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David Kazhdan

Publications and source records attributed to David Kazhdan.

At least 91 records · Page 5Linked to original sources
arXiv2017

On the bias of cubic polynomials

Let $V$ be a vector space over a finite field $k=\mathbb{F} _q$ of dimension $n$. For a polynomial $P:V\to k$ we define the bias of $P$ to be $$b_1(P)=\frac {|\sum _{v\in V}ψ(P(V))|}{q^n}$$ where $ψ:k\to \mathbb{C} ^\star$ is a non-trivial additive character. A. Bhowmick and S. Lovett proved that for any $d\geq 1$ and $c>0$ there exists $r=r(d,c)$ such that any polynomial $P$ of degree $d$ with $b_1(P)\geq c$ can be written as a sum $P=\sum _{i=1}^rQ_iR_i$ where $Q_i,R_i:V\to k$ are non constant polynomials. We show the validity of a modified version of the converse statement for the case $d=3$.

math.NT↗
arXiv2016

Regularization of divergent integrals

We study the Hadamard finite part of divergent integrals of differential forms with singularities on submanifolds. We give formulae for the dependence of the finite part on the choice of regularization and express them in terms of a suitable local residue map. The cases where the submanifold is a complex hypersurface in a complex manifold and where it is a boundary component of a manifold with boundary, arising in string perturbation theory, are treated in more detail.

math-ph↗
arXiv2016

Quantization of Lie bialgebras, I

In the paper "On some unsolved problems in quantum group theory", V.Drinfeld formulated the problem of the existence of a universal quantization for Lie bialgebras. When the paper "Tensor structures arising from affine Lie algebras, III", by Kazhdan and Lusztig, appeared, Drinfeld asked whether its methods could be useful for the problem of universal quantization of Lie bialgebras. In this paper we use these methods to construct the universal quantization, which gives a positive answer to Drinfeld's question. We also show the existence of universal quantization of classical r-matrices, unitary r-matrices, and quasitriangular Lie bialgebras, which answers the corresponding questions of Drinfeld.

q-alg↗
arXiv2016

Divergent integrals, residues of Dolbeault forms, and asymptotic Riemann mappings

We describe the asymptotic behaviour and the dependence on the regularization of logarithmically divergent integrals of products of meromorphic and antimeromorphic forms on complex manifolds. Our formula is expressed in terms of residues of Dolbeault forms, a notion introduced in this paper. The proof is based on a result on the asymptotic behaviour of Riemann mappings of small domains.

math.CV↗
arXiv2016

On Shalika germs

Let $G$ be a reductive group over a local field $F$ satisfying the assumptions of \cite{Deb1}, $G_{reg}\subset G$ the subset of regular elements. Let $T\subset G$ be a maximal torus. We write $T_{reg}=T\cap G_{reg}$. Let $dg ,dt$ be Haar measures on $G$ and $T$. They define an invariant measure $dg/dt$ on $G/T$. Let $\mathcal{H}$ be the space of complex valued locally constant functions on $G$ with compact support. For any $f\in \mathcal{H} ,t\in T_{reg}$ we define $I_t(f)=\int_{G/T}f(\bar gt\bar g^{-1})dg/dt$. Let $P$ be the set of conjugacy classes of unipotent elements in $G$. For any $Ω\in P$ we fix an invariant measure $ω$ on $Ω$. As well known \cite {R} for any $f\in \mathcal{H}$ the integral $$I_Ω(f)=\int_Ωfω$$ is absolutely convergent. Shalika \cite{Sh} has shown that there exist functions $\tilde{j}_Ω(t),Ω\in P$ on $T\cap G_{reg}$ such that $$I_t(f) = \sum_{Ω\in P} \tilde{j}_Ω(t) I_Ω(f)\qquad\qquad (\star)$$ for any $f\in \mathcal{H} ,t\in T$ {\it near} to $e$ where the notion of {\it near} depends on $f$. For any positive real number $r$ one defines an open $Ad$-invariant subset $G_r$ of $G$ and a subspace $\mathcal{H}_r$ as in \cite{Deb1}. In this paper I show that for any $f\in \mathcal{H}_r$ the equality $(\star)$ is true for all $t\in T_{reg}\cap G_r$.

math.RT↗
arXiv2015

Geometry of second adjointness for p-adic groups

We present a geometric proof of Bernstein's second adjointness for a reductive $p$-adic group. Our approach is based on geometry of the wonderful compactification and related varieties. Considering asymptotic behavior of a function on the group in a neighborhood of a boundary stratum of the compactification, we get a "co-specialization" map between spaces of functions on various varieties with $G\times G$ action. These maps can be viewed as maps of bimodules for the Hecke algebra, and the corresponding natural transformations of functors lead to the second adjointness. We also get a formula for the "co-specialization" map expressing it as a composition of the orishperic transform and inverse intertwining operator; a parallel result for $D$-modules was obtained in arXiv:0902.1493. As a byproduct we obtain a formula for the Plancherel functional restricted to a certain commutative subalgebra in the Hecke algebra, generalizing a result by Opdam.

math.RT↗
arXiv2015

On a Theorem of N. Katz and Bases in Irreducible Representations

N. Katz has shown that any irreducible representation of the Galois group of F_q((t)) has unique extension to a special representation of the Galois group of k(t) unramified outside 0 and infinity and tamely ramified at infinity. In this paper we analyze the number of not necessarily special such extensions and relate this question to a description of bases in irreducible representations of multiplicative groups of division algebras.

