The seminar usually takes place on Mondays at 11:00 a.m. in Room 261 of the Ziskind Building.

Link to previous years

Title: Degenerate Whittaker functionals on representations of real reductive groups

Abstract: It was proven by Nakayama that for modules over commutative algebras, the support can be measured by existence of functionals equivariant with respect to different characters of the algebra.
For smooth representations of reductive groups over local fields, one is interested in functionals equivariant with respect to characters of the nilradical of Borel subgroups. Functionals equivariant with respect to non-degenerate characters are called Whittaker functionals. It was shown by Rodier in the p-adic case and Kostant in the archimedean case that an irreducible representation has (non-zero) Whittaker functionals if and only if it has maximal wavefront set / associated variety.
In the p-adic case, the correspondence between existence of other equivariant functionals and the wavefront set was found by Moeglen and Waldspurger, and in the real case there are partial results by Matumoto, establishing in some cases a connection between the associated variety and existence of functionals equivariant w.r.t. non-degenerate characters of nilradicals of (bigger) parabolic subgroups.
In our joint work with S. Sahi, we take a different path and establish a a precise connection between the associated variety of a representation and the existence of functionals equivariant w.r.t. (degenerate) characters of nilradicals of the Borel subgroup.
Remark. This is a "preseason" meeting in an unusual day. The next talk will be on the first Monday of the semester, Nov 7, by Alexander Rahm.

Title: Number-theoretic formulae for the homological torsion of the Bianchi groups

Abstract: The Bianchi groups are the arithmetic groups PSL_2 over rings of integers in imaginary quadratic number fields. We present the recent discovery of formulae for one of their essential invariants, the homological torsion. These formulae depend only on basic number-theoretic information like ideal class numbers, numbers of prime divisors, occurrence of a given norm on the ring of integers of a number field.

Title: Derivatives for smooth representations of GL(n,R) and GL(n,C)

Abstract: The notion of derivatives for smooth representations of GL(n,Q_p) was defined by Bernstein and Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by Sahi and called the ``adduced" representation.
In our joint work with A. Aizenbud and S. Sahi we define derivatives of all orders for smooth admissible Frechet representations (of moderate growth). The real case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation.
We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations.

Title: Cohen-Macaulay modules and questions of Harmonic analysis

Title: Local and global aspects of unitary periods

Abstract: The study of period integrals of automorphic forms is interrelated with the theory of invariant linear forms on representations of local groups. I will present one case study, that of unitary periods, where much is now known both in the local and global aspects. This is an application of a work of Jacquet over many years and a recent joint work with Feigon and Lapid.

Title: Projective Normality of G.I.T. quotient varieties modulo finite solvable groups and Weyl groups

Abstract

Title: On the Jacobian conjecture and related problems

Abstract: I'll report on some new results related to the two-dimensional Jacobian conjecture.

Title: Fourier-Mukai transform and chiral differential operators

Abstract: Let CDO(X) denote the groupoid of algebras of chiral differential operators over an abelian variety X. We show that the groupoids CDO(X) and CDO(X') are equivalent, X' being the dual abelian variety. There exists a family of such equivalences parametrised by classes of non-degenerate algebras of twisted differential operators over X.

Title: Fast Matched Filter in Linear Time and Group Representation: What? Why? How?

Abstract

Title: Polynomial functors and categorifications

Abstract: I will explain how to categorify various Fock space representation structures on the algebra of symmetric functions via the category of strict polynomial functors. We enrich Schur-Weyl duality functor to a morphism of Kac-Moody and Heisenberg categorifications. This is a joint work with O.Yacobi.

Title: Hilbert's irreducibility theorem and Galois representations

Abstract: Hilbert's irreducibility theorem asserts if f is a polynomial in two variables X,Y with integral coefficients that is irreducible and of degree at least 1 in Y, then there exists an irreducible specialization, i.e. a rational number a, such that f(a,Y) is irreducible. A field with irreducible specializations is called Hilbertian. The numerous applications of this theorem makes the question of under what conditions an extension of a Hilbertian field is again Hilbertian. It turns out the the most difficult part is separable algebraic extensions. Jarden conjectured that if K is Hilbertain, A abelian variety over K, and E/K is an extension of K that is contained in the field generated by all torsion points of A, then E is Hilbertian. In this talk I shall discuss a solution of the conjecture using Galois representations.

