On the symmetric square Unstable twisted characters

We provide a purely local computation of the (elliptic) twisted (by “transpose-inverse”) character of the representationπ=I(1) of PGL(3) over ap-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads — see [FK] — to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals. The new case in [FK] is the unstable one. A direct local proof of the fundamental lemma is given in [V].     

A simple trace formula

The Selberg trace formula is of unquestionable value for the study of automorphic forms and related objects. In principal it is a simple and natural formula, generalizing the Poisson summation formula, relating traces of convolution operators with orbital integrals. This paper is motivated by the belief that such a fundamental and natural relation should admit asimple and short proof. This is accomplished here for test functions with a single supercusp-component, and another component which is spherical and “sufficiently-admissible” with respect to the other components. The resulting trace formula is then use to sharpen and extend the metaplectic correspondence, and the simple algebras correspondence, of automorphic representations, to the context of automorphic forms with asingle supercuspidal component, over any global field. It will be interesting to extend these theorems to the context of all automorphic forms by means of a simple proof. Previously a simple form of the trace formula was known for test functions with two supercusp components; this was used to establish these correspondences for automorphic forms with two supercuspidal components. The notion of “sufficiently-admissible” spherical functions has its origins in Drinfeld's study of the reciprocity law for GL(2) over a function field, and our form of the trace formula is analogous to Deligne's conjecture on the fixed point formula in étale cohomology, for a correspondence which is multiplied by by a sufficiently high power of the Frobenius, on a separated scheme of finite type over a finite field. Our trace formula can be used (see [FK′]) to prove the Ramanujan conjecture for automorphic forms with a supercuspidal component on GL(n) over a function field, and to reduce the reciprocity law for such forms to Deligne's conjecture. Similar techniques are used in ['t'F] to establish base change for GL (n) in the context of automorphic forms with a single supercuspidal component. They can be used to give short and simple proofs of rank one lifting theorems forarbitrary automorphic forms; see [″F] for base change for GL(2), [F′] for base change forU(3), and [′F′] for the symmetric square lifting from SL(2) to PGL(3). Partially supported by NSF grants.     

On the metaplectic analog of Kazhdan’s “endoscopic” lifting

A simple “twisted” form of the Selberg trace formula, due to Kazhdan, is used to prove a metaplectic analog of Kazhdan’s lifting of grossencharakters of a cyclic extension of degreer to automorphic representations of GL(r).     

Heteroclinic cycles in bifurcation problems withO(3) symmetry and the spherical Bénard problem

Summary It has been known since a paper of Armbruster and Chossat ([AC91]) that robust heteroclinic cycles between equilibria can bifurcate in differential systems which are invariant under the action of the groupO(3) defined as the sum of its “natural” irreducible representations of degrees 1 and 2 (i.e., of dimensions 3 and 5). Moreover, these cycles can be seen numerically in the simulation of the amplitude equations resulting from a center manifold reduction of the Bénard problem in a nonrotating spherical shell with suitable aspect ratio ([FH86]). In the present work we first generalize the results of [AC91] to the interactions of irreducible representations of degrees ℓ and ℓ+1 for any ℓ>0. Heteroclinic cycles of various types are shown to exist under certain “generic” conditions and are classified. We show in particular that these conditions are satisfied in most cases when the differential system proceeds from a ℓ, ℓ+1 mode interaction bifurcation in the spherical Bénard problem.     

Boundary values of holomorphic functions, singular unitary representations ofO(p,q), and their limits asq→∞

Let Ω be a bounded circular domain in ℂ ^N , let M be a submanifold in the boundary of Ω, and let H be a Hilbert space of holomorphic functions in Ω. We show that, under certain conditions stated in terms of the reproducing kernel of the space H, the restriction operator to the submanifold M is well defined for all functions from H. We apply this result to constructing a family of “singular” unitary representations of the groups SO(p,q). The singular representations arise as discrete components of the spectrum in the decomposition of irreducible unitary highest weight representations of the groups U(p,q) restricted to the subgroups SO(p,q). Another property of the singular representations is that they persist in the limit as q→∞. Bibliography: 70 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 9–91. Translated by B. Bekker.     

The moduli space of (3,3,3) trilinear forms

LetU, V andW be three dimensional vector spaces over ∉ (or an alebraically closed field with characteristic not equal to 2 or 3). We prove that the moduli space of trilinear forms onU ^* ⊗V ^* ⊗W ^* is isomorphic to ℙ² by applying Geometric Invariant Theory to the action ofPGL(U)×PGL(V)×PGL(W) on ℙU⊗V⊗W).     

