Fall 2014-- Summer 2015
Title: Generalized and degenerate Whittaker models.
Abstract. The Whittaker model is a very useful tool in the representation theory of reductive groups and in automorphic forms. However, it is known that only the “largest” representations have Whittaker models. In order to overcome this problem, for other representations various kinds of degenerate or generalized Whittaker models are considered since the 80s. Over non-archimedean fields, Moeglen and Waldspurger characterized the existence and multiplicities of these models in terms of the wave-front set. For GL(n,F) they can be also described in terms of the Bernstein-Zelevinsky derivatives. Over the archimedean fields, only partial results are known in this direction. I want to talk about these partial results, including my recent works with Siddhartha Sahi on degenerate Whittaker models, archimedean Bernstein-Zelevinsky derivatives (joint also with A. Aizenbud) and the connection between them. I will also report on a work in progress with S. Sahi and R. Gomez on embedding between different degenerate Whittaker models. This result is new also over p-adic fields.
Title: Strange Lie superalgebras.
Abstract. Strange Lie superalgebras are two families of finite-dimensional simple Lie superalgebras. In my talk I will describe the main properties of each family. The first part of the lecture will be accessible to students who have taken a course on Lie algebras.
Title: Gamma factors of GL(n,R)-distinguished representations of GL(n,C).
Abstract. An irreducible representation (p,V) of GL_n(C) is called GL_n(R)-distinguished if there exists a non-zero continuous GL_n(R)-invariant functional L from V to C. In the talk we give a necessary condition for GL_n(R)-distinction. As a corollary, we prove that the Rankin-Selberg gamma factor of pX p' at s=1/2 for p,p' distinguished representations of GL_m(C),GL_n(C) respectively equals 1.
Title: Affine generalized root systems and symmetrizable affine Kac-Moody superalgebras.
Abstract. Correspondence between different types of Lie algebras and abstract root systems is a classical and useful tool. In the end of the 19th century E.J. Cartan and W. Killing classified real root systems and finite dimensional complex Lie algebras. They showed the correspondence between reduced root systems and these algebras. I.G. Macdonald classified affine root systems in the beginning of the 1970's. V.G. Kac later realized these systems are, in most cases, real parts of Kac-Moody algebras of affine type. V. Serganova classified generalized root systems in 1996 and showed their almost perfect correspondence to basic classical Lie superalgebras. We defined a generalization we call affine generalized root systems, and studied their correspondence to symmetrizable affine Kac-Moody superalgebras. In the talk we will define the above types of root systems, present their precise correspondences to Lie (super)algebras, and present the main points of our classification of affine generalized root systems.
Title: Root systems of Kac-Moody algebras and suuperalgebras.
Abstract. The talk does not assume any knowledge of representation theory apart of the structure of a complex semisimple Lie algebra.
Title: Deligne categories and Kronecker coefficients.
Abstract. In this talk, I will present an application of the theory of Deligne categories to the study of Kronecker coefficients. Kronecker coefficients are structural constants for the category Rep(S_n) of finite-dimensional representations of the symmetric group; namely, given three irreducible representations \mu, \tau, \lambda of S_n, the Kronecker coefficient Kron( \lambda, \mu, \tau) is the multiplicity of \lambda inside \mu \otimes \tau. The study of Kronecker coefficients has been described as "one of the main problems in the combinatorial representation theory of the symmetric group", yet very little is known about them. I will define a "stable" version of the Kronecker coefficients (due to Murnaghan), which generalizes both Kronecker coefficients and Littlewood-Richardson coefficients (structural constants for general linear groups). It turns out that the stable Kronecker coefficients appear naturally as structural constants in the Deligne categories Rep(S_t), which are interpolations of the categories Rep(S_n) to complex t. I will explain this phenomenon, and show that the categorical properties of Rep(S_t) allow us not only to recover known properties of the stable Kronecker coefficients, but also obtain new identities.
