random field reconstruction with quantization in wireless sensor networks we develop new and novel algorithms for spatial field reconstruction and spatial exceedance level estimation we consider spatial physical phenomena which are partially observed by a wireless sensor network we focus on the practical scenario in which due to bandwidth and power constraints the observations at the sensors are quantized and transmitted over imperfect wireless channels we first develop the spatial best linear unbiased estimator s blue algorithm for the spatial field reconstruction estimation we then develop an algorithm that is based on a multivariate series expansion approach resulting in a saddle point type approximation to solve both problems of spatial field reconstruction and exceedance level estimation we derive the posterior cramer rao lower bound pcrlb and quantify the achievable mse in the estimation the estimation accuracy of the proposed algorithms is quantified via numerical comparisons between the two algorithms and the pcrlb

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

complex geometry and the asymptotics of harish chandra modules for real reductive lie groups i let g be a connected semisimple real matrix group it is now apparent that the representation theory of g is intimately connected with the complex geometry of the flag variety 9 by studying appropriate orbit structures on 9 we are naturally led to representation theory in the category of harish chandra modules x or the representation theory of category 9 the jacquet functor j f 9 has proved a useful tool in converting yfw problems into 9 problems which are often more tractable in this paper we advance the philosophy that the complex geometry of 9 associated to yf and 9 interacts in a natural way with the functor j leading to deep new information on the structure of jacquet modules this in turn gives new insight into the structure of certain nilpotent cohomology groups associated to harish chandra modules our techniques are based upon many of the ideas present in the proof of the kazhdan lusztig conjectures and bernstein s proof of the jantzen conjecture

birational sheets in reductive groups we define the group analogue of birational sheets a construction performed by losev for reductive lie algebras for g semisimple simply connected we describe birational sheets in terms of lusztig spaltenstein induction and we prove that they form a partition of g and that they are unibranch varieties with smooth normalization by means of a local study

arithmetic harmonic analysis on character and quiver varieties ii we study connections between the topology of generic character varieties of fundamental groups of punctured riemann surfaces macdonald polynomials quiver representations hilbert schemes on c x x c x modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups c 2012 elsevier inc all rights reserved

extrapolation methods for vector sequences this is an expository paper that describes and compares five methods for extrapolating to the limit or anti limit of a vector sequence without explicit knowledge of the sequence generator the methods are the minimal polynomial extrapolation mpe studied by cabay and jackson mesina and skelboe the reduced rank extrapolation rre of eddy which we show to be equivalent to mesina s version of mpe the vector and scalar versions of the epsilon algorithm vea sea introduced by wynn and extended by brezinski and gekeler and the topological epsilon algorithm tea of brezinski we cover the derivation and error analysis of iterated versions of the algorithms as applied to both linear and nonlinear problems and we show why these versions tend to converge quadratically we also present samples from extensive numerical testing that has led us to the following conclusions a tea in spite of its role as a theoretical link between the polynomial type and the epsilon type methods has no practical appl

green functions and character sheaves purely combinatorial way using earlier ideas of p hall in the general case they were introduced 3 in terms of l adic cohomology as values of certain virtual representations of r o t is a maximal torus defined over fq at unipotent elements the characters of rg 0 at arbitrary elements were then expressible in a simple form in terms of green functions of g and of smaller groups one drawback of the definition of 3 was that it did not allow one to compute the green functions except in some simple cases another definition of green functions was proposed by springer first in terms of trigonometric sums on the lie algebra and later in geometrical terms 17 this definition was applicable only in large characteristic and in that case it was identified with the definition in 3 by springer 17 and kazhdan 10 in 12 i observed that green functions for gln can be redefined in terms of intersection cohomology this led me to a new definition of green functions

rational rigidity for f4 p abstract we prove the existence of certain rationally rigid triples in f 4 p for good primes p i e p 3 thereby showing that these groups occur as regular galois groups over q t and so also over q we show that these triples give rise to rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic 0

