Translation representations for automorphic solutions of the wave equation in non-euclidean spaces. III

In Part III we show that the translation representations defined in Part I, and shown to be complete in Part II, are incoming and outgoing in the sense of propagation of signals along rays. We give a new proof of the absolute continuity of the spectrum below -¼n², and point out its implications for local energy decay. We give a new proof of completeness in dimensions 2 and 3 when energy is positive. Finally, we define and prove completeness of the translation representations when the metric is perturbed on a compact set.     

Translation representations for automorphic solutions of the wave equation in non-euclidean spaces. I

This paper deals with the spectral theory of the Laplace-Beltrami operator Δ acting on automorphic functions in n-dimensional hyperbolic space Hⁿ. We study discrete subgroups Γ which have a fundamental polyhedron F with a finite number of sides and infinite volume. Concerning these we have shown previously that the spectrum of Δ contains at most a finite number of point eigenvalues in [-(1/2(n - 1))², 0], and none less than (1/2(n -1))². Here we prove that the spectrum of Δ is absolutely continuous and of infinite multiplicity in (-∞, -(1/2(n - 1))²). Our approach uses the non-Euclidean wave equation introduced by Faddeev and Pavlov, Energy E_F is defined as (u_t, u_t)-(u, Lu), where the bracket is the L₂ scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric. Energy is conserved under the group of operator U(t) relating initial data to data at time t. We construct two isometric representations of the space of automorphic data by L₂(R, N) which transmute the action of U(t) into translation. These representations are given explicitly in terms of integrals of the data over horospheres. In Part II we shall show the completeness of these representations. u_tt-Lu = 0, L = Δ + (1/2(n - 1))².     

Nilpotent orbits over ground fields of good characteristic

Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in (F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge. Under some further (fairly mild) assumptions on G, we prove convergence for arbitrary orbital integrals on the Lie algebra and on the group. The convergence of orbital integrals in the case where F has characteristic 0 was obtained by Deligne and Ranga Rao (1972).     

Three‐query PCPs with perfect completeness over non‐Boolean domains

We study non‐Boolean PCPs that have perfect completeness and query three positions in the proof. For the case when the proof consists of values from a domain of size d for some integer constant d ≥ 2, we construct a nonadaptive PCP with perfect completeness and soundness d^?1 + d^?2 + ?, for any constant ? > 0, and an adaptive PCP with perfect completeness and soundness d^?1 + ?, for any constant ? > 0. The latter PCP can be converted into a nonadaptive PCP with perfect completeness and soundness d^?1 + ?, for any constant ? > 0, where four positions are read from the proof. These results match the best known constructions for the case d = 2 and our proofs also show that the particular predicates we use in our PCPs are nonapproximable beyond the random assignment threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

The correlations of finite Desarguesian planes, Part III: The classification (II)

This paper continues the classification of the correlations of planes of odd nonsquare order. Part I (Generalities) included introductory definitions and results (Section 1), algebraic preliminaries (Section 2), as well as a discussion of equivalent correlations (Section 3) and of their general properties (Section 4). The classification proper revolves around a special polynomial which can have one, two, or q + 1 zeros, or no zeros at all, and each of these four possibilities leads to different families of correlations. Part II contained Section 5, devoted to the cases in which the correlation is defined by a diagonal matrix (Subsection 5.1) or the polynomial in the preceding paragraph possesses q + 1 zeros (Subsection 5.2), one zero (Subsection 5.3) and two zeros (Subsection 5.4). Subsection 5.5 presented certain results to be used in the subsequent sections. The present article contains Section 6, devoted to the case in which the above-mentioned polynomial has no zeros.     

Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

Let F be a finite extension of ℚ _p . For each integer n≥1, we construct a bijection from the set ?_F ⁰ (n) of isomorphism classes of irreducible degree n representations of the (absolute) Weil group of F, onto the set ? _F ⁰ (n) of isomorphism classes of smooth irreducible supercuspidal representations of GL _n (F). Those bijections preserve epsilon factors for pairs and hence we obtain a proof of the Langlands conjectures for GL _n over F, which is more direct than Harris and Taylor’s. Our approach is global, and analogous to the derivation of local class field theory from global class field theory. We start with a result of Kottwitz and Clozel on the good reduction of some Shimura varieties and we use a trick of Harris, who constructs non-Galois automorphic induction in certain cases. Oblatum 1-III-1999 & 21-VII-1999 / Published online: 29 November 1999     

On cuspidal representations of general linear groups over discrete valuation rings

We define a new notion of cuspidality for representations of GL _n over a finite quotient o _k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G _λ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GL _n (F). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GL _n (o _k ) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G _λ . A functional equation for zeta functions for representations of GL _n (o _k ) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL₄(o₂) are constructed. Not all these representations are strongly cuspidal.     

The m-ordered real free pro-2-group. Cohomological characterizations ∗

In this paper we investigate representations of simple algebraic groups over an algebraically closed field of characteristic 2 and of their Lie algebras. For the groups of rank 4 or less, we shall determine all of the extensions of simple modules. The central theme will be the study of some intimate connections among the groups of types B_l, C _l and D_l (and F ₄ when l = 4). We also give calculations for those other groups of rank 4 or less which have not already been treated elsewhere ([1], [18]), but this is primarily for the sake of completeness.     

Blow up for a class of quasilinear wave equations in one space dimension

For suitable σ and F, we prove that all classical solutions of the quasilinear wave equation , with initial data of compact support, develop singularities in finite time. The assumptions on σ and F include in particular the model case , for q ⩾ 2,and ϵ = ±1. The starting point of the proof is to write the equation under the form of a first order system of two equations, in which F(ϕ) appears as a nonlocal term. Then, we present a new idea to control the effect of this perturbation term, and we conclude the proof by using well‐known methods developed for 2 × 2 systems of conservation laws. Copyright © 2000 John Wiley & Sons, Ltd.     