Title: From groups to clusters.
Abstract. I will present a new combinatorial construction of finite-dimensional algebras with some interesting representation theoretic properties: they are of tame representation type, symmetric and have periodic modules. The quivers we consider are dual to ribbon graphs and they naturally arise from triangulations of oriented surfaces with marked points. The class of algebras that we get contains in particular the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin-Shapiro-Thurston and Labardini to triangulations of closed surfaces with punctures. Hence our construction may serve as a bridge between modular representation theory of finite groups and cluster algebras. All notions will be explained during the talk.
Title: Geometric degree estimate for a Jacobian mapping of a plane via algebraic degrees.
Abstract. Yitang Zhang who became famous recently thanks to break-through in the twins conjecture wrote his PhD thesis on the plane Jacobian conjecture where he gave an estimate of the geometric degree of the corresponding mapping of a plane via algebraic degrees of the images of the coordinate functions. In my talk I'll explain how to get a better estimate via the Newton polytope approach.
Title: Non-commutative geometry and non-commutative integrable systems
Title: On Lattices over Valuation Rings of Arbitrary Rank
Title:Complex analytic vanishing cycles for formal schemes
Title: Families of Harish Chandra modules connecting compact and noncompact Lie groups.
Abstract. Families of representations naturally appear in representation theory of real reductive Lie groups. In my talk I will demonstrate how the Lie groups themselves come in families and how families of representations (of non-isomorphic groups) play a significant role in representation theory. I will be focusing on the groups SU(1,1), SU(2) and their Cartan motion group. Furthermore, I will show that there exists an algebraic family of Harish Chandra pairs that is associated with these groups. We shall see how families of Harish Chandra modules relate representations of SU(1,1), SU(2) and their Cartan motion group. These families of HC modules will finally be used to provide some insights into the theory of contraction of representations and the Mackey bijection. This talk is based on a joint work with Joseph Bernstein and Nigel Higson.
Title: On the asymptotic of L_2 invariants of arithmetic groups.
Title: Ladder representations and Galois distinction.
Abstract. The space GL_n(E)/GL_n(F), for a quadratic extension E/F of p-adic fields, serves as an approachable case for the study of harmonic analysis on p-adic symmetric spaces on one hand, while having ties with Asai L-functions on the other. It is long known that a GL_n(F)-distinguished representation of GL_n(E) must be contragredient to its own Galois conjugate. Conversely, a conjecture often attributed to Jacquet states that the last-mentioned condition is close to being sufficient for distinction. We show the conjecture is valid for the class of ladder representations which was recently explored by Lapid and Minguez. Along the way, we will suggest a reformulation of the conjecture which concerns standard modules in place of irreducible representations.
Title: gl(\infty) and Deligne categories
Title: TBA
Title: The polyhedral structure of B(infinity): graphs, tableaux and Catalan sets.
Title: Geometry of numbers.
Title: Representation theory of inner forms of GL(n) over a local non-archimedean field - old and new results.
Title: Representation theory of inner forms of GL(n) over a local non-archimedean field - old and new results (cont.)
Title: Prime polynomial values of linear functions in short intervals.
Abstract. In this talk I will present a function field analogue of a conjecture in number theory. This conjecture is a combination of several famous conjectures, including the Hardy-Littlewood prime tuple conjecture, conjectures on the number of primes in arithmetic progressions and in short intervals, and the Goldbach conjecture. I prove an asymptotic formula for the number of simultaneous prime values of n linear functions, in the limit of a large finite field. A key role is played by the computation of some Galois groups.
Title: Conformal and differential Lie algebras.
Title: Square-integrable representations of classical p-adic groups and their Jacquet modules.
Title: Around the Thom-Sebastiani theorem.
