stability in the presence of degeneracy and error estimation given an approximation to a local minimizer to a nonlinear optimization problem and to associated multipliers we obtain a tight error estimate in terms of the violation of the first order conditions our results apply to degenerate optimization problems where independence of the active constraint gradients and strict complementarity can be violated

modifying sqp for degenerate problems most local convergence analyses of the sequential quadratic programming sqp algorithm for nonlinear programming make strong assumptions about the solution namely that the active constraint gradients are linearly independent and that there are no weakly active constraints in this paper we establish a framework for variants of sqp that retain the characteristic superlinear convergence rate even when these assumptions are relaxed proving general convergence results and placing some recently proposed sqp variants in this framework we discuss the reasons for which implementations of sqp often continue to exhibit good local convergence behavior even when the assumptions commonly made in the analysis are violated finally we describe a new algorithm that formalizes and extends standard sqp implementation techniques and we prove convergence results for this method also

dimension formula for the affine deligne lusztig variety x mu b x μ b the study of certain union x mu b of affine deligne lusztig varieties in the affine flag varieties arose from the study of shimura varieties with iwahori level structure in this paper we give an explicit dimension formula for x mu b associated to sufficiently large dominant coweight mu

sqp methods for solving optimal control problems with control and state constraints adjoint variables sensitivity analysis and real time control parametric nonlinear optimal control problems subject to control and state constraints are studied two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which sqp methods provide efficient solution methods it is shown that sqp methods can be used also for a check of second order sufficient conditions and for a postoptimal calculation of adjoint variables in addition sqp methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters numerical sensitivity analysis is the basis for real time control approximations of perturbed solutions which are obtained by evaluating a first order taylor expansion with respect to the parameter the proposed numerical methods are illustrated by the optimal control of a low thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics the examples illustrate the robustness accuracy and efficiency of the proposed numerical algorithms

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

fourier transforms and frobenius eigenvalues for finite coxeter groups abstract lusztig s classification of the unipotent characters of a finite chevalley or steinberg group involves a certain non abelian fourier transformation we construct analogous transformations for the suzuki and ree groups based on a set of axioms derived from lusztig s theory of character sheaves we also determine fourier matrices for the spetses in the sense of broue michel and the second author associated with twisted dihedral groups this completes the determination of fourier matrices for all spetses associated with finite coxeter groups we end by collecting common properties of these fourier matrices and the eigenvalues of frobenius of character sheaves and unipotent characters

the development version of the chevie package of gap3 abstract i describe the current state of the development version of the chevie package which allows to transform the theories of coxeter groups reductive algebraic groups complex reflection groups hecke algebras braid monoids etc into actual computations examples are given showing the code to check some results of lusztig

stabilized sequential quadratic programming for optimization and a stabilized newton type method for variational problems the stabilized version of the sequential quadratic programming algorithm ssqp had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems when the lagrange multipliers associated to a solution are not unique superlinear convergence of ssqp had been previously established under the strong second order sufficient condition for optimality without any constraint qualification assumptions we prove a stronger superlinear convergence result than the above assuming the usual second order sufficient condition only in addition our analysis is carried out in the more general setting of variational problems for which we introduce a natural extension of ssqp techniques in the process we also obtain a new error bound for karush kuhn tucker systems for variational problems that holds under an appropriate second order condition

minimal length elements in twisted conjugacy classes of finite coxeter groups let w be a finite coxeter group and let f be an automorphism of w that leaves the set of generators of w invariant we establish certain properties of elements of minimal length in the f conjugacy classes of w that allow us to define character tables for the corresponding twisted iwahori hecke algebras these results are extensions of results obtained by geck and pfeiffer in the case where f is trivial

on the μ ordinary locus of a shimura variety abstract in this paper we study the ordinary locus of a shimura variety with parahoric level structure under the axioms in 12 we show that ordinary locus is a union of certain maximal ekedahl kottwitz oort rapoport strata introduced in 12 and we give criteria on the density of the ordinary locus

affineness of deligne lusztig varieties for minimal length elements abstract we prove that the deligne lusztig varieties associated to elements of the weyl group which are of minimal length in their twisted class are affine our proof differs from the proof of he and orlik rapoport and it is inspired by the case of regular elements which correspond to the varieties involved in broue s conjectures

