On the nonvanishing of someL-functions

The non-vanishing, at the centre of symmetry, of theL-function attached to an automorphic representation of GL(2) or its twists by quadratic characters has been extensively investigated, in particular by Waldspurger. The purpose of this paper is to outline a new proof of Waldspurger’s results. The automorphic representations of GL(2) and its metaplectic cover are compared in two different ways; one way is by means of a “relative trace formula”; the relative trace formula presented here is actually a generalization of the work of Iwaniec. This is the text of a lecture delivered in June 1987 in Paris at the Symposium in honour of R Godement. This work was supported in part by NSF grant DMS-85-02789.     

Global Jacquet–Langlands correspondence,multiplicity one and classification of automorphic representations

In this paper we generalize the local Jacquet-Langlands correspondence to all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one Theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Mœglin–Waldspurger and Jacquet–Shalika for GL(n).     

Classification of manifolds with weakly 1/4-pinched curvatures

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R ² has non-negative isotropic curvature. The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS-0604960.     

The unitary representation theory of GL(n) of an infinite discrete field

IfK is an infinite field and ifG=GL(n, K) with the discrete topology, then all principal-series representations ofG are irreducible, and any two such with the same central character ψ are weakly equivalent to one another and to the ψ-regular representation. In addition, every irreducible unitary representation ofG which is not one-dimensional weakly contains a representation of the principal series. We deduce that every maximal ideal ofC*(G) is either of codimension 1 or else a kernel of a principal-series representation. In particular, except in the exceptional case whereK is an infinite algebraic extension of a finite field, the reducedC*-algebra of PGL(n, K) is simple, as was also shown by de la Harpe in many cases. Partially supported by NSF Grant DMS-85-06130. It is a pleasure also to acknowledge the hospitality of the Institute for Advanced Studies, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel, from January to August, 1988. Partially supported by NSF Grants DMS-84-00900 and DMS-87-00551. Much of this work was done while visiting at, and partially supported by, the Department of Mathematics and Computer Science, Bar-Ilan University, 52100 Ramat Gan, Israel.     

Central Value Of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindel?f hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting. The first author was partially supported by NSF grant DMS-0070762. The second author was partially supported by NSF grant DMS-0355285. Received: July 2005 Accepted: August 2005     

The orbit method for profinite groups and a p-adic analogue of Brown’s theorem

We develop an approach to the character theory of certain classes of finite and profinite groups based on the construction of a Lie algebra associated to such a group, but without making use of the notion of a polarization which is central to the classical orbit method. Instead, Kirillov’s character formula becomes the fundamental object of study. Our results are then used to produce an alternate proof of the orbit method classification of complex irreducible representations of p-groups of nilpotence class < p, where p is a prime, and of continuous complex irreducible representations of uniformly powerful pro-p-groups (with a certain modification for p = 2). As a main application, we give a quick and transparent proof of the p-adic analogue of Brown’s theorem, stating that for a nilpotent Lie group over ℚ_p the Fell topology on the set of isomorphism classes of its irreducible representations coincides with the quotient topology on the set of its coadjoint orbits. The research of M. B. was partially supported by NSF grant DMS-0401164.     

Analyse Harmonique sur les groupes de Heisenberg généralisés

LetG=(X ₁,X ₂,X ₃) _B be the generalized Heisenberg group as defined in Commen. Math. Helv. 1974 byH. Reiter. Under some natural conditions onG involving not the separability, we classify the unitary irreducible representations ofG, and prove a Fourier inversion formula.     

Interpolation in the unit ball ofC n

A necessary and sufficient condition is given for a discrete multiplicity variety in the unit ballB _n ofC ⁿ to be an interpolating variety for weighted spaces of holomorphic functions inB _n . Partially supported by NSF Grant DMS-9706376.     

Geometric cycles with local coefficients and the cohomology of arithmetic subgroups of the exceptional group <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

We study the cohomology of a compact locally symmetric space attached to an arithmetic subgroup of a rational form of a group of type G ₂ with values in a finite dimensional irreducible representation E of G ₂. By constructing suitable geometric cycles and parallel sections of the bundle [(E)\tilde]{\tilde{E}} we prove non-vanishing results for this cohomology. We prove all possible non-vanishing results compatible with the known vanishing theorems regarding unitary representations with non-zero cohomology in the case of the short fundamental weight of G ₂. A decisive tool in our approach is a formula for the intersection numbers with local coefficients of two geometric cycles.     

Irreducibility of some Faber polynomials

Apply weight 0 Hecke operators to the modular function j and express the result as a polynomial in j. These polynomials were considered long ago in analysis, and recently attracted the attention of number theorists primarily for their connection with Borcherds’ infinite products. In particular, Ken Ono conjectured that all of them are irreducible. We prove a partial result towards this conjecture by presenting infinite families of these polynomials which are proved to be irreducible. Supported by NSF grant DMS-0501225.     

A local version of the two-circles theorem

A necessary and sufficient condition is given so that in a domain Ω there are no functions whose average over all balls contained in Ω of radiir ₁,r ₂ vanish except the zero function. Partially supported by NSF grant DMS-8401356 and by NSF grant OJR 85-OV-108 through the Systems Research Center of the University of Maryland.     