Abstract. For germs of holomorphic functions $f : \mathbf{C}^{m+1} \to \mathbf{C}$, $g : \mathbf{C}^{n+1} \to \mathbf{C}$ having an isolated critical point at 0 with value 0, the classical Thom-Sebastiani theorem describes the vanishing cycles group $\Phi^{m+n+1}(f \oplus g)$ (and its monodromy) as a tensor product $\Phi^m(f) \otimes \Phi^n(g)$, where $(f \oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n)$. I will discuss algebraic variants and generalizations of this result over fields of any characteristic, where the tensor product is replaced by a certain local convolution product, as suggested by Deligne. The main theorem is a K?nneth formula for $R\Psi$ in the framework of Deligne's theory of nearby cycles over general bases.
Title: Autocorrelations of the M"obius function over function fields.
Abstract. In this talk we shall discuss results on autocorrelations of the arithmetic M"obius function of polynomials over finite fields, in the limit of a large base field. Special consideration will be given to base fields of characteristic 2, where both methods and results substantially differ from those applicable in odd characteristics. The methods used are mostly elementary, with a hint of algebraic geometry.
Title: A Family of New-way Integrals for the Standard L-function of Cuspidal Representations of the Exceptional Group of Type G2.
Abstract. In a joint work with N. Gurevich we have constructed a family of Rankin-Selberg integrals representing the standard twisted L-function of a cuspidal representation of the exceptional group of type G2. This integral representations use a degenerate Eisenstein series on the family of quasi-split forms of Spin8 associated to an induction from a character on the Heisenberg parabolic subgroup. This integral representations are unusual in the sense that they unfold with a non-unique model. A priori this integral is not factorizable but using remarkable machinery proposed by I. Piatetski-Shapiro and S. Rallis we prove that in fact the integral does factor. As the local generating function of the local L-factor was unknown to us, we used the theory of C*-algebras in order to approximate it and perform the unramified computation. If time permits, I will discuss the poles of the relevant Eisenstein series and some applications to the theory of CAP representations of G2.
Title: Reciprocity laws and K-theory
Abstract. We associate to a full flag F in an n-dimensional variety X over a field k, a "symbol map" $\mu_F :K(F_X) \to \Sigma^n K(k)$. Here, F_X is the field of rational functions on X, and K(.) is the K-theory spectrum. We prove a "reciprocity law" for these symbols: Given a partial flag, the sum of all symbols of full flags refining it is 0. Examining this result on the level of K-groups, we derive the following known reciprocity laws: the degree of a principal divisor is zero, the Weil reciprocity law, the residue theorem, the Contou-Carr\`{e}re reciprocity law (when X is a smooth complete curve) as well as the Parshin reciprocity law and the higher residue reciprocity law (when X is higher-dimensional). This is a joint work with Evgeny Musicantov.
Title: A description of two-generated subalgebras of a polynomial ring in one variable and a new proof of the AMS theorem.
Abstract. The famous AMS (Abhyankar-Moh-Suzuki) theorem states that if two polynomials $f$ and $g$ in one variable with coefficients in a field $F$ generate all algebra of polynomials, i.e. any polynomial $h$ in one variable can be expressed as $h = H(f, g)$ where $H$ is a polynomial in two variables, then either the degree of $f$ divides the degree of $g$, or the degree of $g$ divides the degree of $f$, or the degree of $f$ and the degree of $g$ are divisible by the characteristic of the field $F$. There were several wrong published proofs of this theorem and there are many correct published proofs of this theorem but all of them either long or not self-contained. Recently I found a (relatively) short and self-contained proof which is not published yet and which I can explain in one-two hours.
Title: Orbits of parabolic subgroups on generalized symmetric spaces.
Abstract. Let G be a connected reductive algebraic group defined over a field k of characteristic not 2, s an involution of G defined over k, H a k-open subgroup of the fixed point group of s and G_k (resp. H_k) the set of k-rational points of G (resp. H). The homogeneous space X_k:=G_k/H_k is a generalization of a real reductive symmetric space to arbitrary fields and is called a generalized symmetric space. Orbits of parabolic k-subgroups on these generalized symmetric spaces occur in various situations, but are especially of importance in the study of representations of G_k related to X_k. In this talk we present a number of structural results for these parabolic k-subgroups that are of importance for the study of these generalized symmetric space and their applications.