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

the g stable pieces of the wonderful compactification let g be a connected simple algebraic group over an algebraically closed field there is a partition of the wonderful compactification of into finite many g stable pieces which was introduced by lusztig in this paper we will investigate the closure of any g stable piece in we will show that the closure is a disjoint union of some g stable pieces which was first conjectured by lusztig we will also prove the existence of cellular decomposition if the closure contains finitely many orbits

local behavior of an iterative framework for generalized equations with nonisolated solutions an iterative framework for solving generalized equations with nonisolated solutions is presented for generalized equations with the structure where is a multifunction and f is single valued the framework covers methods that at each step solve subproblems of the type the multifunction approximates f around s besides a condition on the quality of this approximation two other basic assumptions are employed to show q superlinear or q quadratic convergence of the iterates to a solution a key assumption is the upper lipschitz continuity of the solution set map of the perturbed generalized equation moreover the solvability of the subproblems is required conditions that ensure these assumptions are discussed in general and by means of several applications they include monotone mixed complementarity problems karush kuhn tucker systems arising from nonlinear programs and nonlinear equations particular results deal with error bounds and upper lipschitz continuity properties for these problems

an algorithm for degenerate nonlinear programming with rapid local convergence this paper describes and analyzes an algorithmic framework for solving nonlinear programming problems in which strict complementarity conditions and constraint qualifications are not necessarily satisfied at a solution the framework is constructed from three main algorithmic ingredients the first is any conventional method for nonlinear programming that produces estimates of the lagrange multipliers at each iteration the second is a technique for estimating the set of active constraint indices the third is a stabilized lagrange newton algorithm with rapid local convergence properties results concerning rapid local convergence and global convergence of the proposed framework are proved the approach improves on existing approaches in that less restrictive assumptions are needed for convergence and or the computational workload at each iteration is lower

lossless convexification of non convex optimal control problems for state constrained linear systems this paper analyzes a class of finite horizon optimal control problems with mixed non convex and convex control constraints and linear state constraints a convex relaxation of the problem is proposed and it is proved that a solution of the relaxed problem is also a solution of the original problem this process is called lossless convexification and its generalization for problems with state constraints is the primary contribution of the paper doing so enables the use of interior point methods of convex optimization to obtain global optimal solutions of the original non convex problem the approach is also demonstrated on example problems

character sheaves on certain spherical varieties abstract we study a class of perverse sheaves on some spherical varieties which include the strata of the de concini procesi completion of a symmetric variety this is a generalization of the theory of parabolic character sheaves

brief paper optimal control problems with a continuous inequality constraint on the state and the control we consider an optimal control problem with a nonlinear continuous inequality constraint both the state and the control are allowed to appear explicitly in this constraint by discretizing the control space and applying a novel transformation a corresponding class of semi infinite programming problems is derived a solution of each problem in this class furnishes a suboptimal control for the original problem furthermore we show that such a solution can be computed efficiently using a penalty function method on the basis of these two ideas an algorithm that computes a sequence of suboptimal controls for the original problem is proposed our main result shows that the cost of these suboptimal controls converges to the minimum cost for illustration an example problem is solved

global convergence of a nonsmooth newton method for control state constrained optimal control problems we investigate a nonsmooth newton method for the numerical solution of optimal control problems subject to mixed control state constraints the necessary conditions are stated in terms of a local minimum principle by use of the fischer burmeister function the local minimum principle is transformed into an equivalent nonlinear and nonsmooth equation in appropriate banach spaces this nonlinear and nonsmooth equation is solved by a nonsmooth newton s method we prove the global convergence and the locally superlinear convergence under certain regularity conditions the globalized method is based on the minimization of the squared residual norm numerical examples for the rayleigh problem conclude the article

