Archimedean Case Research Articles

Stable local trace formula

Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg’s trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.

Smooth transfer of Kloosterman integrals (the Archimedean case)

We establish the existence of a transfer, which is compatible with Kloosterman integrals, between Schwartz functions on ${\rm GL}_n(\Bbb{R})$ and Schwartz functions on the variety of non-degenerate Hermitian forms. Namely, we consider an integral of a Schwartz function on ${\rm GL}_n(\Bbb{R})$ along the orbits of the two sided action of the groups of upper and lower unipotent matrices twisted by a non-degenerate character. This gives a smooth function on the torus. We prove that the space of all functions obtained in such a way coincides with the space that is constructed analogously when ${\rm GL}_n(\Bbb{R})$ is replaced with the variety of non-degenerate hermitian forms. We also obtain similar results for $\frak{gl}_n(\Bbb{R})$. The non-Archimedean case was done by H. Jacquet ({\it Duke Math. J.}, 2003) and our proof is based on the ideas of this work. However we have to face additional difficulties that appear only in the Archimedean case. Those results are crucial for the comparison of the Kuznetsov trace formula and the relative trace formula of ${\rm GL}_n$ with respect to the maximal unipotent subgroup and the unitary group, as done by H. Jacquet, and by B. Feigon, E. Lapid, and O. Offen.

Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-archimedean dynamics

We determine when the equidistribution property for possibly moving targets holds for a rational function of degree more than one on the projective line over an algebraically closed field of any characteristic and complete with respect to a non-trivial absolute value. This characterization could be useful in the positive characteristic case. Based on the variational argument, we give a purely local proof of the adelic equidistribution theorem for possibly moving targets, which is due to Favre and Rivera-Letelier, using a dynamical Diophantine approximation theorem by Silverman and by Szpiro--Tucker. We also give a proof of a general equidistribution theorem for possibly moving targets, which is due to Lyubich in the archimedean case and due to Favre and Rivera-Letelier for constant targets in the non-archimedean and any characteristic case and for moving targets in the non-archimedean and 0 characteristic case.

Multiplicity one theorems: the Archimedean case

Let $G$ be one of the classical Lie groups $\GL_{n+1}(\R)$, $\GL_{n+1}(\C)$, $\oU(p,q+1)$, $\oO(p,q+1)$, $\oO_{n+1}(\C)$, $\SO(p,q+1)$, $\SO_{n+1}(\C)$, and let $G'$ be respectively the subgroup $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\oO(p,q)$, $\oO_n(\C)$, $\SO(p,q)$, $\SO_n(\C)$, embedded in $G$ in the standard way. We show that every irreducible Casselman-Wallach representation of $G'$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups $\GL_{n}(\R)\ltimes \oH_{2n+1}(\R)$, $\GL_{n}(\C)\ltimes \oH_{2n+1}(\C)$, $\oU(p,q)\ltimes \oH_{2p+2q+1}(\R)$, $\Sp_{2n}(\R)\ltimes \oH_{2n+1}(\R)$, $\Sp_{2n}(\C)\ltimes \oH_{2n+1}(\C)$, with their respective subgroups $\GL_{n}(\R)$, $\GL_{n}(\C)$, $\oU(p,q)$, $\Sp_{2n}(\R)$, $\Sp_{2n}(\C)$.

Uniqueness of Ginzburg-Rallis models: The Archimedean case

In this paper we prove the uniqueness of Ginzburg-Rallis models in the Archimedean case. As a key ingredient, we introduce a new descent argument based on two geometric notions attached to submanifolds, which we call metrical properness and unipotent χ \chi -incompatibility.

Range inclusion of operators on non-archimedean Banach space

In this paper, we establish some range inclusion theorems for non-archimedean Banach spaces over general valued fields. These theorems provide close relationship among range inclusion, majorization and factorization for bounded linear operators. It is found that these results depend strongly on a continuous extension property, which is always true in the classical archimedean case, but may fail to hold for the nonarchimedean setting. Several counterexamples are given to show that our results are sharp in some sense.

Uniqueness of Bessel Models: The Archimedean Case

In the archimedean case, we prove uniqueness of Bessel models for general linear groups, unitary groups and orthogonal groups.

Multiplicity one theorems

In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an admissible irreducible representation of GL(n+1), when restricted to GL(n) decomposes with multiplicity one. Similar Theorems are obtained for orthogonal or unitary groups.

Uniqueness of Shalika functionals: the archimedean case

Let F be either ℝ or ℂ. Let (π, V) be an irreducible admissible smooth Frechet representation of GL 2n (F). A Shalika functional φ: V → ℂ is a continuous linear functional such that for any g ∈ GL n (F), A ∈ Mat n×n (F) and v ∈ V we have φ [π( g 0 A g )ν] = exp(2π i Re Tr(g -1 A))φ(ν). In this paper we prove that the space of Shalika functionals on V is at most one-dimensional. For nonarchimedean F (of characteristic zero) this theorem was proved by Jacquet and Rallis.

On the asymptotics of Whittaker functions

We study the asymptotics of Whittaker functions on split groups and relate them to the cuspidal exponents of the representation.

On Reducibility and Unitarizability for Classical p-Adic Groups, Some General Results

Abstract. 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.