Title: Oka Principles and the Linearization Problem.
Abstract. Let Q be a Stein space and L a complex Lie group. Then Grauert's Oka Principle states that the canonical map of the isomorphism classes of holomorphic principle L-bundles over Q to the isomorphism classes of topological principle L-bundles over Q is an isomorphism. In particular he showed that if P, P' are holomorphic principle L-bundles and $\Phi\colon P\to P'$ a topological isomorphism, then there is a homotopy $\Phi_t$ of topological isomorphisms with $\Phi_0=\Phi$ and $\Phi_1\colon P\to P'$ a holomorphic isomorphism. Let X and Y be Stein G-manifolds where G is a reductive complex Lie group. Then there is a quotient Stein space Q_X$ and a morphism $\pi_X\colon X\to Q_X$ such that $(\pi_X)^*\O(Q_X)=\O(X)^G$. Similarly we have $p_Y\colon Y\to Q_Y$. Suppose that $\Phi\colon X\to Y$ is a $G$-biholomorphism. Then the induced mapping $\phi\colon Q_X\to Q_Y$ has the following property: for any $z\in Q_X$, $X_z:=\pi_X^{-1}(z)$ is $G$-isomorphic to $Y_{\phi(z)}$ (the fibers are actually affine $G$-varieties). We say that $\phi$ is admissible. Now given an admissible $\phi$, assume that we have a G-equivariant homeomorphism $\Phi\colon X\to Y$ lifting $\phi$. Our goal is to establish an Oka principle, saying that $\Phi$ has a deformation $\Phi_t$ with $\Phi_0=\Phi$ and $\Phi_1$ biholomorphic. We establish this in two main cases. One case is where $\Phi$ is a diffeomorphism that restricts to G-isomorphisms on the reduced fibers of $\pi_X$ and $\pi_Y$. The other case is where $\Phi$ restricts to G-isomorphisms on the fibers and X satisfies an auxiliary condition, which usually holds. Finally, we give applications to the Holomorphic Linearization Problem. Let G act holomorphically on X=C^n. When is there a change of coordinates such that the action of G becomes linear? We prove that this is true, for X satisfying the same auxiliary condition as before, if and only if the quotient Q_X is admissibly biholomorphic to the quotient of a G-module V.
Title: Galois groups and splitting fields of Mori trinomials.
Abstract. We discuss a certain class of irreducible polynomials over the rationals that was introduced by Shigefumi Mori forty years ago in his Master Thesis. We prove that the Galois group of a Mori polynomial coincides with the corresponding full symmetric groups and the splitting field is ``almost" unramified over its quadratic subfield.
Title: Piecewise-linear and non-archimedean geometries.
Abstract. This will be kind of a survey talk (including classical results, more recent ones, and a joint work with Amaury Thuillier which is still in progress ) about the deep links which exist between non-archimedean geometry over a valued field and piecewise linear geometry. I will mainly focus on the properties of some subsets of non-archimedean analytic spaces (in the sense of Vladimir Berkovich), called the skeleta, that inherit a canonical piecewise linear structure.
Title: The representation theory of invariant subalgebras constructed from g-algebras.
Title: Koszul duality for semidirect products.
Title: Double canonical bases.
Title:Possibly a solution of the two dimensional JC (Jacobian Conjecture).
Abstract. Several years ago I introduced Newton polytopes related to the potential counterexamples to the JC. This approach permitted to obtain some additional information which though interesting, was not sufficient to get a contradiction. It seems that a contradiction can be obtained by comparing Newton polytopes for the left and right side of a (somewhat mysterious) equality G_x=-y_F.