a subalgebra of 0 hecke algebra let w i be a finite coxeter group in the case where w is a weyl group berenstein and kazhdan in a berenstein d kazhdan geometric and unipotent crystals ii from unipotent bicrystals to crystal bases in quantum groups in contemp math vol 433 amer math soc providence ri 2007 pp 13 88 constructed a monoid structure on the set of all subsets of i using unipotent linear bicrystals in this paper we will generalize this result to all types of finite coxeter groups including non crystallographic types our approach is more elementary based on some combinatorics of coxeter groups moreover we will calculate this monoid structure explicitly for each type

minimal length elements of coxeter groups let w be a coxeter group and wj a finite parabolic subgroup we present here a geometric new proof of x he s result on minimal length elements of an orbit of wj which acts on w in a certain way

a new variational result for quasi newton formulae the recent measure function of byrd and nocedal siam j numer anal 26 1989 pp 727 739 is considered and simple proofs of some of its properties are given it is then shown that the bfgs and dfp formulae satisfy a least change property with respect to this new measure

decomposition matrices for low rank unitary groups we study the decomposition matrices of the unipotent blocks of finite special unitary groups sun q for unitary primes larger than n up to few unknown entries we give a complete solution for n 2 10 we also prove a general result for two column partitions when divides q 1 this is achieved using projective modules coming from the adic cohomology of deligne lusztig varieties

robust receding horizon control of constrained nonlinear systems we present a method for the construction of a robust dual mode receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error the controller is dual mode in a neighborhood of the origin the control action is generated by a linear feedback controller designed for the linearized system outside this neighborhood receding horizon control is employed existing receding horizon controllers for nonlinear continuous time systems which are guaranteed to stabilize the nonlinear system to which they are applied require the exact solution at every instant of an optimal control problem with terminal equality constraints these requirements are considerably relaxed in the dual mode receding horizon controller presented in this paper stability is achieved by imposing a terminal inequality rather than an equality constraint only approximate minimization is required a variable time horizon is permitted robustness is achieved by employing conservative state and stability constraint sets thereby permitting a margin of error the resultant dual mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear constrained systems

a chebyshev technique for solving nonlinear optimal control problems a numerical technique for solving nonlinear optimal control problems is introduced the state and control variables are expanded in the chebyshev series and an algorithm is provided for approximating the system dynamics boundary conditions and performance index application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the chebyshev coefficients the optimum condition is obtained by applying the method of constrained extremum for linear quadratic optimal control problems the state and control variables are determined by solving a set of linear equations in the chebyshev coefficients applicability is illustrated with the minimum time and maximum radius orbit transfer problems

strictly transversal slices to conjugacy classes in algebraic groups abstract we show that for every conjugacy class o in a connected semisimple algebraic group g over an algebraically closed field of characteristic good for g one can find a special transversal slice to the set of conjugacy classes in g such that o intersects and dim o codim

the character sheaves on the group compactification abstract we give a definition of character sheaves on the group compactification which is equivalent to lusztig s definition in g lusztig parabolic character sheaves ii mosc math j 4 4 2004 869 896 we also prove some properties of the character sheaves on the group compactification

connected components of closed affine deligne lusztig varieties for rese fgln abstract let f be a p adic field and let e f be a finite unramified extension for the weil restriction res e f g l n of the general linear group g l n we determine the connected components of closed affine deligne lusztig varieties of arbitrary parahoric level

differential operators on quantized flag manifolds at roots of unity abstract the quantized flag manifold which is a q analogue of the ordinary flag manifold is realized as a non commutative scheme and we can define the category of d modules on it using the framework of non commutative algebraic geometry however when the parameter q is a root of unity lusztig s frobenius morphism allows us to handle d modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold in this paper we will show that this sheaf of rings on the ordinary flag manifold is an azumaya algebra over its center we also show that its restriction to certain subsets are split azumaya algebras these are analogues of some results of bezrukavnikov mirkovic rumynin on d modules on flag manifolds in positive characteristics

conjugacy classes and straight elements in coxeter groups as a byproduct of our methods we also obtain a characterisation of straight elements of w namely of those elements w w for which l wn nl w for any n z in particular we generalise previous characterisations of straight elements within the class of so called cyclically fully commutative cfc elements and we give a shorter and more transparent proof that coxeter elements are straight 2014 elsevier inc

