Berezin-Toeplitz quantization and the least unsharpness principle
We show that compatible almost-complex structures on symplectic manifolds correspond to optimal quantizations.
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arXiv subjects
David Kazhdan
Publications and source records attributed to David Kazhdan.
Hecke-Hopf algebras
Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{\bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${\bf H}(W)$ have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of $H_{\bf q}(W)$-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the "Demazure" part of ${\bf H}(W)$ is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra.
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A construction of projective bases for irreducible representations of multiplicative groups of division algebras over local fields
Let $F$ be a local non-archimedian field of positive characteristic, $D$ be a skew-field with center $F$ and $ G=D^{\star}$ be the multiplicative group of $D$. The goal of this paper is to provide a canonical decomposition of any complex irreducible representation $V$ of $G$ in a direct sum of one-dimensional subspaces.
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Extending weakly polynomial functions from high rank varieties
Let $k$ be a field, $V$ a $k$-vector space and $X$ be a subset of $V $. A function $f:X\to k$ is weakly polynomial of degree $\leq a$, if the restriction of $f$ on any affine subspace $L\subset X$ is a polynomial of degree $\leq a$. In this paper we consider the case when $X= \mathbb X (k)$ where $\mathbb X$ is a complete intersection of bounded codimension defined by a high rank polynomials of degrees $d, char(k)=0$ or $char (k)>d$ and either $k$ is algebraically closed, or $k=\mathbb F _q,q>ad$. We show that under these assumptions any $k$-valued weakly polynomial function of degree $ \leq a$ on $X$ is a restriction of a polynomial of degree $\leq a$ on $V$. Our proof is based on Theorem 1.11 on fibers of polynomial morphisms $P:\mathbb F _q^n\to \mathbb F _q^m$ of high rank. This result is of an independent interest. For example it immediately implies a strengthening of the result of [4].
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Polynomial functions as splines
Let $V$ be a vector space over a finite field $k$. We give a condition on a subset $A \subset V$ that allows for a local criterion for checking when a function $f:A \to k$ is a restriction of a polynomial function of degree $<m$ on $V$. In particular, we show that high rank hypersurfaces of $V$ of degree $\ge m$ satisfy this condition. In addition we show that the criterion is robust (namely locally testable in the theoretical computer science jargon).
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Remarks on the asymptotic Hecke algebra
Let $G$ be a split reductive $p$-adic group. Let ${\mathcal H}(G)$ be its Hecke algebra and let ${\mathcal C}(G)\supset {\mathcal H}(G)$ be the Harish-Chandra Schwartz algebra. The purpose of this note is to give a spectral interpretation of Lusztig's asymptotic Hecke algebra $J$ (which contains the Iwahori part of ${\mathcal H}(G)$ as a subalgebra), which shows that $J$ is a subalgebra of ${\mathcal C} (G)$. This spectral description also allows to define a version of $J$ beyond the Iwahori component - i.e. we define certain subalgebra ${\mathcal J}(G)$ of ${\mathcal C}(G)$ which contains ${\mathcal H}(G)$. We explain a relation between ${\mathcal J}(G)$ and the Schwartz space of the basic affine space studied by us about 20 years ago.
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Schwartz space of parabolic basic affine space and asymptotic Hecke algebras
Let $F$ be a local non-archimedian field and $G$ be the group of $F$-points of a split connected reductive group over $F$. In a previous aricle we defined an algebra $\mathcal J(G)$ of functions on $G$ which contains the Hecke algebra $\mathcal H(G)$ and is contained in the Harish-Chandra Schwartz algebra $\mathcal C(G)$. We consider $\mathcal J(G)$ as an algebraic analog the algebra $\mathcal C(G)$. Given a parabolic subgroup $P$ of $G$ with a Levi subgroup $M$ and the unipotent radical $U_P$ we write $X_P:=G/U_P$. In this paper we study two versions of the Schwartz space of $X_P$. The first is $\mathcal S(X_P):=\mathcal J({\mathcal S} _c(X_P))$ and the 2nd is the space spanned by functions of the form $Φ_{Q,P}(ϕ)$ where $Q$ is another parabolic with the same Levi subgroup, $ϕ\in \mathcal S_c(X_Q)$ and $Φ_{Q,P}$ is a normalized intertwining operator from $L^2(X_Q)$ to $L^2(X_P)$. We formulate a series of conjectures about these spaces, for example, we conjecture that $\mathcal S'(X_P)\subset \mathcal S(X_P)$ and that this embedding is an isomorphism on $M$-cuspidal part. We give a proof of some of our conjectures.
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Bernstein components via Bernstein center
Let G be a reductive p-adic group. Let $Φ$ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that $Φ$ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modiification of the arguments of J.F.Dat who proved our result in one direction.
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Deligne-Lusztig duality and wonderful compactification
We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for $p$-adic groups and the homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for $G=GL(n)$ by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct we obtain a description of the Serre functor for representations of a p-adic group.
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A categorical approach to the stable center conjecture
The stable center conjecture asserts that the space of stable distributions in the Bernstein center of a reductive p-adic is closed under convolution. It is closely related to the notion of an L-packet and endoscopy theory. We describe a categorical approach to the depth zero part of the conjecture. As an illustration of our method, we show that the Bernstein projector to the depth zero spectrum is stable.
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On the depth r Bernstein projector
In this paper we prove an explicit formula for the Bernstein projector to representations of depth at most r. As a consequence, we show that the depth zero Bernstein projector is supported on topologically unipotent elements and it is equal to the restriction of the character of the Steinberg representation. As another application, we deduce that the depth r Bernstein projector is stable. Moreover, for integral depths our proof is purely local.
