadofsky. The vanishing of the Tate construction can be considered as a natural equivalence between the colimits and limits in Sp_{K(n)} parametrized by finite groupoids. Hopkins and Lurie proved a generalization of this result where finite groupoids are replaced by arbitrary \pifinite 00groupoids.
There is another possible sequence of (stable 00)categories who can be considered as “monochromatic layers”, those are the T(n)local 00categories Sp_{T(n)}. For the Sp_{T(n)} the vanishing of the Tate construction was proved by Kuhn. We shall prove that the analog of Hopkins and Lurie’s result in for Sp_{T(n)}. Our proof will also give an alternative proof for the K(n)local case.
This is a joint work with Shachar Carmeli and Lior Yanovski

Jan 9

Reconstruction of formal schemes using their derived categories

Saurabh Singh (BGU)

