scholarly journals A VARIANT OF HARISH-CHANDRA FUNCTORS

2017 ◽  
Vol 18 (5) ◽  
pp. 993-1049
Author(s):  
Tyrone Crisp ◽  
Ehud Meir ◽  
Uri Onn
Keyword(s):  
Local Fields ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Test Case ◽  
Reductive Groups ◽  
Profinite Groups ◽  
Wide Range ◽  
Clifford Theory ◽  
Iwahori Subgroup ◽  
Suitable Restriction

Harish-Chandra induction and restriction functors play a key role in the representation theory of reductive groups over finite fields. In this paper, extending earlier work of Dat, we introduce and study generalisations of these functors which apply to a wide range of finite and profinite groups, typical examples being compact open subgroups of reductive groups over non-archimedean local fields. We prove that these generalisations are compatible with two of the tools commonly used to study the (smooth, complex) representations of such groups, namely Clifford theory and the orbit method. As a test case, we examine in detail the induction and restriction of representations from and to the Siegel Levi subgroup of the symplectic group $\text{Sp}_{4}$ over a finite local principal ideal ring of length two. We obtain in this case a Mackey-type formula for the composition of these induction and restriction functors which is a perfect analogue of the well-known formula for the composition of Harish-Chandra functors. In a different direction, we study representations of the Iwahori subgroup $I_{n}$ of $\text{GL}_{n}(F)$, where $F$ is a non-archimedean local field. We establish a bijection between the set of irreducible representations of $I_{n}$ and tuples of primitive irreducible representations of smaller Iwahori subgroups, where primitivity is defined by the vanishing of suitable restriction functors.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

scholarly journals The cohomology of semi-infinite Deligne–Lusztig varieties

2020 ◽  
Vol 2020 (768) ◽  
pp. 93-147
Author(s):  
Charlotte Chan
Keyword(s):  
Large Class ◽  
Finite Type ◽  
Cohomology Group ◽  
Geometric Realization ◽  
Local Fields ◽  
Irreducible Representations ◽  
Division Algebras ◽  
Full Generality ◽  
Homology Groups ◽  
Supercuspidal Representations

AbstractWe prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes {X_{h}}. Boyarchenko’s two conjectures are on the maximality of {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant {1/n} in the case {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of {X_{h}} attains its Weil–Deligne bound, so that the cohomology of {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

scholarly journals Alvis–Curtis duality for representations of reductive groups with Frobenius maps

2020 ◽  
Vol 32 (5) ◽  
pp. 1289-1296
Author(s):  
Junbin Dong
Keyword(s):  
Irreducible Representations ◽  
Reductive Groups ◽  
Infinite Type ◽  
Abstract Representations

AbstractWe generalize the Alvis–Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis–Curtis duality of infinite type, which we define in this paper, also interchanges the irreducible representations in the principal representation category.

scholarly journals Plate Finite Element with Physical Shape Functions: Correctness of the Formulation

2017 ◽  
Vol 63 (3) ◽  
pp. 19-37
Author(s):  
W. Gilewski ◽  
M. Sitek
Keyword(s):  
Finite Element ◽  
Good Choice ◽  
Test Case ◽  
Element Formulation ◽  
Structural Elements ◽  
Shape Functions ◽  
Solution Convergence ◽  
Wide Range ◽  
Physical Shape ◽  
Moderately Thick Plates

Abstract The formulation of a plate finite element with so called ‘physical’ shape functions is revisited. The derivation of the ‘physical’ shape functions is based on Hencky-Bolle theory of moderately thick plates. The considered finite element was assessed in the past, and the tests showed that the solution convergence was achieved in a wide range of thickness to in-plane dimensions ratios. In this paper a holistic correctness assessment is presented, which covers three criteria: the ellipticity, the consistency and the inf-sup conditions. Fulfilment of these criteria assures the existence of a unique solution, and a stable and optimal convergence to the correct solution. The algorithms of the numerical tests for each test case are presented and the tests are performed for the considered formulation. In result it is concluded that the finite element formulation passes every test and therefore is a good choice for modeling plate structural elements regardless of their thickness.