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Geometric approach to parabolic induction
In this note we construct a "restriction" map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on a parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig-Spaltenstein on induced unipotent classes to all infinite fields.
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Character values and Hochschild homology
We present a conjecture (and a proof for G=SL(2)) generalizing a result of J. Arthur which expresses a character value of a cuspidal representation of a $p$-adic group as a weighted orbital integral of its matrix coefficient. It also generalizes a conjecture by the second author proved by Schneider-Stuhler and (independently) the first author. The latter statement expresses an elliptic character value as an orbital integral of a pseudo-matrix coefficient defined via the Chern character map taking value in zeroth Hochschild homology of the Hecke algebra. The present conjecture generalizes the construction of pseudo-matrix coefficient using compactly supported Hochschild homology, as well as a modification of the category of smooth representations, the so called compactified category of smooth $G$-modules. This newly defined "compactified pseudo-matrix coefficient" lies in a certain space on which the weighted orbital integral is a conjugation invariant linear functional, our conjecture states that evaluating a weighted orbital integral on the compactified pseudo-matrix coefficient one recovers the corresponding character value of the representation. We also discuss general properties of that space, building on works of Waldspurger and Beuzart-Plessis.
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On Bias and Rank
Given a hypersurface $X\subset \mathbb{P}^{N+1}_{\mathbb{C}}$ Dimca gave a proof showing that the cohomologies of X are the same as the projective space in a range determined by the dimension of the singular locus of X. We prove the analog of Dimca's result case when $\mathbb{C}$ is replaced with an algebraically closed field of finite characteristic and singular cohomology is replaced with $\ell$-adic étale cohomology. The Weil conjectures allow relating results about éatle cohomology to counting problems over a finite field. Thus by applying this result, we are able to get a relationship between the algebraic properties of certain polynomials and the size of their zero set.
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On ranks of polynomials
Let $V$ be a vector space over a field $k, P:V\to k, d\geq 3$. We show the existence of a function $C(r,d)$ such that $rank (P)\leq C(r,d)$ for any field $k,char (k)>d$, a finite-dimensional $k$-vector space $V$ and a polynomial $P:V\to k$ of degree $d$ such that $rank(\partial P/\partial t)\leq r$ for all $t\in V-0$. Our proof of this theorem is based on the application of results on Gowers norms for finite fields $k$. We don't know a direct proof in the case when $k=\mathbb C$.
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Extending linear and quadratic functions from high rank varieties
Let $k$ be a field, $V$ be a $k$-vector space and $X\subset V$ an algebraic irreducible subvariety. We say that a function $f:X(k) \to k$ is weakly linear if its restriction to any two-dimensional linear subspace $W$ of $V$ contained in $X$ is linear and that it is weakly quadratic if its restriction to any three-dimensional linear subspace $W$ of $V$ contained in $X$ is quadratic. We say that $X$ is admissible if any weakly linear function on $X$ is a restriction of a linear function on $V$ and any weakly quadratic function on $X$ is a restriction of a quadratic function on $V$. The main result in the paper concerns the case when the field $k$ is a finite. We show that for any $d,L\geq 1$ there exists $r=r(d,L,k)\in \mathbb Z _+$ such that any complete intersection $X\in V$ in a vector space $V$ of codimension $L$, degree $d$ and rank $\geq r$ is admissible. Moreover we show the existence of a function $r(d,L)$ such that one can take $r(d,L,k)=r(d,L)$ for all finite fields $k$ of characteristic $>d$. The proof of the admissibility for finite fields $k$ is based on bounds on the number of $k$-points on ancillary varieties $E(X)$. These results allow us to bound the dimension of varieties $E(X)$. Using these results we were able to prove the admissibility of complex homogeneous varieties of high rank. Using the results of \cite{br} one can extend our proofs to show the admissibility of varieties of high rank over local non-archimedian fields. Also using Corollary $4.3$ of \cite{cmpv} one can dispense with the assumption that $X$ is a complete intersection.
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Appoximate Cohomology
Let $k$ be a field, $G$ be an abelian group and $r\in \mathbb N$. Let $L$ be an infinite dimensional $k$-vector space. For any $m\in End_k(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty]$ the minimal $R$ such that for any map $A:G \to End_k(L)$ with $r(A(g'+g'')-A(g')-A(g''))\leq r$, $g',g''\in G$ there exists a homomorphism $χ:G\to End_k(L)$ such that $r(A(g)-χ(g))\leq R(G, r, k)$ for all $g\in G$. We show the finiteness of $R(G,r,k)$ for the case when $k$ is a finite field, $G=V$ is a $k$-vector space $V$ of countable dimension. We actually prove a generalization of this result. In addition we introduce a notion of {\it Approximate Cohomology} groups $H^k_{\mathcal F} (V,M)$ (which is a purely algebraic analogue of the notion of $ε$-representation (\cite{ep})) and interperate our result as a computation of the group $H^1_{\mathcal F} (V,M)$ for some $V$-modules $M$.
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A spectral decomposition of orbital integrals for $PGL(2,F)$
Let $F$ be a local non-archimedian field, $G$ a semisimple $F$-group, $dg$ a Haar measure on $G$ and $\mathcal S(G)$ be the space of locally constant complex valued functions $f$ on $G$ with compact support. For any regular elliptic congugacy class $Ω=h^G\subset G$ we denote by $I_Ω$ the $G$-invariant functional on $\mathcal S (G)$ given by $$I_Ω(f)=\int_G f(g^{-1}hg)dg$$ This paper provides the spectral decomposition of functionals $I_Ω$ in the case $G=PGL(2,F)$ and in the last section first steps of such an analysis for the general case.
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