Finite Factor Representations of 2-Step Nilpotent Groups and the Orbit Method

In this paper, we describe factor representations of discrete 2-step nilpotent groups with 2-divisible center in the spirit of the orbit method. We show that some standard theorems of the orbit method are valid for these groups. In the case of countable 2-step nilpotent groups we explain how to construct a factor representation starting with an orbit of the “coadjoint representation.” We also prove that every factor representation (more precisely, every trace) can be obtained by this construction, and prove a theorem on the decomposition of a factor representation restricted to a subgroup. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 120–140.     

Cross ratios,surface groups,PSL(<Emphasis Type="Italic">n</Emphasis>,ℝ) and diffeomorphisms of the circle

This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into PSL(n,ℝ) – known as the n-Hitchin component – to a subset of the set of cross ratios on the boundary at infinity of the group. Similarly, we study some representations into associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains all n-Hitchin components as well as the set of negatively curved metrics on the surface.     

On the reduction to a complete class in multiple decision problems (2)

Summary As a criterion for the reduction to a complete class of decision rule in case where actions, samples and states are finite in number, “regret-relief ratio” criterion and “incremental loss-gain ratio” criterion were introduced in 2-state of nature case [2]. In this paper, “generalized regret-relief ratio” criterion ink-state of nature case is introduced as an extension of “regret-relief ratio” criterion and its usefulness is shown with an example. The Institute of Statistical Mathematics     

Challenging Students to View the Concept of Area in Triangles in a Broad Context: Exploiting the Features of Cabri-II

This study focuses on the constructions in terms of area and perimeter in equivalent triangles developed by students aged 12–15 years-old, using the tools provided by Cabri-Geometry II [Labore (1990). Cabri-Geometry (software), Université de Grenoble]. Twenty-five students participated in a learning experiment where they were asked to construct: (a) pairs of equivalent triangles “in as many ways as possible” and to study their area and their perimeter using any of the tools provided and (b) “any possible sequence of modifications of an original triangle into other equivalent ones”. As regards the concept of area and in contrast to a paper and pencil environment, Cabri provided students with different and potential opportunities in terms of: (a) means of construction, (b) control, (c) variety of representations and (d) linking representations, by exploiting its capability for continuous modifications. By exploiting these opportunities in the context of the given open tasks, students were helped by the tools provided to develop a broader view of the concept of area than the typical view they would construct in a typical paper and pencil environment.     

Algebraic and combinatorial properties of zircons

In this paper we introduce and study a new class of posets, that we call zircons, which includes all Coxeter groups partially ordered by Bruhat order. We prove that many of the properties of Coxeter groups extend to zircons often with simpler proofs: in particular, zircons are Eulerian posets and the Kazhdan-Lusztig construction of the Kazhdan-Lusztig representations can be carried out in the context of zircons. Moreover, for any zircon Z, we construct and count all balanced and exact labelings (used in the construction of the Bernstein-Gelfand-Gelfand resolutions in the case that Z is the Weyl group of a Kac-Moody algebra). Partially supported by the program “Gruppi di trasformazioni e applicazioni”, University of Rome “La Sapienza”. Part of this research was carried out while the author was a member of the Institut Mittag-Leffler of the Royal Swedish Academy of Sciences.     

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.     

Tableau complexes

Let X, Y be finite sets and T a set of functions X → Y which we will call “ tableaux”. We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such tableau complexes have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions [BP79]. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch’s set-valued semistandard tableaux. One vertex decomposition of this “Young tableau complex” parallels Lascoux’s transition formula for vexillary double Grothendieck polynomials [La01, La03]. Consequently, we obtain formulae (both old and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs [Wa85] and Buch [Bu02], each of which implies the classical tableau formula for Schur polynomials.     

Provability interpretations of modal logic

We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev ^* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.     

Langlands duality for representations of quantum groups

We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

Distribution results for lattices in SL(2,ℚ p )

In this paper we study the ergodic properties of the linear action of lattices Γ of SL(2,ℚ_p) on ℚ_p × ℚ_p and distribution results for orbits of Γ. Following Serre, one can define a “geodesic flow” for an associated tree (actually associated to GL(2,ℚ_p)). The approach we use is based on an extension of this approach to “frame flows” which are a natural compact group extension of the geodesic flow.     

Spectra of elements in the group ring of SU(2)

We present a new method for establishing the “gap” property for finitely generated subgroups of SU(2), providing an elementary solution of Ruziewicz problem on S² as well as giving many new examples of finitely generated subgroups of SU(2) with an explicit gap. The distribution of the eigenvalues of the elements of the group ring R[SU(2)] in the N-th irreducible representation of SU(2) is also studied. Numerical experiments indicate that for a generic (in measure) element of R[SU(2)], the “unfolded” consecutive spacings distribution approaches the GOE spacing law of random matrix theory (for N even) and the GSE spacing law (for N odd) as N→∞; we establish several results in this direction. For certain special “arithmetic” (or Ramanujan) elements of R[SU(2)] the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction. Received June 1, 1998 / final version received September 8, 1998