deligne lusztig varieties and period domains over finite fields abstract we prove that the drinfeld space is essentially the only deligne lusztig variety which is at the same time a period domain over a finite field we also discuss an affineness criterion for deligne lusztig varieties and its relation to cohomology vanishing theorems

orthogonal projections on convex sets for newton like methods this paper develops the fixed almost least change secant methods for solving of nonlinear problems the presented results generalize and extend some results given by dennis and walker siam j numer anal 18 1981 pp 949 987 among other things multilevel update algorithms are described

on the affineness of deligne lusztig varieties abstract we prove that the deligne lusztig variety associated to minimal length elements in any conjugacy class of the weyl group is affine which was conjectured by orlik and rapoport in s orlik m rapoport deligne lusztig varieties and period domains over finite fields arxiv 0705 1646

method of evolving junctions a new approach to optimal control with constraints abstract we propose a new strategy called method of evolving junctions mej to compute the solutions for a class of optimal control problems with constraints on both state and control variables our main idea is that by leveraging the geometric structures of the optimal solutions we recast the infinite dimensional optimal control problem into an optimization problem depending on a finite number of points called junctions then using a modified gradient flow method whose dimension can change dynamically we find local solutions for the optimal control problem we also employ intermittent diffusion a global optimization method based on stochastic differential equations to obtain the global optimal solution we demonstrate via a numerical example that mej can effectively solve path planning problems in dynamical environments

green polynomials of weyl groups elliptic pairings and the extended dirac index abstract in this paper we give a uniform construction of irreducible genuine characters of the pin cover w of a weyl group w and put them into the context of theory of springer representations in the process we provide a direct connection between springer theory via green polynomials the irreducible representations of w and an extended dirac operator for graded hecke algebras we also introduce a q elliptic pairing for w with respect to the reflection representation v these constructions are of independent interest the q elliptic pairing is a generalization of the elliptic pairing of w introduced by reeder and it is also related to s kato s notion of graded kostka systems for the semidirect product a w c w s v

fundamental elements of an affine weyl group fundamental elements are certain special elements of affine weyl groups introduced by gortz haines kottwitz and reuman they play an important role in the study of affine deligne lusztig varieties in this paper we obtain characterizations of the fundamental elements and their natural generalizations we also derive an inverse to a version of newton hodge decomposition in affine flag varieties as an application we obtain a group theoretic generalization of oort s results on minimal p divisible groups and we show that in certain good reduction of pel shimura datum each newton stratum contains a minimal ekedahl oort stratum this generalizes a result of viehmann and wedhorn

cohomologie des varietes de deligne lusztig abstract we study the cohomology of deligne lusztig varieties with aim the construction of actions of hecke algebras on such cohomologies as predicted by the conjectures of broue malle and michel ultimately aimed at providing an explicit version of the abelian defect conjecture we develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them we are able to compute the cohomology of varieties associated to possibly twisted rank 2 groups and powers of the longest element w 0 some indeterminacies remain for g 2 we use this to construct hecke algebra actions on the cohomology of varieties associated to w 0 or its square for groups of arbitrary rank in the subsequent work f digne j michel endomorphisms of deligne lusztig varieties nagoya j math 183 2006 we construct actions associated to more general regular elements and we study their traces on cohomology

brief paper lossless convexification of a class of optimal control problems with non convex control constraints we consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost convex state constraints and non convex control constraints we propose a convex relaxation of the non convex control constraints and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem which is referred to as the lossless convexification of the optimal control problem the lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non convex optimal control problem the solution approach is demonstrated on a number of planetary soft landing optimal control problems

character sheaves on the semi stable locus of a group compactification abstract we study the intermediate extension of the character sheaves on an adjoint group to the semi stable locus of its wonderful compactification we show that the intermediate extension can be described by a direct image construction as a consequence we show that the ordinary restriction of a character sheaf on the compactification to a semi stable stratum is a shift of semisimple perverse sheaf and is closely related to lusztig s restriction functor from a character sheaf on a reductive group to a direct sum of character sheaves on a levi subgroup we also provide a conjectural formula for the boundary values inside the semi stable locus of an irreducible character of a finite group of lie type which gives a partial answer to a question of springer 2006 21 this formula holds for steinberg character and characters coming from generic character sheaves in the end we verify lusztig s conjecture lusztig 2004 16 12 6 inside the semi stable locus of the wonderful compactification