Title: Zhelobenko invariants and filtration on the Cartan

Title: Zhelobenko invariants and filtration on the Cartan

Title: Partial flag varieties and nilpotent elements

Abstract: The flag variety of a complex reductive algebraic group G is by definition the quotient G/B by a Borel subgroup. It identifies with the set of Borel subalgebras of Lie(G). Given a nilpotent element x in Lie(G), one calls Springer fiber the subvariety formed by the Borel subalgebras which contain x. Springer fibers have in general a quite complicated structure (in general not irreducible, singular). Nevertheless, a theorem of DeConcini, Lusztig, and Procesi asserts that, when G is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In the talk, we study varieties generalizing the Springer fibers to the context of partial flag varieties, that is, subvarieties of the quotient G/P by a parabolic subgroup (instead of a Borel subgroup). We propose a generalization of DeConcini-Lusztig-Procesi's theorem to this context.

Title: Weyl groups for locally compact groups

Abstract: It is well understood how to associate a Weyl group W(G) with any algebraic group G, such that for any epimorphism of algebraic group G to H one gets a surjection W(G) to W(H). In this talk I will explain how to view this association (over a local field) in such a way that could be extended to any locally compact group. We use transcendental methods, borrowed from ergodic theory and random walks theory, that I will try to explain. We get a natural obstruction on linear representations of groups, that enables us to prove some old and new rigidity results, e.g Margulis' super-rigidity theorem (that I will explain). The talk is based on a joint work with Alex Furman.

Title: Flexible Varieties

Abstract

Title: Generalized Harish-Chandra isomorphism

Abstract: In our joint work with Khoroshkin and Vinberg, for any complex reductive Lie algebra g and any locally finite g-module V, the Harish-Chandra description of g-invariants in the universal enveloping algebra U(g) was extended to the tensor product of U(g) with V. Another proof of this result has been found by Joseph, who then also linked it to Clifford algebra conjecture of Kostant. In this talk, I will explain how this result has been used by Khoroshkin and myself to give explicit realizations of all simple finite-dimensional modules of Yangians and their twisted analogues.

Title: Fusion procedure for representations of Yangians

Abstract: In this talk, I will explain the combinatorics underlying explicit explicit realizations of all simple finite dimensional modules of Yangians and their twisted analogues. In particular, I will develop the fusion procedure for twisted Yangians. For the non-twisted Yangians, this procedure goes back to the works of Cherednik and Rogawski on intertwining operators between the principal series representations of affiine Hecke algebras.

Title: Global spherical functions

Please note the unusual day,time and room.

Abstract: The affine Demazure characters are one of the main objects in the Kac-Moody representation theory. In the level one case, the corresponding quadratic-type generating functions were proven several years ago to be (very remarkable) solutions of the q-Toda eigenvalue problem. It was done only for dominant (W-invariant) Demazure characters; the general case is in progress (for level=1).
Importantly, the simplest way of arriving at this theorem/theory is via its q,t-deformation followed by the limit t-->0. The corresponding generating functions were called global spherical functions due to their analyticity anywhere; they solve the Macdonald-Ruijsenaars eigenvalue problem and go through the Macdonald polynomials for dominant weights (through the Demazure characters in the limit).

Title: On ramified covers of algebraic surfaces, and combinatorics of some infinite discrete groups.

Abstract: Bernhard Riemann studied (complex) algebraic curves by considering ramified covers of the projective line. In dimension two, one of the classical approaches of the Italian school of algebraic geometry was to study (complex) algebraic surfaces by considering ramified covers of the simplest surface -- the projective plane.
Such ramified covers are classified, in any dimensions, by suitable representations of some (usually very large!) discrete group Gamma into a symmetric group Sym_n. Here Gamma is the fundamental group of a complement to the branch divisor.
Now, a striking difference appears between dimensions one and two. It was discovered by Enriques, Castelnuovo and, especially, Zariski.
First, any collection of points on a projective line is a ramification locus of some ramified cover. The corresponding fundamental group is the free group F_n, which always has lots of representations into the symmetric group Sym_n. In particular, the construction gives lots of complex curves. There were many studies of the corresponding representations.
The case of dimension two is very different. Only very special plane curves are ramification divisors, and only very special groups are fundamental groups. Moreover, there are restrictions on representations.
We argue that the groups Gamma appearing here are (a very broad) generalizations of the Artin's braid group, and define the corresponding class of discrete groups. We call them twisted Artin-Brieskorn groups. Then we discuss combinatorics of their representations into symmetric groups Sym_n, and discuss the related Chisini conjecture of algebraic geometry.

