Archimedean Components Research Articles

Decomposition of commutative ordered Γ-semigroups into archimedean components

We can see that any semigroup can be considered as a Γ-semigroup. In this paper, we first introduce the relation “η” on ordered Γ-semigroups and prove that for commutative ordered Γ-semigroups, we have the usual relation N is equal to the relation η. It is shown that if M is commutative, then M is, uniquely, an ordered semilattice of archimedean sub-Γ-semigroups which means that it is decomposable, in a unique way, into its archimedean components.

Limits of discrete series with infinitesimal character zero

For a connected linear semisimple Lie group , this paper considers those nonzero limits of discrete series representations having infinitesimal character 0, calling them totally degenerate. Such representations exist if and only if has a compact Cartan subgroup, is quasisplit, and is acceptable in the sense of Harish-Chandra. Totally degenerate limits of discrete series are natural objects of study in the theory of automorphic forms: in fact, those automorphic representations of adelic groups that have totally degenerate limits of discrete series as archimedean components correspond conjecturally to complex continuous representations of Galois groups of number fields. The automorphic representations in question have important arithmetic significance, but very little has been proved up to now toward establishing this part of the Langlands conjectures. There is some hope of making progress in this area, and for that one needs to know in detail the representations of under consideration. The aim of this paper is to determine the classification parameters of all totally degenerate limits of discrete series in the Knapp-Zuckerman classification of irreducible tempered representations, i.e., to express these representations as induced representations with nondegenerate data. The paper uses a general argument, based on the finite abelian reducibility group attached to a specific unitary principal series representation of . First an easy result gives the aggregate of the classification parameters. Then a harder result uses the easy result to match the classification parameters with the representations of under consideration in representation-by-representation fashion. The paper includes tables of the classification parameters for all such groups

Spinor L-functions, theta correspondence, and Bessel coefficients

In this paper we prove two seemingly unrelated theorems. First we establish the entireness of the spinor L-functions of certain automorphic cuspidal representations of the similitude symplectic group of order four over the rational numbers. We also prove a theorem related to the existence of Bessel models for generic discrete series representations of the same group over the real numbers. The two results are linked by the method of proof; in both cases it is based on the pull-back of an appropriately chosen global Bessel functional via the theta correspondence for the dual pair (GO(2, 2),GSp(4)). The first main theorem is related to analytic properties of spinor L-functions. We prove the entireness of the spinor L-function for those generic automorphic cuspidal representation which satisfy a condition at the archimedean place (see below). Our study of the spinor L-function is based on an integral representation which works for generic representations. These integrals which were introduced by M. Novodvorsky in the Corvallis conference [26] serve as one of the few available integral representations for the Spinor L-function of GSp(4). Some of the details missing in Novodvorsky’s original paper have been reproduced in Daniel Bump’s survey article [4]. Further details have been supplied by [40]. Novodvorsky’s integral was first generalized by Ginzburg [10], and further generalized by Soudry [39], to orthogonal groups of arbitrary odd degree. In light of the results of [40], it is sufficient to study the integral of Novodvorsky at the archimedean place. Archimedean computations are often forbidding, and unless one expects major simplifications due to the nature of the parameters, the resulting integrals are often quite hard to manage. In our case of interest, the work of Moriyama [25] benefits from exactly such simplifications when he treats the case of cuspidal representations with archimedean components in the generic (limit of) discrete series. In this work, we concentrate on those archimedean representations for which direct computations have yielded very little. For this reason, our methods are a bit indirect, in fact somewhat more indirect than what at first seems necessary. Our method is based on the theta correspondence. First we observe in Lemma 2.2 that Novodvorsky’s integral is in fact a split Bessel functional. Then in 2.1 we pull the Bessel functional back via the theta correspondence for the dual reductive pair (GO(2, 2),GSp(4)), and prove that the resulting functional on GO(2, 2) is Eulerian. On the other hand,

The structure of n-contractive t-norms

Archimedean components of t-norms are shown to determine the degree of contractivity of the investigated t-norms. It is shown that Archimedean components play an important role also in the case of torsion t-norms. A special class of generated n-contractive left-continuous t-subnorms is introduced, thus allowing to construct left-continuous n-contractive t-norms by means of ordinal sums. Several examples of 3- and 4-contraction t-norms are given.