computing green functions in small characteristic abstract let g q be a finite group of lie type over a field with q elements where q is a prime power the green functions of g q as defined by deligne and lusztig are known in almost all cases by work of beynon spaltenstein lusztig und shoji open cases exist for groups of exceptional type e 6 2 e 7 e 8 in small characteristics we propose a general method for dealing with these cases which proceeds by a reduction to the case where q is a prime and then uses computer algebra techniques in this way all open cases in type e 6 2 e 7 are solved as well as at least one particular open case in type e 8

on the stack of semistable g bundles over an elliptic curve in a recent paper ben zvi and nadler proved that the induction map from b bundles of degree 0 to semistable g bundles of degree 0 over an elliptic curve is a small map with galois group isomorphic to the weyl group of g we generalize their result to all connected components of text bun g for an arbitrary reductive group g we prove that for every degree i e topological type there exists a unique parabolic subgroup such that any semistable g bundle of this degree has a reduction to it and moreover the induction map is small with galois group the relative weyl group of the levi this provides new examples of simple automorphic sheaves which are constituents of eisenstein sheaves for the trivial local system

generic fibers of the generalized springer resolution of type a abstract it is well known that when the lie algebra is of type a d e the springer fiber above a subregular nilpotent element is described by the dynkin diagram and is called the dynkin curve of the lie algebra on the other hand the closure of the minimal nilpotent orbit is obtained by collapsing the zero section of a cotangent bundle of a projective space p k in this article we are interested in the study of the generalized springer resolution of type a we give a complete description of the generalized springer fiber above a generic singularities showing that it is isomorphic to a dynkin curve or to a projective space

unipotent classes and special weyl group representations abstract we show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the weyl group

the geometry of generalized steinberg varieties for a reductive algebraic group g the steinberg variety of g is the set of all triples consisting of a unipotent element u in g and two borel subgroups of g that contain u we define generalized steinberg varieties that depend on four parameters and analyze in detail two special cases that turn out to be related to distinguished double coset representatives in the weyl group using one of the two special cases we define a parabolic version of a map from the weyl group to a set of nilpotent orbits of g in lie g defined by joseph and study some of its properties

euler characteristics of the real points of certain varieties of algebraic tori let g be a complex connected reductive group which is defined over r let g be its lie algebra and t the variety of maximal tori of g for g r let t be the variety of tori in t whose lie algebra is orthogonal to with respect to the killing form we show using the fourier sato transform of conical sheaves on real vector bundles that the weighted euler characteristic see below of t r is zero unless is nilpotent in which case it equals 1 dimt 2 this is a real analogue of a result over finite fields which is connected with the steinberg representation of a reductive group

characters of finite reductive lie algebras abstract for any finite group of lie type g q deligne and lusztig p deligne g lusztig representations of reductive groups over finite fields ann of math 2 103 1976 103 161 defined a family of virtual q l characters r t g of g q such that any irreducible character of g q is an irreducible constituent of at least one of the r t g in this paper we study analogues of this result for characters of the finite reductive lie algebra g q where g lie g motivated by the results of e letellier fourier transforms of invariant functions on finite reductive lie algebras lecture notes in math vol 1859 springer verlag 2005 and g lusztig representations of reductive groups over finite rings represent theory 8 2004 1 14 we define two families r t g and r t g of virtual q l characters of g q we prove that they coincide when is in general position and that they differ in general we verify that any character of g q appears in some r t g we conjecture that this is also true if r t g is replaced by r t g

unitary equivalences for reductive p adic groups we establish a transfer of unitarity for a bernstein component of the category of smooth representations of a reductive p adic group to the associated hecke algebra in the framework of the theory of types whenever the hecke algebra is an affine hecke algebra with geometric parameters in the sense of lusztig possibly extended by a group of automorphisms of the root datum it is known that there is a large class of such examples detailed in the paper as a consequence we establish relations between the unitary duals of different groups in the spirit of endoscopy