scholarly journals Inferring Biological Mechanisms by Data-Based Mathematical Modelling: Compartment-Specific Gene Activation during Sporulation in Bacillus subtilis as a Test Case

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Dagmar Iber
Keyword(s):  
Bacillus Subtilis ◽  
Gene Activation ◽  
Experimental Information ◽  
Specific Gene ◽  
Model Parameters ◽  
Test Case ◽  
Detailed Model ◽  
Wide Range ◽  
Technical Advances ◽  
Biological Functionality

Biological functionality arises from the complex interactions of simple components. Emerging behaviour is difficult to recognize with verbal models alone, and mathematical approaches are important. Even few interacting components can give rise to a wide range of different responses, that is, sustained, transient, oscillatory, switch-like responses, depending on the values of the model parameters. A quantitative comparison of model predictions and experiments is therefore important to distinguish between competing hypotheses and to judge whether a certain regulatory behaviour is at all possible and plausible given the observed type and strengths of interactions and the speed of reactions. Here I will review a detailed model for the transcription factor , a regulator of cell differentiation during sporulation in Bacillus subtilis. I will focus in particular on the type of conclusions that can be drawn from detailed, carefully validated models of biological signaling networks. For most systems, such detailed experimental information is currently not available, but accumulating biochemical data through technical advances are likely to enable the detailed modelling of an increasing number of pathways. A major challenge will be the linking of such detailed models and their integration into a multiscale framework to enable their analysis in a larger biological context.

NOx Emissions of a Premixed Partially Vaporized Kerosene Spray Flame

2006 ◽  
Author(s):  
Stefan Baessler ◽  
Klaus G. Mo¨sl ◽  
Thomas Sattelmayer
Keyword(s):  
Equivalence Ratio ◽  
Glass Tube ◽  
Gas Analysis ◽  
Test Case ◽  
Nox Emissions ◽  
Air Temperatures ◽  
Spray Flames ◽  
Wide Range ◽  
Phase Doppler Anemometer ◽  
Analysis System

An important question for future aero-engine combustors is how partial vaporization influences the NOx emissions of spray flames. In order to address this question an experimental study of the combustion of partially vaporized kerosene/air mixtures was conducted, which assesses the influence of the degree of fuel vaporization on the NOx emissions in a wide range of equivalence ratios covering the entire lean burning regime. The tests were performed at atmospheric pressure, inlet air temperatures of 313 to 376K, a reference mean air velocity of 1.35m/s, and equivalence ratios of 0.6, 0.7 and 0.9 using Jet A1 fuel. An ultrasonic atomizer was used to generate a fuel spray with a Sauter Mean Diameter of approximately 50μm. The spray and the heated air were mixed in a glass tube of 71mm diameter and a variable length of 0.5 to 1m. The temperature of the mixing air and the length of the preheater tube were used for the control of the degree of vaporization. Downstream of the vaporizing section, the mixture was ignited and the flame was stabilized with a hot wire ring that is electrically heated. For local exhaust measurements a temperature controlled suction probe in combination with a conventional gas analysis system were used. The vaporized ratio of the injected fuel was determined by a Phase Doppler Anemometer (PDA). In order to optimize the accuracy of these measurements, extensive validation tests with a patternator method were performed and a calibration curve was derived. The data collected in this study illustrates the effect of the vaporization rate Ψ upstream of the flame front on the NOx emissions, which changes with varying equivalence ratio and degree of vaporization. In the test case with low pre-vaporization, the equivalence ratio only has a minor influence on the NOx emissions. Experiments made with air preheat and higher degrees of vaporization show two effects: With increasing preheat air temperature, NOx emissions increase due to higher effective flame temperatures. However, with an increasing degree of vaporization, emissions become lower due to the dropping number and size of burning droplets, which act as hot spots. A correction for the effect of the preheat temperature was developed. It reveals the effect of the degree of pre-vaporization and shows that the NOx emissions are almost independent of Ψ for near-stoichiometric operation. At overall lean conditions the NOx emissions drop nonlinearly with Ψ. This leads to the conclusion that a high degree of vaporization is required in order to achieve substantial NOx abatement.