Archimedean components of triangular norms

AbstractThe Archimedean components of triangular norms (which turn the closed unit interval into anabelian, totally ordered semigroup with neutral element 1) are studied, in particular their extension to triangular norms, and some construction methods for Archimedean components are given. The triangular norms which are uniquely determined by their Archimedean components are characterized. Using ordinal sums and additive generators, new types of left-continuous triangular norms are constructed.

H-transformation of t-norms

Cancellative t-norms are generated only if they do not posses anomalous pairs and then they are isomorphic to subsemigroups of ([0, ∞], +). A new construction method for t-norms, so-called H-transformation of t-norms, is introduced and studied, generalizing a recent example of Hájek. H-transformation allows to construct non-generated cancellative (left-continuous) t-norms with prescribed number of non-trivial Archimedean components. The unique t-norm stable under H-transformation is also described. In an analogous way, H-transformation of residual implications is given. Finally, a general H-transformation based on cancellative discrete t-conorms is proposed and examined.

On the Eisenstein cohomology of arithmetic groups

The cohomology $H ^* (\Gamma, E)$ of an arithmetic subgroup $\Gamma$ of a connected reductive algebraic group $G$ defined over some algebraic number field $F$ can be interpreted in terms of the automorphic spectrum of $\Gamma$. With this framework in place, there is a sum decomposition of the cohomology into the cuspidal cohomology (i.e., classes represented by cuspidal automorphic forms for $G$) and the so-called Eisenstein cohomology constructed as the span of appropriate residues or derivatives of Eisenstein series attached to cuspidal automorphic forms on the Levi components of proper parabolic $F$-subgroups of $G$. The main objective of this paper is to isolate a specific structural part in the Eisenstein cohomology. This pertains to regular Eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered. It is shown that the cohomological degree of these classes is bounded from below by the constant $q_0(G(\mathbb{R}))=((1/2) [\dim X_{G(\mathbb{R})}-(\rk (G(\mathbb{R}))-\rk(K_{\mathbb{R}}))]$, where $K_{\mathbb{R}}$ denotes a maximal compact subgroup of the real Lie group $G(\mathbb{R})$, where $X_{G(\mathbb{R})}$ is the associated symmetric space. This investigation has various applications. One of these is a vanishing result for the cohomology in the generic case (i.e., where the representation determining the coefficient system $E$ has regular highest weight) in degrees below $q_0(G(\mathbb{R}))$. This is a sharp bound depending only on the underlying real Lie group $G(\mathbb{R})$ (Corollary 5.6, Proposition 5.8). This result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular Eisenstein cohomology classes.

On the number of factorizations of an element in an atomic monoid

Let S be a reduced commutative cancellative atomic monoid. If s is a nonzero element of S, then we explore problems related to the computation of η(s), which represents the number of distinct irreducible factorizations of s∈S. In particular, if S is a saturated submonoid of Nd, then we provide an algorithm for computing the positive integer r(s) for which 0<limn→∞η(sn)nr(s)−1<∞. We further show that r(s) is constant on the Archimedean components of S. We apply the algorithm to show how to compute limn→∞η(sn)nr(s)−1 and also consider various stability conditions studied earlier for Krull monoids with finite divisor class group.

Quelques relations entre les cohomologies des variétés de Shimura et celles de Griffiths–Schmid (cas du groupe SU(2,1))

Our aim is to study the cohomology groups of some coherent sheaves on a Griffiths–Schmid variety associated with an anisotropic Q-from of the unitary group SU(2,1). We define some transforms relating this cohomology to the coherent cohomology groups of some sheaves defined on certain threefolds, which are fibered in projective lines over Picard modular surfaces. In particular, we give a complete and explicit description, in terms of classical Picard modular forms, of the holomorphic (resp. anti-holomorphic) part of the 1-cohomology of the Griffiths-Schmid variety. From this, it results an explicit generating system for the part of the 2–cohomology which correspond to those automorphic representations whose archimedean component is a degenerate limit of discrete series.