macdonald functions associated to complex reflection groups let w be the complex reflection group sn z ez n in the author s previous paper j algebra 245 2001 650 694 hall littlewood functions associated to w were introduced in the special case where w is a weyl group of type bn they are closely related to green polynomials of finite classical groups in this paper we introduce a two variables version of the above hall littlewood functions as a generalization of macdonald functions associated to symmetric groups a generalization of macdonald operators is also constructed and we characterize such functions by making use of macdonald operators assuming a certain conjecture

on special pieces the springer correspondence and unipotent characters let g be a connected reductive algebraic group over the algebraic closure of a finite field bbb f q of good characteristic in this paper we demonstrate a remarkable compatibility between the springer correspondence for g and the parametrization of unipotent characters of g bbb f q in particular we show that in a suitable sense large portions of these two assignments in fact coincide this extends earlier work of lusztig on springer representations within special pieces of the unipotent variety

hamiltonian reduction and nearby cycles for mirabolic d modules abstract we study holonomic d modules on sl n c c n called mirabolic modules analogous to lusztig s character sheaves we describe the supports of simple mirabolic modules we show that a mirabolic module is killed by the functor of hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus we introduce an analogue of verdier s specialization functor for representations of cherednik algebras which agrees on category o with the restriction functor of bezrukavnikov and etingof in type a we also consider a verdier specialization functor on mirabolic d modules we show that hamiltonian reduction intertwines specialization functors on mirabolic d modules with the corresponding functors on representations of the cherednik algebra this allows us to apply known hodge theoretic purity results for nearby cycles in the setting considered by bezrukavnikov and etingof

characterizations of character sheaves for complex reductive algebraic groups character sheaves are geometric objects associated to a reductive algebraic group they are closely related to irreducible characters of finite chevalley groups the existence of character sheaves was conjectured by g lusztig in the book characters of reductive groups over finite fields 1984 there he described what one should expect character sheaves to be for complex reductive groups a definition of character sheaves for an arbitrary characteristic quite different from this description was given by him later 1985 it is shown in this thesis in particular that the original 1984 lusztig s suggestion for describing character sheaves is equivalent to the later definition we use alternative characterizations of character sheaves for a connected complex reductive algebraic group g that are obtained in this thesis and also the one conjectured by g lusztig and g laumon that was obtained in 1988 by v ginsburg and independently by i mirkovic and k vilonen these characterizations involve microsupport and microlocalization of a sheaf a microsupport ss cal f of a complex cal f of sheaves on g is a closed conic subset of t sp g g times bf g it tells us about singularities of cal f a microlocalization mu sb m cal f of cal f along a submanifold m of g is a complex of sheaves on the conormal bundle t sbsp m g which allows us to analyze cal f on the infinitesimally small neighbourhood of m theorem let g be a complex connected reductive algebraic group cal f be a g equivariant for conjugation irreducible perverse sheaf on g let cal n be the nilpotent cone of the lie algebra g of g s may denote either g or the set of all semisimple elements of g the following are equivalent i cal f is a character sheaf ii ss cal f vert sb s subset s times cal n iii supp mu sb cal o cal f subset cal o times cal n where cal o subset s is a conjugacy class iv supp mu sb x cal f subset cal n x in s characterization ii when s g is the one given by ginsburg mirkovic and vilonen characterization iii with s being the set of semisimple elements was conjectured by j bernstein for arbitrary characteristic this kind of characterizations essentially simplifies the development and our understanding of the theory currently there is still no such description for positive characteristic an important intermediate result is the description of the microsupport of a doubc constructible sheaf cal f on a manifold x in terms of microlocalizations to the points of this manifold ss cal f overline bigcup limits sb x in x rm supp mu sb x cal f

on unipotent supports of reductive groups with a disconnected centre let g be a simple algebraic group defined over a finite field of good characteristic with associated frobenius endomorphism f in this article we extend an observation of lusztig which gives a numerical relationship between an ordinary character of gf and its unipotent support to the case where z g is disconnected we then use this observation in some applications to the ordinary character theory of gf 2013 elsevier inc

