Representations of double coset hypergroups and induced representations

The principal goal of this paper is to investigate the representation theory of double coset hypergroups. IfK=G//H is a double coset hypergroup, representations ofK can canonically be obtained from those ofG. However, not every representation ofK originates from this construction in general, i.e., extends to a representation ofG. Properties of this construction are discussed, and as the main result it turns out that extending representations ofK is compatible with the inducing process (as introduced in [7]). It follows that a representation weakly contained in the left-regular representation ofK always admits an extension toG. Furthermore, we realize the Gelfand pair (where are a local field andR its ring of integers) as a polynomial hypergroup on ℕ₀ and characterize the (proper) subset of its dual consisting of extensible representations.     

Sugli zeri delle funzioni zeta di Dedekind

Riassunto Scopo di questo lavoro è dare una formula asintotica per il numero degli zeri di ReF _K(λ+it) e di ImF _K(λ+it), dove eζ _K(8) è la funzione zeta di Dedekind associata al campo numericoK, con 0<t<T e λ numero reale fissato tale che 1−1/n<λ<1 doven è il grado diK.

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Summary The aim of this paper is to give an asymptotic formula for the number of zeros of ReF _K(λ+it) and ImF _K(λ+it), where andζ _K(8) is the Dedekind zeta function for a number fieldK, with 0<t<T and λ fixed real number such that 1−1/n<λ<1, wheren is the degree ofK.

     

Congruences for the class numbers of real cyclic sextic number fields

LetK ₆ be a real cyclic sextic number field, andK ₂,K ₃ its quadratic and cubic subfield. Leth(L) denote the ideal class number of fieldL. Seven congruences forh ^- =h (K ₆)/(h(K ₂)h(K ₃)) are obtained. In particular, when the conductorf ₆ ofK ₆ is a primep, , whereC is an explicitly given constant, andB _n is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields. Project supported by the National Natural Science Foundation of China (Grant No. 19771052).     

Unitary units in modular group algebras

Letp be a prime,K a field of characteristicp, G a locally finitep-group,KG the group algebra, andV the group of the units ofKG with augmentation 1. The anti-automorphismg→g ⁻¹ ofG extends linearly toKG; this extension leavesV setwise invariant, and its restriction toV followed byv→v ⁻¹ gives an automorphism ofV. The elements ofV fixed by this automorphism are calledunitary; they form a subgroup. Our first theorem describes theK andG for which this subgroup is normal inV. For each elementg inG, let denote the sum (inKG) of the distinct powers ofg. The elements 1+(g-1) withh,hεG are thebicyclic units ofKG. Our second theorem describes theK andG for which all bicyclic units are unitary. Research partly supported by the Hungarian National Foundation for Scientific Research grant no. T4265. The second author is indebted to the ‘Universitas’ Foundation and the Lajos Kossuth University of Debrecen, Hungary, for warm hospitality and generous support during the period when this work began. This article was processed by the authors using the Springer-Verlag T_EX mamath macro package 1990.     

Amitsur cohomology of cubic extensions of algebraic integers

LetK be the rational fieldQ or a complex quadratic number field other than . LetL be a normal three-dimensional field extension onK. IfR andS are the rings of algebraic integers ofK andL respectively, then the Amitsur cohomology groupH ² (S/R, U) is trivial. Inflation and class numbers give information about cohomology arising from certain nonnormal cubic extensions. This work was supported in part by NSF Grant GP-28409.     

A solution to one of Kano's conjectures concerningk-th maximal spanning trees

Let (G) be the collection of all spanning trees of a connected and weighted graphG, andF ₁,F ₂, ...,F _m the partition of (G) such thatF _n the set ofi-th maximal spanning trees ofG. Kano^[1] conjectured that for anyA F ₁ and every integerk, 1km, there existsT F _k such that |T/A|k–1. This paper gives the conjecture a very simple proof, and related results.     

The continuous spectrum of the relative trace formula for GL(3) over a quadratic extension

LetE/F be a quadratic extension of number fields,G the group GL(3,E) regarded as an algebraic group overF andU a quasi-split unitary group in three variables. Let alsoϑ be a generic character of a maximal unipotent subgroupN ofG. We derive an explicit expression for the integral whereK _cont is the continuous part of the kernel attached to a smooth function of compact support onG(A). In particular, we prove that this expression is absolutely convergent. The result can be used to show that a cuspidal representation ofG contains a vectorφ such thatεφ(u)du≠0 if and only if it is a base change from a representation of GL(3,F). Partially supported by NSF Grant DMS-91-01637.     

