Hecke algebras with unequal parameters and Vogan's left cell invariants

In 1979, Vogan introduced a generalised $\\tau$ -invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells in the sense of Kazhdan and Lusztig. Although it is not a complete invariant, it is extremely useful in describing left cells. Here, we propose a general framework for defining such invariants which also applies to Hecke algebras with unequal parameters.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1405.573

Dirac cohomology, elliptic representations and endoscopy

The first part (Sections 1-6) of this paper is a survey of some of the recent developments in the theory of Dirac cohomology, especially the relationship of Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology; the second part (Sections 7-12) is devoted to understanding the unitary elliptic representations and endoscopic transfer by using the techniques in Dirac cohomology. A few problems and conjectures are proposed for further investigations.Comment: This paper will appear in `Representations of Reductive Groups, in Honor of 60th Birthday of David Vogan', edited by M. Nervins and P. Trapa, published by Springe

Transfer of K-types on local theta lifts of characters and unitary lowest weight modules

In this paper we study representations of the indefinite orthogonal group O(n,m) which are local theta lifts of one dimensional characters or unitary lowest weight modules of the double covers of the symplectic groups. We apply the transfer of K-types on these representations of O(n,m), and we study their effects on the dual pair correspondences. These results provide examples that the theta lifting is compatible with the transfer of K-types. Finally we will use these results to study subquotients of some cohomologically induced modules

Highest weight categories arising from Khovanov's diagram algebra II: Koszulity

This is the second of a series of four articles studying various generalisations of Khovanov's diagram algebra. In this article we develop the general theory of Khovanov's diagrammatically defined "projective functors" in our setting. As an application, we give a direct proof of the fact that the quasi-hereditary covers of generalised Khovanov algebras are Koszul.Comment: Minor changes, extra sections on Kostant modules and rigidity of cell modules adde

Mechanical Testing of 3D Printed Prosthetics

The Rapid Orthotics for CURE Kenya team as a whole aims to empower the orthopedic technicians in the CURE Kenya hospital by creating, optimizing, and testing 3D printed prosthetics and orthotics. Our team started in 2016 by creating a 3D printing process for below the knee prosthetic sockets. Since then, we had adapted to the hospital\u27s needs over the years, expanding the capabilities of the system itself. Presently, a section of our team has worked specifically with these leg sockets to ensure the safety and functionality for patients. They have done testing to make sure the sockets are strong enough and to make sure the silicone liners are safe for use in developing countries. In addition to safety testing, over the years we have created ankle-foot orthotics and prosthetic hands. The design part of our team works to create new 3D printed devices to help our clients reach more patients. By 2024 we hope to fully integrate our expanded system in the orthopedic workshop in Kijabe, Kenya.https://mosaic.messiah.edu/engr2020/1018/thumbnail.jp

Longitudinal Examination of Everyday Executive Functioning in Children With ASD: Relations With Social, Emotional, and Behavioral Functioning Over Time

Executive functioning (EF) deficits are well-documented in Autism Spectrum Disorder (ASD), yet little is known about the longitudinal trajectory of “everyday” EF and links to social, emotional and behavioral outcomes in ASD. This study examined the profile of everyday EF utilizing parent-reported measures over 2 years, and explored whether prior estimates of EF were related to later co-morbid psychopathology and social functioning in 39 children with ASD and 34 typically developing (TD) children (ages 7–14 years). According to parent reports, children with ASD had impaired scores of EF in all domains at both time points, and showed no significant improvement across 2 years, compared to controls. Regression analyses showed that prior estimates of behavior regulation difficulties at time 1 uniquely predicted later emotional (i.e., symptoms of anxiety/depression) and behavioral (i.e., oppositionality/aggressiveness) problems in children with ASD 2 years later. Furthermore, an improvement of metacognitive skills predicted a reduction of social difficulties over 2 years in ASD. These results imply that EF may be a potential target of intervention for preventing and reducing co-morbid psychopathology and promoting social competence in youth with ASD. Furthermore, the findings that EF related to behavior is more critical for later emotional and behavioral functioning, whereas EF related to cognition is more critical for social functioning, indicates that it may be beneficial to tailor treatment. Future studies investigating the effectiveness of EF-based interventions in improving the cognitive, psychological and social outcomes in ASD are of high priority

The Relationship Between Daytime Sedentary Behavior and Sleep Health in Desk-based Workers

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On unitarizability in the case of classical p-adic groups

In the introduction of this paper we discuss a possible approach to the unitarizability problem for classical p-adic groups. In this paper we give some very limited support that such approach is not without chance. In a forthcoming paper we shall give additional evidence in generalized cuspidal rank (up to) three.Comment: This paper is a merged and revised version of ealier preprints arXiv:1701.07658 and arXiv:1701.07662. The paper is going to appear in the Proceedings of the Simons Symposium on Geometric Aspects of the Trace Formul

Derivatives for smooth representations of GL(n,R) and GL(n,C)

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12].Comment: First version of this preprint was split into 2. The proofs of two theorems which are technically involved in analytic difficulties were separated into "Twisted homology for the mirabolic nilradical" preprint. All the rest stayed in v2 of this preprint. v3: version to appear in the Israel Journal of Mathematic