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This paper is a continuation of three recent articles concerning the structure of hyperinvariant subspace lattices of operators on a (separable, infinite dimensional) Hilbert space . We show herein, in particular, that there exists a “universal” fixed block-diagonal operator B on such that if ε > 0 is given and T is an arbitrary nonalgebraic operator on , then there exists a compact operator K of norm less than ε such that (i) Hlat(T) is isomorphic as a complete lattice to Hlat(B + K) and (ii) B + K is a quasidiagonal, C00, (BCP)-operator with spectrum and left essential spectrum the unit disc. In the last four sections of the paper, we investigate the possible structures of the hyperlattice of an arbitrary algebraic operator. Contrary to existing conjectures, Hlat(T) need not be generated by the ranges and kernels of the powers of T in the nilpotent case. In fact, this lattice can be infinite.

In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra B over Q; he then proved an analog of his result with Zagier for these curves. In Gross’ paper, the curves were defined in a somewhat ad hoc manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on B×, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a “multiplication” of automorphic forms that arises from the representation ring of B×. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to CM points on these curves, and are thus isogenous to a product E × E, where E is an elliptic curve with complex multiplication. For these CM points one can make a relation between the action of the p-th Hecke operator and Frobenius at p, similar to the well-known congruence relation of Eichler and Shimura.

This paper expresses the character of certain depth-zero supercuspidal representations of the rank-2 symplectic group as the Fourier transform of a finite linear combination of regular elliptic orbital integrals—an expression which is ideally suited for the study of the stability of those characters. Building on work of F. Murnaghan, our proof involves Lusztig’s Generalised Springer Correspondence in a fundamental way, and also makes use of some results on elliptic orbital integrals proved elsewhere by the author using Moy-Prasad filtrations of p-adic Lie algebras. Two applications of the main result are considered toward the end of the paper.

In this paper we consider the following problem: Given a unital C*- algebra A and a collection of elements S in the identity component of the invertible group of A, denoted inv0(A), characterize the group of finite products of elements of S. The particular C*-algebras studied in this paper are either unital purely infinite simple or of the form (A ⊗ K)+, where A is any C*-algebra and K is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents (1+ nilpotent), positive invertibles and symmetries (s2 = 1). First we determine the groups of finite products for each collection of elements in (A ⊗ K)+. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for (A ⊗ K)+, is that for A unital purely infinite and simple, inv0(A) is generated by each of these collections of elements.

In a recent paper [2] we defined four classes of infinite dimensional simple Lie algebras over a field of characteristic 0 which we called W*, S*, H*, and K*. As the names suggest, these classes generalize the Lie algebras of Cartan type. A second paper [3] investigates the derivations of the algebras W* and S*, and the possible isomorphisms between these algebras and the algebras defined by Block [1]. In the present paper we investigate the automorphisms of the algebras of type W*.

This paper is concerned with a combinatorial construction which mysteriously “mimics” or “models” the decomposition of certain reducible representations of orthogonal groups. Although no knowledge of representation theory is needed to understand the body of this paper, a little familiarity is necessary to understand the representation theoretic motivation given in the introduction. Details of the proofs will most easily be understood by people who have had some exposure to Schensted's algorithm or jeu de tacquin.

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. LetGbe a compact group andHa closed subgroup ofG. Letμbe the normalizedG-invariant measure over the compact homogeneous spaceG/Hassociated with Weil's formula and. We then present a structured class of abstract linear representations of the Banach convolution function algebrasLp(G/H,μ).

The purpose of this paper is to list all of the characters of An, the alternating group, mentioned in the title. The same problem for the symmetric group, Sn, was dealt with by the authors in [1]. We showr here that, apart from a few exceptions, the imprimitive, irreducible complex characters of An fall naturally into two infinite families. (Throughout this paper characters are taken over the complex numbers.)

The transformations we will discuss in this paper are the Hankel transformation Hυ defined for f ∊ C 0, the collection of continuous functions compactly supported in (0, ∞), by (1.1) and the and transformations defined for such f by (1.2) and (1.3) where Jv >and Yv are the Bessel functions of the first and second kinds respectively, and H v is the Struve function; for the theory of these functions see [1, Chapter VII]. These transformations were studied extensively by one of us in [5] and [6] on the spaces defined in [7; Sections 1 & 5]. In those papers the boundedness of the three transformations was fully given on the spaces for 1 < p < ∞, but not for p = 1. Also inversion formulae were given for the transformations only for portions of their respective ranges of boundedness.