sensor centric data reduction for estimation with wsns via censoring and quantization consider a wireless sensor network wsn with a fusion center fc deployed to estimate signal parameters from noisy sensor measurements if the wsn has a large number of low cost battery operated sensor nodes with limited transmission bandwidth then conservation of transmission resources power and bandwidth is paramount to this end the present paper develops a novel data reduction method which requires no inter sensor collaboration and results in only a subset of the sensor measurements transmitted to the fc using interval censoring as a data reduction method each sensor decides separately whether to censor its acquired measurements based on a rule that promotes censoring of measurements with least impact on the estimator mean square error mse leveraging the statistical distribution of sensor data the censoring mechanism and the received uncensored data fc based estimators are derived for both deterministic via maximum likelihood estimation and random parameters via maximum a posteriori probability estimation for a linear gaussian model quantization of the uncensored measurements at the sensor nodes offers an additional degree of freedom in the resource conservation versus estimator mse reduction tradeoff cramer rao bound analysis for the different censor estimators and censor quantizer estimators is also provided to benchmark and facilitate mse based performance comparisons numerical simulations corroborate the analytical findings and demonstrate that the proposed censoring estimation approach performs competitively with alternative methods under different sensing conditions while having lower computational complexity

on distributed linear estimation with observation model uncertainties we consider distributed estimation of a gaussian source in a heterogenous bandwidth constrained sensor network where the source is corrupted by independent multiplicative and additive observation noises we assume the additive observation noise is zero mean gaussian with known variance however the system designer is unaware of the distribution of multiplicative observation noise and only knows its first and second order moments for multibit quantizers we derive an accurate closed form approximation for the mean square error mse of the linear minimum mse estimator at the fusion center for both error free and erroneous communication channels we propose several rate allocation methods named as longest root to leaf path greedy integer relaxation and individual rate allocation to minimize the mse given a network bandwidth constraint and minimize the required network bandwidth given a target mse we also derive the bayesian cramer rao lower bound crlb for an arbitrarily distributed multiplicative observation noise and compare the mse performance of our proposed methods against the crlb our results corroborate that for the low power multiplicative observation noise and adequate network bandwidth the gaps between the mse of greedy and integer relaxation methods and the crlb are negligible while the mse of individual rate allocation and uniform methods is not satisfactory through analysis and simulations we also explore why maximum likelihood and maximum a posteriori estimators based on one bit quantization perform poorly for the low power additive observation noise

remarks on computing irreducible characters 0 1 let g be a connected reductive algebraic group defined over a finite field fq and let g fq be the finite group of all fq rational points of g we would like to present here a strategy for computing the character table of g fq under the assumption that p the characteristic of fq is sufficiently large we can assume that g has a simply connected derived group indeed in the general case we can find a connected reductive group g with simply connected derived group and a surjective homomorphism f g g whose kernel is a central torus such that g and f are defined over fq then the character table

representations of finite lie algebras abstract we develop an approach to investigate representations of finite lie algebras g f over a finite field f q through representations of lie algebras g with frobenius morphisms f over the algebraic closure k f q as an application we first show that frobenius morphisms on classical simple lie algebras can be used to determine easily their f q forms and hence reobtain a classical result given in g b seligman modular lie algebras springer verlag berlin 1967 we then investigate representations of finite restricted lie algebras g f regarded as the fixed point algebra of a restricted lie algebra g with restricted frobenius morphism f by introducing the f orbital reduced enveloping algebras u g associated with a reduced enveloping algebra u g we partition simple g f modules via f orbits of their p characters we further investigate certain relations between the categories of g modules with p character g f modules with p character and g f modules with p character for an automorphism of g finally we illustrate the theory with the example of sl 2 f q