Skew loops and quadric surfaces

A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R ³ with a point of positive curvature and no skew loops are the quadrics. In particular: Ellipsoids are the only closed surfaces without skew loops. Our efforts also yield results about skew loops on cylinders and positively curved surfaces. Received: January 7, 2002 RID="*" ID="*"The first author was partially supported by the NSF grant DMS-0204190.     

Unitary mappings between multiresolution analyses of L2 (R) and a parametrization of low-pass filters

This article provides classes of unitary operators of L²(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L²(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We use these unitary operators to provide an interesting class of scaling functions. We show that the Dai-Larson unitary parametrization of orthonormal wavelets is not suitable for the study of scaling functions. These operators give an interesting relation between low-pass filters corresponding to scaling functions, which is implemented by a special class of unitary operators acting on L²([−π, π)), which we characterize. Using this characterization we recapture Daubechies' orthonormal wavelets bypassing the spectral factorization process. Acknowledgements and Notes. Partially supported by NSF Grant DMS-9157512, and Linear Analysis and Probability Workshop, Texas A&M University Dedicated to the memory of Professor Emeritus Vassilis Metaxas.     

Some remarks on lengths of propositional proofs

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(m ^d+1) symbols. We show that renaming Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. Supported in part by NSF grant DMS-9205181     

Contractions ofSU(1, n) andSU(n+1) via Berezin quantization

We establish contractions of discrete series representations ofSU(1,n) and of unitary irreducible representations ofSU(n+1) to the unitary irreducible representations of the (2n+1)-dimensional Heisenberg group by use of the Berezin calculus on the coadjoint orbits associated to these representations by the Kirillov-Kostant method of orbits.     

Universal non-completely-continuous operators

A bounded linear operator between Banach spaces is calledcompletely continuous if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators fromL ₁ into an arbitrary Banach space, namely, the operator fromL ₁ into ⊆_∞ defined byT ₀(f) = (∫r _n f d μ) _n>-0, wherer _n is thenth Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach spaces. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space. Supported in part by NSF grant DMS-9306460. Participant, NSF Workshop in Linear Analysis & Probability, Texas A&M University (supported in part by NSF grant DMS-9311902). Supported in part by NSF grant DMS-9003550.     

Bessel functions for GL 2

In classical analytic number theory there are several trace formulas or summation formulas for modular forms that involve integral transformations of test functions against classical Bessel functions. Two prominent such are the Kuznetsov trace formula and the Voronoi summation formula. With the paradigm shift from classical automorphic forms to automorphic representations, one is led to ask whether the Bessel functions that arise in the classical summation formulas have a representation theoretic interpretation. We introduce Bessel functions for representations of GL ₂ over a finite field first to develop their formal properties and introduce the idea that the γ-factor that appears in local functional equations for L-functions should be the Mellin transform of a Bessel function. We then proceed to Bessel functions for representations of GL ₂(?) and explain their occurrence in the Voronoi summation formula from this point of view. We briefly discuss Bessel functions for GL ₂ over a p-adic field and the relation between γ-factors and Bessel functions in that context. We conclude with a brief discussion of Bessel functions for other groups and their application to the question of stability of γ-factors under highly ramified twists.     

Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms

The theory of zeta functions associated with prehomogeneous vector spaces (p.v. for short) provides us a unified approach to functional equations of a large class of zeta functions. However the general theory does not include zeta functions related to automorphic forms such as the HeckeL-functions and the standardL-functions of automorphic forms on GL(n), even though they can naturally be considered to be associated with p.v.’s. Our aim is to generalize the theory to zeta functions whose coefficients involve periods of automorphic forms, which include the zeta functions mentioned above. In this paper, we generalize the theory to p.v.’s with symmetric structure ofK _ε-type and prove the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character. In another paper, we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary representations of compact groups. Dedicated to the memory of Professor K G Ramanathan     

A spectral identity between symmetric spaces

Let π be a cuspidal automorphic representation ofGL _2n . We prove an identity between two spectral distributions onSp _2n andGL _2n respectively. The first is the spherical distribution with respect toSp _n×Sp _nof the residual Eisenstein series induced from π. The second is the weighted spherical distribution of π with respect toGL _n×GL _nand a certain degenerate Eisenstein series. A similar identity relates the pair (U _2n ,Sp _n) and (GL _n/E,GL _n/F) whereE/F is the quadratic extension defining the quasi-split unitary groupU _2n . We also have a Whittaker version of these trace identities. First-named author partially supported by NSF grant DMS 0070611. Second-named author partially supported by NSF grant DMS 9970342.     

A series of unitary irreducible representations induced from a symmetric subgroup of a semisimple Lie group

LetG/H be a semisimple symmetric space. Generalizing results of Flensted-Jensen we give a sufficient condition for the existence of irreducible closed invariant subspaces of the unitary representations ofG induced from unitary finite dimensional representations ofH. This provides a method of constructing unitary irreducible representations ofG, and we show by examples that for some irreducible admissible representations ofG, this method exhibits not previously known unitarity.This work was supported by the Danish Natural Science Research Council.