Throughout this paper, S will denote a finite semigroup and Z+ the set of positive integers. E = E(S) denotes the set of idempotents of S. Let . If , then let AB = {ab| a ∈ A, b ∈ B}. has been studied by many authors, including [2, 3, 5, 6, 7]. If X is a set, then |X| denotes the cardinality of X. For undefined terms in this paper, see [1,4].THEOREM 1. Let I be an ideal of S, a subgroup of . Then has a normal subgroups such that is isomorphic to a subgroup of and is isomorphic to a subgroup of .

This paper is a direct continuation of the article “Proximal analysis and boundaries of closed sets in Banach space, Part I: Theory”, by the same authors. It is devoted to a detailed analysis of applications of the theory presented in the first part and of its limitations.Theorem 2.1 has important consequences for geometry of Banach spaces. We start the presentation with a discussion of density and existence of R-proper points (Definition 1.3) for closed sets in Banach spaces. Our considerations will be based on the “lim inf” inclusions proven in the first part of our paper.

In this paper we continue our effort [11], [12], [13], [14] to interpret geometrically the harmonic forms on certain locally symmetric spaces constructed by using the theta correspondence. The point of this paper is to prove an integral formula, Theorem 2.1, which will allow us to generalize the results obtained in the above papers to the finite volume case (the previous papers treated only the compact case). We then apply our integral formula to certain finite volume quotients of symmetric spaces of orthogonal groups. The main result obtained is Theorem 4.2 which is described below. We let (,) denote the bilinear form associated to a quadratic form with integer coefficients of signature (p, q). We assume that the fundamental group Γ ⊂ SO(p, q) of our locally symmetric space is the subgroup of the integral isometries of (,) congruent to the identity matrix modulo some integer N. We assume that N is chosen large enough so that Γ is neat (the multiplicative subgroup of C* generated by the eigenvalues of the elements of Γ has no torsion), Borel [2], 17.1 and that every element in Γ has spinor norm 1, Millson-Raghunathan [15], Proposition 4.1. These conditions are needed to ensure that our cycles Cx (see below) are orientable. The methods we will use apply also to unitary and quaternion unitary locally symmetric spaces, see [13].

In [16] Freese showed that the word problem for the free modular lattice on 5 generators is unsolvable. His proof makes essential use of Mclntyre's construction of a finitely presented field with unsolvable word problem [30]. (We follow Cohn [7] in calling what is commonly called a division ring a field, and what is commonly called a field a commutative field.) In this paper we will use similar ideas to obtain unsolvability results for varieties of modular ortholattices. The material in this paper is fairly wide ranging, the following are recommended as reference texts.

The problem which motivated the writing of this paper is that of finding structure behind the decomposition of the sl3 representation spaces V* ⊗ W = Hom(V, W) for finite dimensional irreducible sl3-modules V and W. For sl2 this extends the classical Clebsch-Gordon problem. The question has been considered for sl3 in a computational way in [5]. In this paper we build a conceptual algebraic framework going beyond the enveloping algebra of sl3.For each dominant integral weight α let Vα be an irreducible representation of sl3 of highest weight α. It is well known that, for weights α, μ, λ, the multiplicity of Vλ in Hom(Vα, Vα+μ) is bounded by the multiplicity of μ in Vλ, with equality for generic α. This suggests the possibility of a single construction of highest weight vectors of weight X in Hom(Vα, Vα+μ) which is valid for all a.

The aim of this paper is to extend to a suitable class of topological semigroups parts of well-defined theory of representations of topological groups. In attempting to obtain these results it was soon realized that no general theory was likely to be obtainable for all locally compact semigroups. The reason for this is the absence of any analogue of the group algebra Ll(G). So the theory in this paper is restricted to a certain family of topological semigroups. In this account we shall only give the details of those parts of proofs which depart from the standard proofs of analogous theorems for groups.On a locally compact semigroup S the algebra of all μ ∊ M(S) for which the mapping and of S to M(S) (where denotes the point mass at x) are continuous when M(S) has the weak topology was first studied in the sequence of papers [1, 2, 3] by A. C. and J. W. Baker.