scholarly journals Composite UHBR Fan for Forced Response and Flutter Investigations

2021 ◽  
Author(s):  
Torben Eggers ◽  
Jens Friedrichs ◽  
Jan Goessling ◽  
Joerg R. Seume ◽  
Nunzio Natale ◽  
...  
Keyword(s):  
Forced Response ◽  
Operating Conditions ◽  
Rotor Blades ◽  
Future Research ◽  
Test Case ◽  
Composite Blade ◽  
Wide Range ◽  
Open Test ◽  
Performance Specifications ◽  
Transonic Fan

Abstract In the CA3ViAR (Composite fan Aerodynamic, Aeroelastic, and Aeroacoustic Validation Rig) project, a composite low-transonic fan is designed and tested. The aim is a scaled ultra-high bypass ratio (UHBR) fan with state-of-the-art aerodynamic performance and composite rotor blades, which features aeroelastic phenomena, e.g. forced response by inlet distortions and flutter, under certain operating points within the wind tunnel. In this paper, the aerodynamic and aeroelastic design process starting from the overall performance specifications to a threedimensional numerical model is described. A target of eigen-frequency and twist-to-plunge ratio is specified such that flutter occurs at desired operating conditions with a sufficient margin with respect to the working line. Different materials and layups of the composite blade are analyzed to reach the structural target. The fan should serve as an open test case to advance the future research on aerodynamic, aeroelastic, and aeroacoustic performance investigations in a wide range of operating conditions. A preliminary fan stage design is presented in this paper.

Multiphysics Numerical Investigation of an Aeronautical Lean Burn Combustor

2019 ◽  
Author(s):  
D. Bertini ◽  
L. Mazzei ◽  
A. Andreini ◽  
B. Facchini
Keyword(s):  
Fluid Dynamics ◽  
Pressure Ratio ◽  
Computational Cost ◽  
Engine Performance ◽  
Pollutant Emissions ◽  
Strong Interactions ◽  
Inlet Temperature ◽  
Test Case ◽  
Lean Burn ◽  
Wide Range

Abstract The importance of the combustion chamber has been underestimated for years by aeroengine manufacturers that focused their research efforts mainly on other components, such as compressor and turbine, to improve the engine performance. Nevertheless, stricter requirements on pollutant emissions have contributed to increase the interest on combustor development and, nowadays, new design concepts are widely investigated. To meet the goals of ACARE FlightPath 2050 and future ICAO-CAEP standards one of the most promising results is provided by the Lean Burn technology. As this combustion mode is based on a lean Primary Zone, the air devoted to liner cooling is restricted and advanced cooling systems must be exploited to obtain higher overall effectiveness. The pushing trends of Turbine Inlet Temperature and Overall Pressure Ratio in modern aeroengine are not supported enough by the development of materials, thus making the research branch of liner cooling increasingly relevant. In this context, Computational Fluid Dynamics is able to predict the flow field and the complex interactions between the involved phenomena, supporting the design of modern Lean Burn combustors in all stages of the process. RANS approaches provide a solution of the problem with low computational cost, but can lack in accuracy when the flow unsteadiness dominates the fluid dynamics and the strong interactions, as in aeroengine combustors. Even if steady simulations can be easily employed in the preliminary design, their inaccuracy can be detrimental for an optimized combustor design and Scale-Resolving methods should be preferred, at least, in the final stages. Unfortunately, having to deal with a multiphysics problem as Conjugate Heat Transfer (CHT) in presence of radiation, these simulations can become computationally expensive and some numerical treatments are required to handle the wide range of time and space scales in an unsteady framework. In the present work the metal temperature distribution is investigated from a numerical perspective on a full annular aeronautical lean burn combustor operated at real conditions. For this purpose, the U-THERM3D multiphysics tool was developed in ANSYS Fluent and applied on the test case. The results are compared against RANS and experimental data to assess the tool capability to handle the CHT problem in the context of scale-resolving simulations.

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