centralizers of iwahori hecke algebras to date integral bases for the centre of the iwahori hecke algebra of a finite coxeter group have relied on character theoretical results and the isomorphism between the iwahori hecke algebra when semisimple and the group algebra of the finite coxeter group in this paper we generalize the minimal basis approach of f1 to provide a way of describing and calculating elements of the minimal basis for the centre of an iwahori hecke algebra which is entirely combinatorial in nature and independent of both the above mentioned theories this opens the door to further generalization of the minimal basis approach to other cases in particular we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the coxeter group we show here that these properties hold for groups of type a and b giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of iwahori hecke algebra

minimal length elements in some double cosets of coxeter groups abstract we study the minimal length elements in some double cosets of coxeter groups and use them to study lusztig s g stable pieces and the generalization of g stable pieces introduced by lu and yakimov we also use them to study the minimal length elements in a conjugacy class of a finite coxeter group and prove a conjecture in m geck s kim g pfeiffer minimal length elements in twisted conjugacy classes of finite coxeter groups j algebra 229 2 2000 570 600

bar operators for quasiparabolic conjugacy classes in a coxeter group abstract the action of a coxeter group w on the set of left cosets of a standard parabolic subgroup deforms to define a module m j of the group s iwahori hecke algebra h with a particularly simple form rains and vazirani have introduced the notion of a quasiparabolic set to characterize w sets for which analogous deformations exist a motivating example is the conjugacy class of fixed point free involutions in the symmetric group deodhar has shown that the module m j possesses a certain antilinear involution called the bar operator and a certain basis invariant under this involution which generalizes the kazhdan lusztig basis of h the well known significance of this basis in representation theory makes it natural to seek to extend deodhar s results to the quasiparabolic setting in general the obstruction to finding such an extension is the existence of an appropriate quasiparabolic analogue of the bar operator in this paper we consider the most natural definition of a quasiparabolic bar operator and develop a theory of quasiparabolic kazhdan lusztig bases under the hypothesis that such a bar operator exists giving content to this theory we prove that a bar operator in the desired sense does exist for quasiparabolic w sets given by twisted conjugacy classes of twisted involutions finally we prove several results classifying the quasiparabolic conjugacy classes in a coxeter group

sharp primal superlinear convergence results for some newtonian methods for constrained optimization as is well known q superlinear or q quadratic convergence of the primal dual sequence generated by an optimization algorithm does not in general imply q superlinear convergence of the primal part primal convergence however is often of particular interest for the sequential quadratic programming sqp algorithm local primal dual quadratic convergence can be established under the assumptions of uniqueness of the lagrange multiplier associated to the solution and the second order sufficient condition at the same time previous primal q superlinear convergence results for sqp required strengthening of the first assumption to the linear independence constraint qualification in this paper we show that this strengthening of assumptions is actually not necessary specifically we show that once primal dual convergence is assumed or already established for primal superlinear rate one needs only a certain error bound estimate this error bound holds for example under the second order sufficient condition which is needed for primal dual local analysis in any case moreover in some situations even second order sufficiency can be relaxed to the weaker assumption that the multiplier in question is noncritical our study is performed for a rather general perturbed sqp framework which covers in addition to sqp and quasi newton sqp some other algorithms as well for example as a byproduct we obtain primal q superlinear convergence results for the linearly constrained augmented lagrangian methods for which no primal q superlinear rate of convergence results were previously available another application of the general framework is sequential quadratically constrained quadratic programming methods finally we discuss some difficulties with proving primal superlinear convergence for the stabilized version of sqp

geometric and homological properties of affine deligne lusztig varieties this paper studies ane deligne lusztig varieties x w b in the ane ag variety of a quasi split tamely ramied group we describe the geometric structure of x w b for a minimal length element w in the conjugacy class of an extended ane weyl group we then provide a reduction method that relates the structure of x w b for arbitrary elements w in the extended ane weyl group to those associated with minimal length elements based on this reduction we establish a connection between the dimension of ane deligne lusztig varieties and the degree of the class polynomial of ane