math.NT↗
arXiv2014

Integrable clusters

The goal of this note is to study quantum clusters in which cluster variables (not coefficients) commute which each other. It turns out that this property is preserved by mutations. Remarkably, this is equivalent to the celebrated sign coherence conjecture recently proved by M. Gross, P. Hacking, S. Keel and M. Kontsevich

math.QA↗
arXiv2014

Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders

Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every $d$ there are unbounded degree simplicial complexes of dimension $d$ with these properties. However, a major open problem, formulated by Gromov, is whether bounded degree high dimensional expanders exist for $d \geq 2$. We present an explicit construction of bounded degree complexes of dimension $d=2$ which are topological expanders, thus answering Gromov's question in the affirmative. Conditional on a conjecture of Serre on the congruence subgroup property, infinite sub-family of these give also a family of bounded degree coboundary expanders. The main technical tools are new isoperimetric inequalities for Ramanujan Complexes. We prove linear size bounds on $F_2$ systolic invariants of these complexes, which seem to be the first linear $F_2$ systolic bounds. The expansion results are deduced from these isoperimetric inequalities.

math.CO↗
arXiv2014

Ramanujan Complexes and bounded degree topological expanders

Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expanders. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov, is whether bounded degree high dimensional expanders, according to these definitions, exist for d >= 2. We present an explicit construction of bounded degree complexes of dimension d = 2 which are high dimensional expanders. More precisely, our main result says that the 2-skeletons of the 3-dimensional Ramanujan complexes are topological expanders. Assuming a conjecture of Serre on the congruence subgroup property, infinitely many of them are also coboundary expanders.

math.CO↗
arXiv2014

Iwahori-Hecke algebras for p-adic loop groups

This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dimensional case.

math.RT↗
arXiv2014

Cartan decomposition for complex loop groups

In this note we formulate and prove a version of Cartan decomposition for holomorphic loop groups, similar to Cartan decomposition for $p$-adic loop groups, discussed proved by Garland (and later by the authors by geometric mathods). The main technical tool that we use is the (well-known) interpretation of twisted conjugacy classes in the holomorphic loop group in terms of principal holomorphic bundles on an elliptic curve.

math.RT↗
arXiv2013

The value ring of geometric motivic integration and the Iwahori Hecke algebra of SL_2

In \cite{HK}, an integration theory for valued fields was developed with a Grothendieck group approach. It was shown that the semiring of semi-algebraic sets with measure preserving morphisms is isomorphic to a certain semiring formed out of twisted varieties over the residue field and rational polytopes over the value group. With a view to representation-theoretic applications, we require a simpler description of the possible values of the integration, and in particular natural homomorphisms into fields. In the present paper we obtain such results after tensoring with Q. Since this operation trivializes the full semiring, we restrict to bounded sets. We show that the resulting Q-algebra is generated by its one-dimensional part. In the "geometric" case, i.e. working over an elementary submodel as a base, we determine the structure precisely. As a corollary we obtain useful canonical homomorphisms in the general case. In the appendix we define the Iwahori Hecke algebra of $SL_2$ over an algebraically closed valued field using motivic integration (to replace the Haar measure) and study its structure.

math.AG↗
arXiv2013

The classical master equation

We formalize the construction by Batalin and Vilkovisky of a solution of the classical master equation associated with a regular function on a nonsingular affine variety (the classical action). We introduce the notion of stable equivalence of solutions and prove that a solution exists and is unique up to stable equivalence. A consequence is that the associated BRST cohomology, with its structure of Poisson_0-algebra, is independent of choices and is uniquely determined up to unique isomorphism by the classical action. We give a geometric interpretation of the BRST cohomology sheaf in degree 0 and 1 as the cohomology of a Lie-Rinehart algebra associated with the critical locus of the classical action. Finally we consider the case of a quasi-projective varieties and show that the BRST sheaves defined on an open affine cover can be glued to a sheaf of differential Poisson_0-algebras.

math.AG↗
arXiv2013

When is the Fourier transform of an elementary function elementary?

Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on V. In this paper we study when the Fourier transform of an elementary function is elementary. If Q has a nonzero Hessian, a necessary condition for this is that the Legendre transform Q_* of Q is rational. The basic example is a nondegenerate quadratic form. We study such functions Q, give examples, and find all of them such that both Q and Q_* are of the form f(x)/t, where f is a cubic form in many variables (the simplest case after quadratic forms). It turns out that this classification is closely related to Zak's classification of Severi varieties. The second half of the paper is devoted to finding and classifying elementary functions with elementary Fourier transforms when Q is a fixed function with rational Q_*. We consider the simplest case when Q is a monomial, and classify combinations of multiplicative characters that can arise. The answer (for real and complex fields) is given in terms of exact covering systems. We also describe examples related to prehomogeneous vector spaces. Finally, we consider examples over p-adic fields, and in particular give a local proof of an integral formula of D.K. that could previously be proved only by a global method.

math.AG↗
arXiv2013

Kirillov models and the Breuil-Schneider conjecture for GL_2(F)

Let F be a local field of characteristic 0. The Breuil-Schneider conjecture for GL_2(F) predicts which locally algebraic representations of this group admit an integral structure. We extend the methods of [K-dS12], which treated smooth representations only, to prove the conjecture for some locally algebraic representations as well.

math.RT↗
arXiv2012

An affine Gindikin-Karpelevich formula

In this paper we give an elementary proof of certain finiteness results about affine Kac-Moody groups over a local non-archimedian field K. Our results imply those proven earlier by Braverman-Kazhdan, Braverman-Finkelberg-Kazhdan and Gaussent-Rousseau using either algebraic geometry or a Kac-Moody version of the Bruhat-Tits building. The above finiteness results allow one to formulate an affine version of the Gindikin-Karpelevich formula, which coincides with the one discussed by Braverman-Finkelberg-Kazhdan in the case when K has positive characteristic. We deduce this formula from an affine version of the Macdonald formula for the spherical function, which will be proved in a subsequent publication.

math.RT↗