Weakly symmetric spaces and bounded symmetric domains

LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space. Consider the principal fiber bundleM=G/K _s G/K, whereK _s is the semisimple part ofK=K _s ·Z _K ⁰ andZ _K ⁰ is the connected center ofK. The natural action ofG onM extends to an action ofG ¹=G×Z _K ⁰ . We prove as the main result thatM is weakly symmetric with respect toG ¹ and complex conjugation. In the case whereD is an irreducible classical bounded symmetric domain andG is a classical matrix Lie group under a suitable quotient, we provide an explicit construction ofM=D×S ¹ and determine a one-parameter family of Riemannian metrics onM invariant underG ¹. Furthermore,M is irreducible with respect to . As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf. [M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.Research partially supported by an NSF grant. The author wishes to thank the International Erwin Schroedinger Institute for its hospitality during the preparation of this paper.     

Minimal presentations for certain group extensions

IfG is a finite group thend(G) denotes the minimal number of generators ofG. IfH andK are groups then the extension, 1 →H →G →K → 1, is called an outer extension ofK byH ifd(G)=d(H)+d(K). Let be the class of groups containing all finitep-groupsG which has a presentation withd(G) = dimH ¹(G,z _p ) generators andr(G)=dimH ² (G,Z _p ) relations: in this article it is shown that ifK is a non cyclic group belonging to andH is a finite abelian p-group then any outer extension ofK byH belongs to .     

Reduction theory for a rational function field

LetG be a split reductive group over a finite field F_q. LetF = F_q(t) and let A denote the adèles ofF. We show that every double coset inG(F)/G(A)/K has a representative in a maximal split torus ofG. HereK is the set of integral adèlic points ofG. WhenG ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.     

Building ofGL(m, D) and centralizers

LetG=GL(m, D) whereD is a central division algebra over a commutative nonarchimedean local fieldF. LetE/F be a field extension contained inM(m, D). We denote byI (resp.I _E) the nonextended affine building ofG (resp. of the centralizer ofE ^x inG). In this paper we prove that there exists a uniqueG _E-equivariant affine mapj _EI_EI. It is injective and its image coincides with the set ofE ^x-fixed points inI. Moreover, we prove thatj _E is compatible with the Moy-Prasad filtrations.This author's contribution was written while he was a post-doctoral student at King's College London and supported by an european TMR grant     

Frattini closed groups and adequate extensions of global fields

LetL be a finite Galois extension of a global fieldF. It is shown that if the Galois groupG=Gal(L/F) satisfies a certain condition, thenL is a maximal commutative subfield of someF-division algebra if and only if the intermediate field corresponding to the Frattini subgroup ofG is also a maximal commutative subfield of someF-division algebra. In particular this condition holds ifG is a supersolvable group. The third author was supported in part by the NSF under Grant DMS 97-01253.     

Hadamard matrices generated by an adaptation of generalized quaternion type array

The purpose of this paper is to prove that ifq 1 (mod 4) andq – 2 are both prime powers, then there exists an Hadamard matrix of order 4q. We rely on relative Gauss sums and generalized quaternion type array. Under the same assumption onq, E. Spence has obtained an Hadamard matrix of order 4q by using a relative difference set and the Goethals-Seidel array. We believe that the matrix constructed here is inequivalent to Spence's matrix, in general.Notation q a power of a primep - Z the rational integer ring - F = GF(q) a finite field withq elements - K = GF(q ^s ) an extension ofF of degrees 2 - F ^× multiplicative group ofF - a primitive element ofK - S _F absolute trace fromF - S _K/F relative trace fromK toF - N _K/F relative norm fromK toF - I _m the unit matrix of orderm - J _m the matrix of orderm with every element + 1 - e the column vector of ordern with every element + 1 - A ^* the transpose of a matrixA - J _m (x) 1 +x + x ² + ... +x ^m-1     

Elementary Abelianp-extensions of algebraic function fields

LetK be a field of characteristicp>0 andF/K be an algebraic function field. We obtain several results on Galois extensionsE/F with an elementary Abelian Galois group of orderp ⁿ.