minimum time speed optimisation over a fixed path in this paper we investigate the problem of optimising the speed of a vehicle over a fixed path for minimum time traversal we utilise a change of variables that has been known since the 1980s although the resulting convexity of the problem was not noted until recently the contributions of this paper are three fold first we extend the convexification of the problem to a more general framework second we identify a wide range of vehicle models and constraints which can be included in this expanded framework third we develop and implement an algorithm that allows these problems to be solved in real time on embedded systems with a high degree of accuracy

on the local convergence of quasi newton methods for constrained optimization we consider the application of a general class of quasi newton methods to the solution of the classical equality constrained nonlinear optimization problem specifically we develop necessary and sufficient conditions for the q superlinear convergence of such methods and present a companion linear convergence theorem the essential conditions relate to the manner in which the hessian of the lagrangian function is approximated

a quasi newton strategy for the ssqp method for variational inequality and optimization problems the quasi newton strategy presented in this paper preserves one of the most important features of the stabilized sequential quadratic programming method the local convergence without constraint qualifications assumptions it is known that the primal dual sequence converges quadratically assuming only the second order sufficient condition in this work we show that if the matrices are updated by performing a minimization of a bregman distance which includes the classic updates the quasi newton version of the method converges superlinearly without introducing further assumptions also we show that even for an unbounded lagrange multipliers set the generated matrices satisfies a bounded deterioration property and the dennis more condition

representations of quantum groups at roots of unity whittaker vectors and q w algebras abstract let u e g be the standard simply connected version of the drinfeld jumbo quantum group at an odd m th root of unity e the center of u e g contains a huge commutative subalgebra isomorphic to the algebra z g of regular functions on a finite covering of a big cell in a complex connected simply connected algebraic group g with lie algebra g let v be a finite dimensional representation of u e g on which z g acts according to a non trivial character g given by evaluation of regular functions at g g then v is a representation of the finite dimensional algebra u g u e g u e g ker g we show that in this case under certain restrictions on m u g contains a subalgebra u g m of dimension m 1 2 dim o where o is the conjugacy class of g and u g m has a one dimensional representation c g we also prove that if v is not trivial then the space of whittaker vectors hom u g m c g v is not trivial and the algebra w g end u g u g u g m c g naturally acts on it which gives rise to a schur type duality between representations of the algebra u g and of the algebra w g called a q w algebra

lifting of elements of weyl groups abstract suppose g is a reductive algebraic group t is a cartan subgroup of g n norm t and w n t is the weyl group if w w has order d it is natural to ask about the orders lifts of w to n it is straightforward to see that the minimal order of a lift of w has order d or 2d but it can be a subtle question which holds we first consider the question of when w itself lifts to a subgroup of n in which case every element of w lifts to an element of n of the same order we then consider two natural classes of elements regular and elliptic in the latter case all lifts of w are conjugate and therefore have the same order we also consider the twisted case

murnaghan nakayama rules for characters of iwahori hecke algebras of classical type in this paper we give murnaghan nakayama type formulas for computing the irreducible characters of the iwahori hecke algebras of types an 1 bn and dn our method is a generalization of a derivation of the murnaghan nakayama formula for the irreducible characters of the symmetric group given by curtis greene greene s approach is to sum up the diagonal entries of the matrices of certain cycle permutations in young s seminormal representations the analogues of the young seminormal representations for the iwahori hecke algebras of types an 1 bn and dn were given by hoefsmit

hirota equations associated with simply laced affine lie algebras the cuspidal class e6 of e6 1 in this paper we derive hirota equations associated with the simply laced affine lie algebras g 1 where g is one of the simply laced complex lie algebras an dn e6 e7 or e8 defined by finite order automorphisms of g which we call lepowsky automorphisms in particular we investigate the hirota equations for lepowsky automorphisms of e6 defined by the cuspidal class e6 of the weyl group w e6 of e6 we also investigate the relationship between the lepowsky automorphisms of the simply laced complex lie algebras g and the conjugate canonical automorphisms defined by kac this analysis is applied to identify the canonical automorphisms for the cuspidal class e6 of e6