This paper is concerned with the description of principal congruence relations. Given elements a and b of a universal algebra , let θ(a, b) denote the smallest congruence relation on containing the pair 〈a, b〉. One of the earliest characterizations of θ(a, b) is Mal'cev's well-known result [5, Theorem 1.10.3], which says that c ≡ d(θ(a, b)) if and only if there exists a sequence z0, z1, …, zn of elements of and a sequence f1, f2, …, fn of unary algebraic functions such that c = z0, d = zn, and for each i = 1, …, n,Although this describes θ(a, b) in terms of a set of unary algebraic functions, it is not possible to predict the number or complexity of the unary functions used independently of the choice of a, b, c and d. Several recent papers ([1], [2], [3], [4], [6]) investigate classes of algebras in which principal congruences are simpler.

1. Introduction. The remarkable 1927 paper by J. H. Redfield [13] which anticipated many recent combinatorial results in Polya counting theory and, in fact, predated Polya's theorem by ten years has been discussed at length by Harary and Palmer [8], Foulkes [5; 6], Sheehan [15; 16] and Read [12], not to mention de Bruijn [3] and others. We shall, in this paper, demonstrate how multilinear techniques may be used in this context. The Redfield superposition theorem and decomposition theorem turn out to be statements about a group acting on finite function spaces, and may thus be dealt with in multilinear terms. We shall prove Redfield's results and an extension due to Foulkes [5].

This paper is motivated by work in two fields, the theory of strict topologies and topological measure theory. In [1], R. C. Buck began the study of the strict topology for the algebra C*(S) of continuous, bounded real-valued functions on a locally compact Hausdorff space S and showed that the topological vector space C*(S) with the strict topology has many of the same topological vector space properties as C0(S), the sup norm algebra of continuous realvalued functions vanishing at infinity. Buck showed that as a class, the algebras C*(S) for S locally compact and C*(X), for X compact, were very much alike. Many papers on the strict topology for C*(S), where S is locally compact, followed Buck's; e.g., see [2; 3].

In recent years there has been some interest in trying to improve the behaviour of maps by extending their domains. For example, in 1953 Whyburn showed that every map is the restriction of a compact map [17]. Similarly, Krolevec proved in 1967 that each locally perfect map can be extended to a perfect map [12] and in an as yet unpublished paper, Dickman obtained the same result for arbitrary maps [4]. In this paper we show that every map can be extended to an open map so that certain properties of the domain and range are preserved in the new domain. These results are then used to obtain analogues and improvements of recent theorems of Arhangel'skiï, Ĉoban, Hodel, and Proizvolov.

In two previous papers (see4;5) O. T. O'Meara and I investigated the problem of generating the integral orthogonal group of a quadratic form by symmetries in the case where the underlying ring of integers was the integers of a dyadic local field of characteristic not 2. In this paper, the investigation is concerned with a local field of characteristic 2. As in (5), only the unimodular case is treated. As in (4) and (5), groupsS(L), Xh(L), andO(L) are introduced for a unimodular latticeLand the relationship betweenS(L) andO(L) studied. As in the previously cited papers, generation by symmetries means thatS(L) =O(L). The following result is obtained.

A ring A has finite norm properly, abbreviated FNP, if each proper homomorphic image of A is finite. In [3], Chew and Lawn described some of the structural properties of FNP rings with identity, which they called residually finite rings. The twofold aim of this paper is to extend the results of [3] to arbitrary rings with FNP and to give characterizations of FNP rings independent of the results of [3].If A is a ring, let A+ denote A regarded as an abelian group. In the first section of this paper, we explore the effects of FNP upon the structure of A+. The following theorem is typical of the results in this section.

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

The Szegö polynomials are defined on T, the real numbers modulo 1. In this paper and in its sequel we give a generalization of Szegö polynomials in which T is replaced by an arbitrary locally compact abelian group θ on whose dual there has been distinguished a measurable order relation compatible with the group structure. The present paper is devoted to the case where θ is compact and therefore discrete. The general case will be taken up in the sequel mentioned above. It is desirable to proceed in this way because the case θ compact is much simpler and much more like the classical situation than is the general case, in which various measure-theoretic difficulties obtrude. Moreover, as it happens, it is possible to develop the theory in this way with relatively little repetition.

In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly.

This paper is a partial solution of problem 24 in (2) which suggests that the finiteness of the partially ordered semigroups generated by various combinations of operators on classes of universal algebras be investigated. The main result is that the semigroups generated by the following sets of operators (for definitions see §2) are finite: {H, S, P, Ps}, {C, H, S, P, PF} {C, H, S, PU, PF}.This paper is part of the author's Master's thesis written in the Department of Mathematics at McMaster University. The author is indebted to the referee for his helpful suggestions.