(GLn+1(F), GLn(F)) is a Gelfand pair for any local field F

AbstractLet F be an arbitrary local field. Consider the standard embedding $\mathrm {GL}_n(F) \hookrightarrow \mathrm {GL}_{n+1}(F)$ and the two-sided action of GLn(F)×GLn(F) on GLn+1(F). In this paper we show that any GLn(F)×GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), $\dim Hom_{\mathrm {GL}_n(F)}(E,\mathbb {C}) \leqslant 1$. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

Admissible metrics for line bundles on curves and abelian varieties over non-archimedean local fields

Following the approach in the archimedean case, we introduce the notion of admissible metrics for line bundles on curves and abelian varieties over non-archimedean local fields. Several properties of admissible metrics are considered and we show that this approach yields the same notion of admissible metrics over curves as doing harmonic analysis on the reduction graph of the curve.

Are copulas unimodal?

Three types of unimodality (central convex, block, and star) are considered and the corresponding sets of unimodal copulas determined. Examples of star unimodal copulas, absolutely continuous, with a nonnull singular part, and even singular, are given. Necessary and sufficient conditions for a diagonal to be the diagonal section of a star unimodal copula are also indicated. Attention is also paid to the Archimedean case.

Diophantine approximations for a constant related to elliptic functions

This paper is devoted to the study of rational approximations of the ratio η(λ)/ω(λ), where ω(λ) and η(λ) are the real period and real quasi-period, respectively, of the elliptic curve y2=x(x-1)(x-λ). Using monodromy principle for hypergeometric function in the logarithm case we obtain rational approximations of (η/ω)(λ) with λ∊Q and we shall find new measures of irrationality, both in the archimedean and non archimedean case.

Ideal Theory of Right Cones and Associated Rings

Right cones are semigroups with a left cancellation law such that for any two elementsa,bthere exists an elementcwithb=acora=bc. Valuation rings, cones of ordered or left ordered groups, semigroups of ordinal numbers, and Hjelmslev rings are examples. The ideal theory of these semigroups is described in terms of prime and completely prime ideals, and a classification of prime segments is given that can be used to solve a problem raised by Skornyakov. The Archimedean case can be dealt with in a satisfactory way with the help of Hölder's theorem. Right cones of rank 1 are classified. We then consider the problem of constructing for a given right coneHa right chain ringRwith the same right ideal and ideal structure asH.

Aggregation of decomposable measures with application to utility theory

This paper investigates the eventwise aggregations of decomposable measures preserving the same decomposable property. These operations are obtained by solving a functional equation closely related to the bisymmetry property. Known results for probability as well as possibility measures can be derived as particular cases of our approach. In addition, the unicity of weighted consensus functions is proved in the Archimedean case. An extension of Von Neumann-Morgenstern utility theory is outlined, where probabilities are changed into decomposable measures.

On a family of quasi-arithmetic means

The main goal of this paper is to solve the idempotency equationF(x, x) = x, x ∈ [0, 1] for a class of functions of the type “convex linear combination of at-norm and at-conorm”. In the non-strict Archimedean case and for eachk ∈ (0, 1), we obtain a unique solutionFk for this equation. While these functionsFk are not associative, we also prove that they satisfy the bisymmetry condition.

Analytic continuation of Archimedean Whittaker integrals

We prove the analytic continuation of a certain family of Archimedean integrals that arise as local factors of global L-functions associated to the standard representation of certain classical groups. INTRODUCTION In this paper we use a variation of Wallach [W] to prove the meromorphic continuation of a certain type of integrals. Our motivating example comes from the construction in [PS-R] generalizing Andrianov's construction of the L-function for Siegel modular forms [A]. The work in [PS-R] also explains the local-global structure of Andrianov's proof by introducing a new principle which enables us to get Euler products in some cases when there is no uniqueness of some appropriate Whittaker model. In this paper we are concerned with the meromorphic continutation (in the Archimedean case) of integrals of such type as those encountered in [PS-R]. Let us review the basic construction in [PS-R] so that we can keep this example in mind. Let G = Sp(n), the symplectic group of rank n, regarded as an algebraic over Q (we take Q instead of a global field just for the sake of simplicity). Assume that n is even. Let T E M(n, Q) be symmetric and nondegenerate, and OT the orthogonal group of T. Consider the oscillator representation w, which corresponds to the reductive dual pair (OT, Sp(n)) and a fixed nontrivial character ,v of Q\A (A is the ring of adeles of Q); w may be realized on S(M(n A)), the space of Schwartz-Bruhat functions on M(n, A) . For q in S(M(n A)), consider the 0-series 0 (g) =) E (g)q$(x), g E G(A). xEM(n ,Q) Received by the editors March 15, 1987 and, in revised form, Feburary 23, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 OD40; Secondary 1 OD 12. Both authors were supported by US-Israel Binational Grant 84-00139. The first author was also supported by NSF grant DMS-8401947. ? 1989 American Mathematical Society 0002-9939/89 $1.00 + $ 25 per page

Two integrals and some modified versions — Critical remarks

The aim of this paper is to discuss different constructions of integrals (Sections 3 and 4) based on @?-decomposable measures (Section 1). According to the classification of the continuous t-conorms @? in essentially two types namely v and Archimedean t-conorms, there exist mainly two types of integrals namely the constructions of Sugeno (Section 3) and of Weber (Section 4). Further constructions corresponding to the Archimedean case result to be special cases or not well defined (Section 4). In all cases a crucial property is some restricted distribution law for the pair (@?, ) with an appropriate operation(Section 2). Some applications shall illustrate the use of the two integrals (Section 5).