Tilting on non-commutative rational projective curves

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy ² = x ³ + x ² z.     

The simplicity of the branching of representations of the groups GL(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>) under the parabolic restrictions

We present a direct proof of the simplicity of the branching of representations of the groups GL(n, q) under the parabolic restrictions. The proof consists of three steps. First, we reduce the problem to the statement that a certain pair of finite groups is a Gelfand pair. Then, we obtain a criterion for establishing this fact, which generalizes the classical Gelfand’s criterion. Finally, we check the obtained criterion with the help of some matrix computations. Bibliography: 7 titles.     

Tensor products of singular representations and an extension of the $ \theta $-correspondence

In this paper we consider the problem of decomposing tensor products of certain singular unitary representations of a semisimple Lie group G. Using explicit models for these representations (constructed earlier by one of us) we show that the decomposition is controlled by a reductive homogeneous space . Our procedure establishes a correspondence between certain unitary representations of G and those of . This extends the usual -correspondence for dual reductive pairs. As a special case we obtain a correspondence between certain representations of real forms of E ₇ and F ₄.     

The small dispersion limit of the korteweg-de vries equation. ii

In Part I* we have shown, see Theorem 2.10, that as the coefficient of u^xxx tends to zero, the solution of the initial value problem for the KdV equation tends to a limit u in the distribution sense. We have expressed u by formula (3.59), where ψ^x is the partial derivative with respect to x of the function ψ* defined in Theorem 3.9 as the solution of the variational problem formulated in (2.16), (2.17). ψ* is uniquely characterized by the variational condition (3.34); its partial derivatives satisfy (3.51) and (3.52), where I is the set I^o defined in (3.36). In Section 4 we show that for t<t^b, I consists of a single interval, and the u satisfies u ^t — 6uu ^x = 0; here t^b is the largest time interval in which (12) has a continuous solution. In Section 5 we show that when I consists of a finite number of intervals, u can be described by Whitham's averaged equation or by the multiphased averaged equations of Flaschka, Forest, and McLaughlin. Equation numbers refer to Part I.     

Polynomial Approximation of Conformal Maps

Let G be a finite domain, bounded by a Jordan curve Γ , and let f ₀ be a conformal map of G onto the unit disk. We are interested in the best rate of uniform convergence of polynomial approximation to f ₀ , in the case that Γ is piecewise-analytic without cusps. In particular, we consider the problem of approximating f ₀ by the Bieberbach polynomials π _n and derive results better than those in [5] and [6] for the case that the corners of Γ have interior angles of the form π/N . In the proof, the Lehman formulas for the asymptotic expansion of mapping functions near analytic corners are used. We study the question when these expansions contain logarithmic terms. December 6, 1995. Date revised: August 5, 1996.     

Lines imply spaces in density Ramsey theory

Some results of geometric Ramsey theory assert that if F is a finite field (respectively, set) and n is sufficiently large, then in any coloring of the points of Fⁿ there is a monochromatic k-dimensional affine (respectively, combinatorial) subspace (see [9]). We prove that the density version of this result for lines (i.e., k = 1) implies the density version for arbitrary k. By using results in [3, 6] we obtain various consequences: a “group-theoretic” version of Roth's Theorem, a proof of the density assertion for arbitrary k in the finite field case when ∥F∥ = 3, and a proof of the density assertion for arbitrary k in the combinatorial case when ∥F∥ = 2.     

Writing representations over minimal fields

The chief aim of this paper is to describe a procedure which, given a d-dimensional absolutely irreducible matrix representation of a finite group over a finite field E, produces an equivalent representation such that all matrix entries lie in a subfield F of E which is as small as possible. The algorithm relies on a matrix version of Hilbert's Theorem 90, and is probabilistic with expected running time O(|E:F|d³) when |F| is bounded. Using similar methods we then describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.     

Simplicity of the branching of representations of the groups GL(<Emphasis Type="Italic">n,q</Emphasis>) under the parabolic restriction: An elementary proof

It is well known that the branching of representations of the groups GL(n, q) under the parabolic restriction is simple. Apparently, an elementary proof of this important fact has not been found so far. We present such a proof, which uses a method not standard for the representation theory of finite groups.     

Properties of solutions ofu″+g(t) u 2n−1 =0, II

Summary Part I [7] of this paper appeared in this journal in 1962. In Sections 1, 2 of this paper we consider the equation of the title wheng(t)>0. In Section 1 we examine the strength of the hypothesis of continuity ofg(t) in known theorems. In Section 2 we determine the combinations of boundedness and oscillation that may occur for solutions of this equation. Section 3 is devoted to properties of solutions of the equation wheng(t)<0. In general, differentiability ofg(t) is not required in the paper.     

Three theorems on common splitting fields of central simple algebras

LetA ₁, …A _n be central simple algebras over a fieldF. Suppose that we possess information on the Schur indexes of some tensor products of (some tensor powers of) the algebras. What can be said (in general) about possible degrees of finite field extensions ofF splitting the algebras? In Part I, we prove a positive result of that kind. In Part II, we prove a negative result. In Part III, we develop a general approach.     

The taming of recurrences in computability logic through cirquent calculus, Part I

This paper constructs a cirquent calculus system and proves its soundness and completeness with respect to the semantics of computability logic. The logical vocabulary of the system consists of negation ${\neg}$ , parallel conjunction ${\wedge}$ , parallel disjunction ${\vee}$ , branching recurrence ?, and branching corecurrence ?. The article is published in two parts, with (the present) Part I containing preliminaries and a soundness proof, and (the forthcoming) Part II containing a completeness proof.