The theorem that every properly primitive binary quadratic form is capable of representing infinitely many prime numbers was first proved completely by H. Weber (5). The purpose of this paper is to give an elementary proof of the case where the form is ax2 + 2bxy + cy2, with a > 0, (a, 2b, c) = 1, and D = b2 — ac not a square. The cases where the form is ax2 + bxy + cy2 with b odd, and the case where the form is ax2+ 2bxy + cy2 with D a square, can be settled very simply once the first case is taken care of, and this is done in a page and a half in the Weber paper.

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).

This paper is primarily a brief elaboration on the axioms for a bicategory introduced in (3). From this point of view, the main aim is the development of the structure of certain systems of topological and uniform spaces, and the present paper merely points out some very general properties which follow from axioms so weak that they are satisfied by any system likely to be considered.

In this paper, we continue the study of integral extensions begun in [7]. Whereas in the previous paper, we dealt exclusively with the extension A[x]/(α(x)), α(x) a monic polynomial over A, we now deal with arbitrary integral extensions. Applications of the results presented herein will be made in subsequent papers.To simplify our presentation, we make the following conventions. By an algebra, we will always mean a commutative complex algebra with an identity element, usually denoted by e

Properties and characterizations for prime and semiprime rings have been provided by A. W. Goldie (2, 3). In a previous paper (1), the authors used the results of (2) to characterize prime and uniform prime modules. It is the aim of the present paper to generalize Goldie's work on semi-prime rings (3) to modules. In this setting certain new properties will appear.Notationally, in the work to follow, the symbol R always denotes a ring and all R-modules will be right R-modules.In the theory of rings an ideal C is said to be prime if and only if whenever AB ⊆ C for ideals A and B, then either A ⊆ C or B ⊆ C. A ring is prime if the zero ideal is prime.

Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.

In a paper (5) published in the Proceedings of the Cambridge Philosophical Society, Primrose obtained the formulae for the number of points contained in a non-degenerate quadric in PG(n, s), the finite projective geometry of n dimensions based on a Galois field GF(s). In § 3 of the present paper the formulae for the number of p-flats contained in a non-degenerate quadric in PG(n, s) are obtained. In § 4 an interesting property of a non-degenerate quadric in PG(2k, 2m) is proved. These properties of a quadric will be used in solving some combinatorial problems of statistical interest in a later paper.

It is the purpose of this paper to establish a closer connection between the logarithmic capacity of sets and double trigonometric series. In (9), closed sets of logarithmic capacity zero were established as sets of uniqueness for a particular class of double trigonometric series under circular (C, 1) summability. By slightly changing this class of series but still maintaining closed sets of logarithmic capacity zero as sets of uniqueness, it is shown in this paper that closed sets of positive logarithmic capacity form sets of multiplicity.

Let (𝒳, d, μ) be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that μ(﹛x﹜) = 0 for all x ∈ 𝒳. In this paper, we show that the boundedness of a Calderón–Zygmund operator T on L2(μ) is equivalent to that of T on Lp(μ) for some p ∈ (1,∞), and that of T from L1(μ) to L1,∞(μ). As an application, we prove that if T is a Calderón–Zygmund operator bounded on L2(μ), then its maximal operator is bounded on Lp(μ) for all p ∈ (1,∞) and from the space of all complex-valued Borel measures on 𝒳 to L1,∞(μ). All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

Let Gn = Spn(F) be the rank n symplectic group with entries in a nondyadic p-adic field F. We further let be the metaplectic extension of Gn by defined using the Leray cocycle. In this paper, we aim to demonstrate the complete list of reducibility points of the genuine principal series of . In most cases, we will use some techniques developed by Tadić that analyze the Jacquetmodules with respect to all of the parabolics containing a fixed Borel. The exceptional cases involve representations induced from unitary characters χ with χ2 = 1. Because such representations π are unitary, to show the irreducibility of π, it suffices to show that . We will accomplish this by examining the poles of certain intertwining operators associated to simple roots. Then some results of Shahidi and Ban decompose arbitrary intertwining operators into a composition of operators corresponding to the simple roots of . We will then be able to show that all such operators have poles at principal series representations induced from quadratic characters and therefore such operators do not extend to operators in for the π in question.

