Dirac cohomology for the degenerate affine Hecke Clifford algebra

We define an analogue of the Dirac operator for the degenerate affine Hecke-Clifford algebra. A main result is to relate the central characters of the degenerate affine Hecke-Clifford algebra with the central characters of the Sergeev algebra via Dirac cohomology. The action of the Dirac operator on certain modules is also computed. Results in this paper could be viewed as a projective version of the Dirac cohomology of the degenerate affine Hecke algebra.Comment: v2: 38 pages, corrected misprints; degenerates affine Hecke-Clifford algebras of other classical types are also considered; accepted by Transformation Group

Duality for Ext-groups and extensions of discrete series for graded Hecke algebras

In this paper, we study extensions of graded affine Hecke algebra modules. In particular, based on an explicit projective resolution on graded affine Hecke algebra modules, we prove a duality result for Ext-groups. This duality result with an Ind-Res resolution gives an algebraic proof of the fact that all higher Ext-groups between discrete series vanish.Comment: 36 pages, v2: Some results and proofs are improved. Sections 5,6,7 in v2 are new. Section 5 in v1 is completely removed and may appear elsewhere; v3: close to published versio

Densest columnar structures of hard spheres from sequential deposition

The rich variety of densest columnar structures of identical hard spheres inside a cylinder can surprisingly be constructed from a simple and computationally fast sequential deposition of cylinder-touching spheres, if the cylinder-to-sphere diameter ratio D is within [1,2.7013]. This provides a direction for theoretically deriving all these densest structures and for constructing such densest packings with nano-, micro-, colloidal or charged particles, which all self-assemble like hard spheres.Comment: 4 pages, 5 figure

First extensions and filtrations of standard modules for graded Hecke algebras

In this paper, we establish connections between the first extensions of simple modules and certain filtrations of of standard modules in the setting of graded Hecke algebras. The filtrations involved are radical filtrations and Jantzen filtrations. Our approach involves the use of information from the Langlands classification as well as some deeper analysis on structure of some modules. Such modules arise from the image of a Knapp-Stein type intertwining operator and is a quotient of a generalized standard module.Comment: v2: 38pages, minor changes and modification

Bernstein-Zelevinsky derivatives: a Hecke algebra approach

Let $G$ be a general linear group over a $p$-adic field. It is well known that Bernstein components of the category of smooth representations of $G$ are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of $G$ by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.Comment: 21 pages, extending part of arXiv:1605.05130. v2: a new appendix is added for the projectivity of the Gelfand-Graev representation, and references are update