Dynamical systems arising from elliptic curves

We exhibit a family of dynamical systems arising from rational points on elliptic curves in an attempt to mimic the familiar toral automorphisms. At the non-archimedean primes, a continuous map is constructed on the local elliptic curve whose topological entropy is given by the local canonical height. Also, a precise formula for the periodic points is given. There follows a discussion of how these local results may be glued together to give a map on the adelic curve. We are able to give a map whose entropy is the global canonical height and whose periodic points are counted asymptotically by the real division polynomial (although the archimedean component of the map is artificial). Finally, we set out a precise conjecture about the existence of elliptic dynamical systems and discuss a possible connection with mathematical physics.

On Perfect Semigroups

It is shown that in order to characterize perfect semigroups in general, it suffices to characterize perfect semigroups among semigroups S such that S is a subsemigroup of a rational vector space, carries the identical involution, and has an archimedean component H such that S = H ∪ {0}.

Limites dégénérées de séries discrètes, formes automorphes et variétés de Griffiths–Schmid: le cas du groupe U (2, 1)

Let , and π be an automorphic representation of G(A) whose archimedean component π∞ is a degenerate limit of discrete series; such a π never occurs in the cohomology (coherent or étale) of a Shimura variety. We show that however it does ‘appear’ in the coherent cohomology of some line bundle over an associated Griffiths-Schmid variety. Moreover we study cup products between such cohomology classes and some other automorphic cohomology classes and we prove some non-vanishing results.

On the congruence extension property for compact semigroups

A topological semigroupS is said to have thecongruence extension property (CEP) provided that for each closed subsemigroupT ofS and each closed congruence σ onT, σ can be extended to a closed congruence\(\bar \sigma \) onS. (That is,\(\bar \sigma \)∩(T xT=σ). The main result of this paper gives a characteriation of Γ-compact commutative archimedean semigroups with the congruence extension property (CEP). Consideration of this result was motivated by the problem of characterizing compact commutative semigroups with CEP as follows. It is well known that every commutative semigroup can be expressed as a semilattice of archimedean components each of which contains at most one idempotemt. The components of a compact commutative semigroup need not be compact (nor Γ-compact) as the congruence providing the decomposition is not necessarily closed. However, any component with CEP which is Γ-compact is characterized by the afore-mentioned result. Characterization of components of a compact commutative semigroup having CEP is a natural step towar characterization of the entire semigroup since CEP is a hereditary property. Other results prevented in this paper give a characterization of compact monothetic semigroups with CEP and show that Rees quotients of compact semigroups with CEP retain CEP.

The semigroup of nonempty finite subsets of rationals

Let Q be the additive group of rational numbers and let ℛ be the additive semigroup of all nonempty finite subsets of Q. For X ∈ ℛ, define AX to be the basis of 〈X − min(X)〉 and BX the basis of 〈max(X) − X〉. In the greatest semilattice decomposition of ℛ, let 𝒜(X) denote the archimedean component containing X. In this paper we examine the structure of ℛ and determine its greatest semilattice decomposition. In particular, we show that for X, Y ∈ ℛ, 𝒜(X) = 𝒜(Y) if and only if AX = AY and BX = BY. Furthermore, if X is a non‐singleton, then the idempotent‐free 𝒜(X) is isomorphic to the direct product of a power joined subsemigroup and the group Q.

Commutative semigroups having greatest regular images

Let S be a commutative semigroup. S has a greatest regular image if and only if each of its Archimedean components contains an idempotent. S has a greatest group-with-zero image if and only if S has precisely two Archimedean components and the upper component contains an idempotent. The existence and structure of these images and of greatest group images is related to tensor products.