Showing papers in "Duke Mathematical Journal in 1997"

On quantum Galois theory

Abstract: The goals of the present paper are to initiate a program to systematically study and rigorously establish what a physicist might refer to as the “operator content of orbifold models.” To explain what this might mean, and to clarify the title of the paper, we will assume that the reader is familiar with the algebraic formulation of 2-dimensional CFT in the guise of vertex operator algebras (VOA), see [B], [FLM] and [DM] for more information on this point. In the paper [DVVV], several ideas are proposed concerning the structure of a holomorphic orbifold. In other words, if V is a holomorphic VOA and if G is a finite group of automorphisms of V, then the sub VOA V G of G-invariants is itself a VOA and the subject of [DVVV] is very much concerned with speculation on the nature of the V -modules. It turns out to be more useful − at least for purpose of inductive proofs − to take V to be a simple VOA. We will then see that V G is also simple whenever G is a finite group of automorphisms of V. One consequence of our main results is the following:

Geometric construction of crystal bases

Abstract: We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample to the conjecture of Kazhdan--Lusztig on the irreducibility of the characteristic variety of the intersection cohomology sheaves associated with the Schubert cells of type A and also to the similar problem asked by Lusztig on the characteristic variety of the perverse sheaves corresponding to canonical bases.

Weyl-Heisenberg Frames and Riesz Bases in L2(Rd).

Abstract: : We study Weyl-Heisenberg (= Gabor) expansions for either L2(Rd) or a subspace of it. These are expansions in terms of the spanning set, involving K and L are some discrete lattices in Rd, P, in L2(Rd), is finite, E is the translation operator, and M is a modulation operator. Such sets X are known as WH systems. The analysis of the 'basis' properties of WH systems (e.g. being a frame or a Riesz basis) is our central topic, with the fiberization-decomposition techniques of shift-invariant systems, developed in a previous paper of us, being the main tool. Of particular interest is the notion of the adjoint of a WH set, and the duality principle which characterizes a WH (tight) frame in term of the stability (orthonormality) of its adjoint. The actions of passing to the adjoint and passing to the dual system commute, hence the dual WH frame can be computed via the dual basis of the adjoint. Estimates for the underlying frame/basis bounds are obtained by two different methods. The Gramian analysis applies to all WH systems, albeit provides estimates that might be quite crude. This approach is invoked to show how, under only mild conditions on X, a frame can be obtained by oversampling a Bessel sequence. Finally, finer estimates of the frame bounds, based on the Zak transform, are obtained for a large collection of WH systems.

A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$

Abstract: For a function $\varphi$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $\varphi(nx)$, $n=1,2,3,\ldots$, constitutes a Riesz basis or a complete sequence in $L^2(0,1)$. The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space $\cal H$ of Dirichlet series $f(s)=\sum_n a_nn^{-s}$, where the coefficients $a_n$ are square summable. It proves useful to model $\cal H$ as the $H^2$ space of the infinite-dimensional polydisk, or, which is the same, the $H^2$ space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given $f$ in $\cal H$ and characters $\chi$, $f_\chi(s)=\sum_na_n\chi(n)n^{-s}$ is a vertical limit function of $f$. We study certain probabilistic properties of these vertical limit functions.

Harmonic measure on locally flat domains

Abstract: We will review work with Tatiana Toro yielding a characterizati on of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation.

Cycles of quadratic polynomials and rational points on a genus-$2$ curve

Abstract: It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N=4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X$_1$(16), whose rational points had been previously computed. We prove there are none with N=5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Galois-stable 5-cycles, and show that there exist Galois-stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N=6.

Group systems, groupoids, and moduli spaces of parabolic bundles

Abstract: Moduli spaces of homomorphisms or, more generally, twisted homomorphisms from fundamental groups of surfaces to compact connected Lie groups, were connected with geometry through their identification with moduli spaces of holomorphic vector bundles (see [29]). Atiyah and Bott [2] initiated a new approach to the study of these moduli spaces by identifying them with moduli spaces of projectively fiat constant central curvature connections on principal bundles over Riemann surfaces, which they analyzed by methods of gauge theory. In particular, they showed that an invariant inner product on the Lie algebra of the Lie group in question induces a natural symplectic structure on a certain smooth open stratum. Although this moduli space is a finite-dimensional object, generally a stratified space which is locally semialgebraic [19] but sometimes a manifold, its symplectic structure (on the stratum just mentioned) was obtained by applying the method of symplectic reduction to the action of an infinite-dimensional group (the group of gauge transformations) on an infinite-dimensional symplectic manifold (the space of all connections on a principal bundle).

On-diagonal lower bounds for heat kernels and Markov chains

Abstract: pt(x, y) = 1 (4πt)n/2 exp −|x− y| 4t  which shows that pt(x, y) behaves like t− n 2 for fixed x and y as t→ ∞. On other manifolds, its behaviour may be described by different functions of t, depending on the geometry of the manifold. The major question on complete non-compact manifolds is: What geometric terms are adequate to describe the long time behaviour of the heat kernel. Much has been known about upper bounds. The seminal works of Nash [N], Aronson [Ar], Varopoulos [V2], Carlen, Kusuoka, Stroock [CKS] and Davies [D1] have brought the understanding that the uniform upper bounds of the heat kernel are closely related to isoperimetric type inequalities including the Sobolev’s, Nash’s and the logarithmic Sobolev inequalities. More recent works [G2], [Carr], [C2] revealed the importance of a Faber-Krahn type inequality and of a generalized Nash inequality (see also the surveys [G4] and [C3]). The situation is quite different with lower bounds of the heat kernel. Until recently, only two methods were known: • a comparison type theorem ([DGM], [ChY]) which requires a pointwise restriction on the Ricci curvature;

Enumerative geometry for the real Grassmannian of lines in projective space

Abstract: We extend the classical Schubert calculus of enumerative geometry for the Grassmann variety of lines in projective space from the complex realm to the real. Specifically, given any collection of Schubert conditions on lines in projective space which generically determine a finite number of lines, we show there exist real generic conditions determining the expected number of real lines. Our main tool is an explicit description of rational equivalences which also constitutes a novel determination of the Chow rings of these Grassmann varieties.

The unitary dual of $p$-adic $G_2$

Abstract: In this paper is completly solved the unitarizability problem for the p-adic reductive group of type G_2: the unitary dual of p-adic G_2 is described in terms of supercuspidal representations of its Levi subgroups.

Twisted symmetric-square $L$-functions and the nonexistence of Siegel zeros on $GL(3)$

Abstract: §

Discrete series characters and the Lefschetz formula for Hecke operators

Abstract: This paper consists of three independent but related parts. In the rst part (xx1{ 6) we give a combinatorial formula for the constants appearing in the umerators" of characters of stable discrete series representations of real groups (see x3) as well as an analogous formula for individual discrete series representations (see x6). Moreover we give an explicit formula (Theorem 5.1) for certain stable virtual characters on real groups; by Theorem 5.2 these include the stable discrete series characters, and thus we recover the results of x3 in a more natural way. In the second part (x7) we use the character formula given in Theorem 5.1 to rewrite the Lefschetz formula of [GM] (for the local contribution at a single xed point component to the trace of a Hecke operator on weighted cohomology) in the same spirit as that of Arthur's Lefschetz formula [A]: in terms of stable virtual characters on real groups (see Theorem 7.14.B). We then sum the contributions of the various xed point components and show that, in the case of middle weighted cohomology, the resulting global Lefschetz xed point formula agrees with Arthur's Lefschetz formula. This gives a topological proof of Arthur's formula.