1. E can be generated overF by some elementy whose minimal polynomial has the specific formT ^pn?T?z.
2. A formula for the genus ofE is given.
3. IfK is finite, then the genus ofE grows much faster than the number of rational points (as [E∶F] → ∞).
4. We present a new example of a function fieldE/K whose gap numbers are nonclassical.

     

Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications

LetF andG denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functionsf andg, respectively. A measure of the closeness betweenF andG is defined by: . Based on two independent samples it is proposed to estimate λ by , whereF _n (x) andG _n (x) are the empirical distribution functions ofF(x) andG(x) respectively and and are taken to be the so-called kernel estimates off(x) andg(x) respectively, as defined by Parzen [16]. Large sample theory of is presented and a two sample goodness-of-fit test is presented based on . Also discussed are estimates of certain modifications of λ which allow us to propose some test statistics for the one sample case, i.e., wheng(x)=f ₀ (x), withf ₀ (x) completely known and for testing symmetry, i.e., testingH ₀:f(x)=f(−x).     

Bemerkungen zu einer metrischen Invarianten algebraischer Zahlkörper

LetK be an algebraic number field andp any positive, real number orp=. In this paper the optimal constantc _p (K) in the well-known inequality containing the imbeddings ofK into and the norm of K is studied. For anlyp an explicit method for the determination ofc _p (K) is given and the interdependence ofp andc _p (K) is investigated. Finally an application ofc _p (K) on problems concerning vectors with a metrical extremal condition in a special class of lattices is provided.     

Regular and normal closure operators and categorical compactness for groups

For a class of groupsF, closed under formation of subgroups and products, we call a subgroupA of a groupG F-regular provided there are two homomorphismsf, g: G » F, withF F, so thatA = {x G |f(x) =g(x)}.A is calledF-normal providedA is normal inG andG/A F. For an arbitrary subgroupA ofG, theF-regular (respectively,F-normal) closure ofA inG is the intersection of allF-regular (respectively,F-normal) subgroups ofG containingA. This process gives rise to two well behaved idempotent closure operators.A groupG is calledF-regular (respectively,F-normal) compact provided for every groupH, andF-regular (respectively,F-normal) subgroupA ofG × H, ₂(A) is anF-regular (respectively,F-normal) subgroup ofH. This generalizes the well known Kuratowski-Mrówka theorem for topological compactness.In this paper, theF-regular compact andF-normal compact groups are characterized for the classesF consisting of: all torsion-free groups, allR-groups, and all torsion-free abelian groups. In doing so, new classes of groups having nice properties are introduced about which little is known.     

Maximal deviation theory of some estimators of prior distribution functions

Summary Let be a sequence of independent identically distributed random variables withθ ₁∼G and the conditional distribution ofx ₁ givenθ ₁=θ given by . HereG is unknown andF _θ(·) is known. This paper provides estimators ofG based onx ₁, …,x _n such that the random variable sup has an asymptotic distribution asn→∞ under certain on conditionsG and for certain choices ofF _θ. A simulation model has been discussed involving the uniform distribution on (0, θ) forF _θ and an exponential distribution forG. Research supported by the National Science Foundation under Grant #MCS77-26809.     

K-invariants in the universal enveloping algebra of the desitter group

Let G be a non-compact connected semisimple Lie group with finite center and let G^K denote the centralizer of a maximal compact subgroup K of G inG, the universal enveloping algebra over of the Lie algebra of G. In [4] Lepowsky defines an injective anti-homo morphism P:G ^KK ^MA, where M is the centralizer in K of a Cartan subalgebraa of the symmetric pair (G,K),K andA are the universal enveloping algebras over corresponding to K anda, respectively, andK ^M is the centralizer of M inK. The subalgebra P(G ^K) ofK ^MA has considerable significance in the infinite dimensional representation theory of G. In this paper we explicitly compute P(G ^K) when G=S0_o(4,1), and show how this result leads to the determination of all irreducible representations of G and its universal covering group Spin(4,1).Partially supported by CONICET (Argentina) grants.