Representations of various one-dimensional central extensions of quantum tori (called quantum torus Lie algebras) were studied by several authors. Now we define a central extension of quantum tori so that all known representations can be regarded as representations of the new quantum torus Lie algebras . The center of now is generally infinite dimensional.In this paper, Z-graded Verma modules over and their corresponding irreducible highest weight modules are defined for some linear functions . Necessary and sufficient conditions for to have all finite dimensional weight spaces are given. Also necessary and sufficient conditions for Verma modules e to be irreducible are obtained.

Let 𝔽q be a finite field of q elements, 𝓕 a p-adic field, and D a quaternion division algebra over 𝓕. This paper proves uniqueness of Shalika models for GL2n(𝔽q) and GL2n(D), and re-obtains uniqueness of Shalika models for GL2n(𝔽q) and GL2n(D), and re-obtains uniqueness of Shalika models for GL2n (𝓕) for any n ∈ ℕ.

Suppose that P = MN is amaximal parabolic subgroup of a quasisplit, connected, reductive classical group G defined over a non-Archimedean field and A is the standard intertwining operator attached to a tempered representation of G induced from M . In this paper we determine all the cases in which Lie(N ) is prehomogeneous under Ad(m) when N is non-abelian, and give necessary and sufficient conditions for A to have a pole at 0.

The aim of this paper is to prove two general results on parabolic induction of classical p-adic groups (actually, one of them holds also in the archimedean case), and to obtain from them some consequences about irreducible unitarizable representations. One of these consequences is a reduction of the unitarizability problem for these groups. This reduction is similar to the reduction of the unitarizability problem to the case of real infinitesimal character for real reductive groups.

This paper shows the existence and uniqueness of Klyachko models for irreducible unitary representations of GL5 (ℱ), where ℱ is a p-adic field. It is an extension of the work of Heumos and Rallis on GL4(ℱ).

Let A = (aj,k)j,k≥1 be a non-negative matrix. In this paper, we characterize those A for which ∥A∥E,F are determined by their actions on decreasing sequences, where E and F are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: ℓp, d(w, p), and ℓp(w). The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

In this paper we prove the unitarity of duals of tempered representations supported onminimal parabolic subgroups for split classical p-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups.

The decomposability number of a von Neumann algebra ℳ (denoted by dec(ℳ)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in ℳ. In this paper, we explore the close connection between dec(ℳ) and the cardinal level of the Mazur property for the predual ℳ* of ℳ, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L1(G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)*, etc. We show that for any of these von Neumann algebras, say ℳ, the cardinal number dec(ℳ) and a certain cardinal level of the Mazur property of ℳ* are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: the compact covering number κ(G) of G and the least cardinality ᙭(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)**.

In this paper we generalise the concept of a Steinberg cross section to non–connected affine Kac–Moody groups. This Steinberg cross section is a section to the restriction of the adjoint quotient map to a given exterior connected component of the affine Kac–Moody group. (The adjoint quotient is only defined on a certain submonoid of the entire group, however, the intersection of this submonoid with each connected component is non-void.) The image of the Steinberg cross section consists of a “twisted Coxeter cell”, a transversal slice to a twisted Coxeter element. A crucial point in the proof of the main result is that the image of the cross section can be endowed with a 𝕔*-action.

We investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study themin this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.

In 1999 V. Arnol’d introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a 1-form to a singular submanifold. We prove that a germ of any subset N of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to N of the contact structure. This is a generalization of the Darboux-Givental’ theoremfor smooth submanifolds of a contactmanifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V. Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold N in terms of the algebraic restriction of the contact structure to N. In particular, the algebraic restriction is zero if and only if N is contained in a smooth Legendrian submanifold of M.

In this paper we describe reducibility of non-unitary generalized principal series for classical p-adic groups in terms of the classification of discrete series due to Moeglin and Tadić.

In this paper we make explicit all L-functions in the Langlands–Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local L-functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for Re(s) ≥ 1/2 in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrumand determining poles of automorphic L-functions.

In this paper, we construct a duality for p-adic orthogonal groups.

This paper contains a comparison of several definitions of equivariant formality for actions of torus groups. We develop and prove some relations between the definitions. Focusing on the case of the circle group, we use S1-equivariant minimal models to give a number of examples of S1-spaces illustrating the properties of the various definitions.