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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.)

In two previous papers [1, 2] we investigated the zeros of certain arithmetic functions. Using units of cubic fields, we also succeeded to construct, almost by accident, and as a by-product so to speak, entirely new and comparatively complicated combinatorial identities. In an interesting paper combinatorialist L. Carlitz [10] proved those identities in an elementary way. In a/m, we had to prove that the units used were fundamental ones.Encouraged by these results, we took a closer look at this method that had led to the construction of combinatorial identities. Since the latter are such an important tool in mathematics, we thought it would be “einer Messe wert“ to generalize these results and lay the theoretical foundations of a new method for the construction of highly sophisticated combinatorial identities.

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.

One of the original applications of catastrophe theory envisaged by Thom was that of discussing the local structure of the focal set for a (generic) smooth submanifold M ⊆ Rn + 1. Thom conjectured that for a generic M there would be only finitely many local topological models, a result proved by Looijenga in [4]. The objective of this paper is to extend Looijenga's result from the smooth category to the algebraic category (in a sense explained below), at least in the case when M has codimension 1.Looijenga worked with the compactified family of distance-squared functions on M (defined below), thus including the family of height functions on M whose corresponding catastrophe theory yields the local structure of the focal set at infinity. For the family of height functions the appropriate genericity theorem in the smooth category was extended to the algebraic case in [1], so that the present paper can be viewed as a natural continuation of the first author's work in this direction.

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).

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.

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

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 (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.

In the first part of this paper we define a partial ordering on the set of all homogeneous identities and find necessary and sufficient conditions that an identity does not imply any identity lower than it in the partial ordering (we call such an identity irreducible). Perhaps the most interesting property established for irreducible identities is that they are skew-symmetric in any two variables of the same odd degree and symmetric in any two variables of the same even degree. The results of the first section are applied to commutative algebras in the remainder of the paper.

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.

1. In this paper, X will always denote a locally compact Hausdorff space, C0(X) the algebra of all complex-valued continuous functions vanishing at infinity on X and B(X) the algebra of all bounded continuous complex-valued functions defined on X. If X is compact, C0(X) is identical to B (X) and all the results of this paper are obvious. Therefore, we will assume at the outset that X is not compact. If A represents an algebra of functions, AR will denote the algebra of all real-valued functions in A.

In this paper we apply various results and methods of previous papers on the kernel to four-person games.Section 2 contains the basic definitions needed. In §3 we prove that the kernel of the general-sum four-person game consists of a line segment (which may shrink to a point). A method for classifying games according to their kernels is suggested in §4 and is used there to characterize all four-person games whose kernel consists of a non-degenerate interval. In the last section, §5, we offer a bargaining procedure, based on principles established in (1), which leads to the kernel in the case of a non-degenerate interval.

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.

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.

A topological group G is called monothetic if it contains a dense cyclic subgroup. An element x of G is called a generator of G if x generates a dense cyclic subgroup of G. We denote by E(G) the set of generators of G; the complement of E(G) in G, consisting of the “non-generators” of G, we write as N(G) Throughout this paper we consider only locally compact abelian (LCA) groups satisfying the T2 separation axiom (note that a monothetic group is automatically abelian). In [1] certain problems of measurability concerning the set E(G) are discussed. In this paper we shall consider some algebraic and topological properties of the sets E(G) and N(G)

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.

This paper continues work begun in a previous paper of the same title (7), which will be called I; results from I will be referred to as Theorem 1.4, Axiom 1.1 etc. The notation used in the present paper will, except where noted, be that of I, to which reference should be made for further details.In § 2, certain ideas presented in I are modified to give a neater and more general theory and then some new results of this theory are added. The remaining two sections develop some of the examples mentioned in I, § 5.

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.

We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group.

The class S of functions under study in this paper was introduced by V. I. Smirnov in 1932. This class was subsequently investigated by various authors, a pertinent paper to the present wrork being that of Tumarkin and Havinson [2], who showed that a plane compact set of logarithmic capacity zero is 5-removable. Another important development, due to Yamashita [3], wras that the class 5 could be characterized as those analytic functions ƒ for which log+ |ƒ| has a quasi-bounded harmonic majorant.In what follows, we discuss the Smirnov class in the context of planar surfaces, exploiting some ideas in the work of Hejhal [1] to establish that a closed, bounded, totally disconnected set is S-removable if and only if its complement belongs to the null class Os.

In this paper we construct an analogue of Iwahori–Hecke algebras of SL2 over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on SL2, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori–Matsumoto type relations.

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.

In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundh¨all for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.