Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthogonal groups. All may be obtained from suitable matrix groups G by taking G′/Z(G′). Our main aims again are to define these groups, prove they are simple, and describe their automorphisms, subgroups and covering groups.

The classical groups

Page 55, proof of Lemma 3.13. This proof is incorrect as it stands because it involves an interchange of limits that has not been justified. A naive attempt to fix the proof might involve assuming that the given representation is continuous into the uniform operator topology, but this assumption is not necessarily valid. Instead, a different approach is needed, and such an approach may be found in the paper by M. Welleda Baldoni, “General representation theory of real reductive groups,” Proc. Sumposia in Pure Math. 61 (1997), 61–72. The relevant proof is the proof of Proposition 1 on page 63. The argument is straightforward enough, but it makes use of the fact that g → π(g)v is of class C1 if and only if for each w in the Hilbert space, g → hπ(g)v, wi is of class C1. She attributes this result to Grothendieck and points to J. B. Neto, Trans. Amer. Math. Soc. 111 (1964), 381–391, for “discussion and results.” Reading Proposition 1 and its proof, one sees that by approaching the development in a slightly different order, one gets around the temptation to use the uniform operator topology and obtains a valid proof of Proposition 3.14.

REPRESENTATION THEORY OF SEMISIMPLE GROUPS: An Overview Based on Examples

* Introduction* Symmetry breaking for the spherical principal series representations* Symmetry breaking operators* More about principal series representations* Double coset decomposition $P'\setminus G/P$* Differential equations satisfied by the distribution kernels of symmetry breaking operators $K$-finite vectors and regular symmetry breaking operators $\widetilde {\mathbb {A}} _{\lambda , \nu }$* Meromorphic continuation of regular symmetry breaking operators ${K}_{{\lambda },{\nu }}^{\mathbb {A}}$* Singular symmetry breaking operator $\widetilde {\mathbb {B}} _{\lambda ,\nu }$* Differential symmetry breaking operators* Classification of symmetry breaking operators* Residue formulae and functional identities* Image of symmetry breaking operators* Application to analysis on anti-de Sitter space* Application to branching laws of complementary series* Appendix* References* List of symbols

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Symmetry Breaking for Representations of Rank One Orthogonal Groups

Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits a Kahler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kahler manifold (M4n, g). If Z also admits a second complex contact structure , then Z=CP2n+1. As an application, we give several new characterizations of the Riemannian manifold HPn= Sp(n+1)/(Sp(n)×Sp(1)).

Fano Manifolds, Contact Structures, and Quaternionic Geometry

In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1 vertical bar 2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator.

https://biblio.ugent.be/publication/3032075/file/3032076.pdf

Dunkl operators and a family of realizations of osp(1|2)

In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1|2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator.

We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules [T. Kobayashi, this http URL {Transf. Groups (2012)}], we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan--Holder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant differential operators [Juhl, this http URL {Progr. Math. 2009}] and its generalizations. This article is the first in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries.

Branching laws for Verma modules and applications in parabolic geometry. I

https://www.ms.u-tokyo.ac.jp/~toshi/texpdf/tk2013i-Fm-I_final_revision_2.pdf

The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|2 A deformation of the Laplace operator was recently introduced by ChF Dunkl It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group) It was shown by B Orsted and S Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P Cerejeiras, U Kahler and G Ren The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs

Petr Somberg

Papers

Dunkl operators and a family of realizations of osp(1|2)

Dunkl operators and a family of realizations of osp(1|2)

Branching laws for Verma modules and applications in parabolic geometry. I

Branching laws for Verma modules and applications in parabolic geometry. I

The Howe Duality for the Dunkl Version of the Dirac Operator