Title: A New Perspective on the İnönϋ-Wigner Contractions

Abstract: Many physical theories approximate other theories under certain limits. Segal, İnönü and Wigner were the first to consider what are the implications of these limits on the corresponding symmetry groups. Contraction is a formal way of applying these limits to Lie groups, Lie algebras and their representations. In recent work we have shown that any contraction of Lie algebra representations is intrinsically a direct limit construction. Moreover, for any İnönü-Wigner contraction of a real three dimensional Lie algebra, we obtained the corresponding contractions of the irreducible representations in a canonical way by pointwise convergence of differential operators. In this talk I’ll review contraction of Lie algebras and their representations focusing on the methods of İnönü and Wigner. I will present several examples, some of which are new and show how the direct limit construction arises naturally. This work was done in collaboration with E. M. Baruch, J. L. Birman and A. Mann.

Title: On Dixmier-Duo isomorphism in positive characteristic - the classical nilpotent case

Abstract

Title: The Capelli identity for Grassmannians

Abstract: The classical Capelli identity [1887] is a certain identity of differential operators on the space of n x n matrices. It played a crucial role in Herman Weyl's approach to 19th century invariant theory and has continued to find modern day applications, e.g. in the work of Atiyah-Bott-Patodi on the index theorem. In the early 1990s the identity was reinterpreted by Kostant and the speaker, as an eigenvalue problem for a certain invariant differential operator, and generalized to the setting of Jordan algebras. Let Gr(n,k) denote the Grassmannian of k-planes in n-space. In recent work, Howe-Lee solve an analogous Capelli type eigenvalue problem for differential operators on Gr(n,2). Their method is elementary but involves fairly intricate computations, and it is not clear how to extend this to other cases. In this talk we explain how to solve the problem for all k using ideas from representation theory.

http://www.math.rutgers.edu/~sahi/Preprints/Capelli4-18.pdf

Title: An algebraic Sato-Tate group and Sato-Tate conjecture

Abstract: The purpose of this lecture is the discussion of the algebraic Sato-Tate group and algebraic Sato-Tate conjecture for abelian varieties over number fields. The existence of the algebraic Sato-Tate group over $\mathbb{Q}$ can be applied to define the Sato-Tate group which is useful in search of the precise formulation of the Sato-Tate conjecture in the framwork of abelian varieties as shown by results of F. Fit{\' e}, K. Kedlaya, V. Rotger, A. Sutherland. I will explain the relation of the algebraic Sato-Tate conjecture to the Mumford-Tate conjecture. This relation allows one to prove the algebraic Sato-Tate conjecture for some families of abelian varieties investigated in papers I wrote jointly with W. Gajda and P. Krason. This lecture presents results of a joint paper with Kiran Kedlaya.

Title: Brauer-Grothendieck groups and Brauer-Manin sets

Abstract: We discuss finiteness properties of Brauer-Grothendieck groups with special reference to abelian varieties and K3 surfaces. This is a report on a joint work with Alexei Skorobogatov.

Title: TBA

Abstract: TBA

Title: Fourier Transform of Algebraic Measures

Abstract: We study the Fourier transform of an absolute value of a polynomial on a finite-dimensional vector space. We prove that this transform is smooth on an open dense set. Our proof is based on Hironaka's desingularization theorem and on the study of the wave front set of the Fourier Transform. Our method suits, both the Archimedean and the non-Archimedean case. We also give some bounds on the open dense set where the Fourier transform is smooth and more generally on its wave front set. These bounds are explicit in terms of resolution of singularities. We also prove the same result on Fourier transform of other measures of algebraic origins. Similar (but less general and explicit) results was proven earlier by Bernstein, Cluckers- Loeser and Hrushovski-Kazhdan using diffident methods.

Title: The Categorical Weil Representation and the Sign Problem

Abstract

Title: Detecting linear dependence in Mordell-Weil groups of abelian varieties

Abstract: I will explain our recent work on “local to global principle” for abelian varieties. I will discuss certain sufficient numeric condition we found. This allows us to determine linear dependence in Mordell-Weil groups of abelian varieties via reduction maps. In particular we try to determine the conditions for detecting linear dependence in Mordell-Weil groups via finite number of reductions. I will also talk about possible generalizations. This work is joint with Grzegorz Banaszak.