Fourier inversion on a reductive symmetric space

Let X be a semisimple symmetric space. In previous papers, [8] and [9], we have dened an explicit Fourier transform for X and shown that this transform is injective on the space C 1 c (X) ofcompactly supported smooth functions on X. In the present paper, which is a continuation of these papers, we establish an inversion formula for this transform

Оцінка і її зв’язок з емоційністю, експресивністю, модальністю

Простежується взаємозв’язок оцінки з іншими мовними категоріями: емоційністю, експресивністю і суб’єктивною модальністю. Встановлено точки перетину оцінки з цими категоріями, визначено параметри їх розмежування. Ключові слова: оцінний компонент, категорія, емоційність, експресивність, суб’єктивна модальність, цінність.Рассматривается взаимосвязь категории оценки с другими языковыми категориями: эмоциональностью, экспрессивностью и субъективной модальностью. Установлены точки пересечения оценки с этими категориями, а также определены параметры их различия. Ключевые слова: оценочный компонент, категория, эмоциональность, экспрессивность, субъективная модальность, ценность.Correlation of an assessment category and other language categories is considered: emotionality, expressiveness and subjective modality. Common aspects regarded in the framework of assessment category in it’s correlation with the other ones have been distinguished, and besides, distinction parameters have been defined. Keywords: assessment component, category, emotionality, expressiveness, subjective modality, value

Convexity for invariant differential operators on semisimple symmetric spaces

Let X = G=H be a homogeneous space of a Lie group G, and let D : C 1 (X) ! C 1 (X) be a non-trivial G-invariant dierential operator. One of the natural questions one can ask for the operator D is whether it is solvable, in the sense that DC 1 (X) =C 1 (X). If G is the group of translations of X = R n and H is trivial, then D has constant coecients, and it is a well known result of Ehrenpreis and Malgrange that hence D is solvable

Multiplicities in the Plancherel decomposition of a semisimple symmetric space

Let G=H be a semisimple symmetric space, where G is a connected semisimple Lie group provided with an involution ?; and H = G ? is the subgroup of xed points for ?: Assume moreover that G is linear (for the purpose of the introduction, the assumptions on G and H are stronger than necessary). Then G has a ?-stable maximal compact subgroup K; the associated Cartan involution commutes with ?: Let g = h + q and g = k +p be the decompositions of the Lie algebra g induced by ? and , then h is the Lie algebra of H and k is the Lie algebra of K

Fourier transforms on a semisimple symmetric space

Let G=H be a semisimple symmetric space, that is, G is a connected semisimple real Lie group with an involution ?, and H is an open subgroup of the group of xed points for ? in G. The main purpose of this paper is to study an explicit Fourier transform on G=H. In terms of general representation theory the (`abstract') Fourier transform of a compactly supported smooth function f 2 C 1 c (G=H) is given by (see [6]) (1) ^ f(?)() =?(f) = ZG=H f(x)?(x) dx; for (?; H?) a unitary irreducible representation of G and 2 (H ? an H-invariant distribution vector for ?. Here dx is the invariant measure on G=H. Thus ^ f f?)() is a smooth vector for H?, depending linearly on . Our goal is to obtain an explicit version of the restriction of this Fourier transform to representations (?; H?) in the (minimal) unitary principal series (??;; H?;) for G=H, under the assumption that the center of G is nite. In the sequel [10] to this paper it is proved that a function f 2 C 1 c (G=H) is uniquely determined by the restriction of ^ f to this series (a priori it is known that f is determined by ^ f )

Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries

Background Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres. Methods This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries. Results In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia. Conclusion This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries

The Action of Intertwining Operators on Spherical Vectors in the Minimal Principal Series of a Reductive Symmetric Space

this paper was (implicitly) announced some time ago in the survey paper [6] (cf. Theorem 11). In recent work ([7]) P. Delorme has established MaassSelberg relations in the more general context of Eisenstein integrals for non-minimal oe`

Induced representations and the Langlands classification

. In these lecture notes 1 we discuss the concept of induction and some of its applications to the representation theory of a real semisimple Lie group. In particular, we give an introduction to parabolic induction, Bruhat theory, the asymptotic behavior of matrix coefficients, the subrepresentation theorem, characterization of discrete series and tempered representations, and, finally, the Langlands classification of irreducible admissible representations. 1 Induced representations 1.1 Homogeneous vector bundles The process of induction allows us to create representations of a Lie group, starting from representations of a subgroup. We start by recalling the notion of an associated vector bundle. Let G be a Lie group, H a closed subgroup, and (¸; V ) a finite dimensional continuous representation of H (here and in the following, representation spaces are always assumed to be complex linear). The group H acts on the product G \Theta V by h \Delta (g; v) = (gh \Gamma1 ; ¸(h)v); ..

The Plancherel formula for a reductive symmetric space.

In these notes, G is a reductive Lie group, i.e., its Lie algebra g is a real reductive Lie algebra. At a later stage we will impose the restrictive condition that G belongs to a certain class R, see appendix