about henrici s transformation for accelerating vector sequences abstract the aim of this paper is to study henrici s transformation for accelerating a vector sequence a convergence acceleration theorem is proved and two different implementations are given the paper ends with some numerical examples

vanishing of odd dimensional intersection cohomology ii for a variety where a connected linear algebraic group acts with only finitely many orbits each of which admits an attractive slice we show that the stratification by orbits is perfect for equivariant intersection cohomology with respect to any equivariant local system this applies to provide a relationship between the vanishing of the odd dimensional intersection cohomology sheaves and of the odd dimensional global intersection cohomology groups for example we show that odd dimensional intersection cohomology sheaves and global intersection cohomology groups vanish for all complex spherical varieties

deligne lusztig restriction of gelfand graev characters abstract let g be a connected reductive algebraic group defined over a finite field f q in f digne g i lehrer j michel on gel fand graev characters of reductive groups with disconnected centre j reine angew math 491 1997 131 147 it is proved that the deligne lusztig restriction of a gelfand graev character of the finite group g f q is still a gelfand graev character however an ambiguity remains on the gelfand graev character obtained in this paper we describe the deligne lusztig restrictions of the gelfand graev characters of the finite group g f q using the theory of kostant slodowy transversal slices for the nilpotent orbits of the lie algebra of g

a unipotent support for irreducible representations 1 1 let g f be the group of fq rational points of a connected reductive algebraic group g defined over a finite field f throughout this paper we shall assume that the characteristic of the ground field is large the following is part of problem ii in l9 let p be an irreducible complex character of g f show that there is a unique umpotent class c in g which has the property that cga c cfqi p g 0 and has maximal dimension among classes with this property one of the results of this paper is a solution of this problem we also prove the following refinement if g e gf is such that tr g p 0 then the unipotent part of g lies in c or in a conjugacy class of dimension dim c we call c the unipotent support of p another map from the set of irreducible representations of gf to unipotent classes in g has been defined in l8 13 43 this was the composition of two maps one from irreducible representations of gf to a set of parameters in the langlands dual of g and one from this set of parameters to unipotent classes in g another conjectural definition of this map which did not pass through the langlands dual has been proposed by kawanaka kl k2 k3 in terms of generalized gelfand graev representations in analogy with real lie groups he uses the name wave front set for the unipotent class attached to a representation one of the results of this paper is a proof see section 11 of kawanaka s conjecture that his wave front set is well defined and that it coincides with the one defined in l8 this has been known earlier from the work of kawanaka in certain special cases we also show that the unipotent support of an irreducible representation coincides with the wave front set of the dual representation

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

convergence of vectorial sequences applications a transformation of vectorial sequences is given analogous to the shank s transform in the scalar case this is possible because of the link with the vector pad e approximants defined for a power series with vectorial coefficients algorithms different from recursive computation of determinants can be defined some numerical examples are given

robust opportunistic inference from non homogeneous distribution free measurements a class of inference problems arising in wireless sensor network contexts is addressed where nodes aim at estimating cooperatively a parameter based on opportunistically gathered measurements to allow for generality non homogeneities are taken into account in the formulation furthermore knowledge of the data distribution is not assumed a novel linear estimation approach is devised based on a hierarchical modeling able to cope with local conditions e g due to manufacturing differences diverse classes of nodes and or spatial variability and heterogeneity of the samples which cannot be controlled in opportunistic inference the total variance of the hierarchy is derived then used to optimize the weights of the linear estimator obtaining in closed form the minimum variance solution of the general non homogeneous inference problem the proposed approach induces in the optimal weights a scalar parameter as additive term to the local sample sizes this acts like a tikhonov regularization on the variance contributions brought by the cooperating nodes which reflect local conditions and are inversely proportional to the sample size as a result of this regularization effect robustness against heavy tailed variations of the number of measurements available at the different nodes is achieved a thorough theoretical analysis corroborated by numerical results reveals that the proposed distribution free approach is superior to both linear and weighted least squares and can perform closely to the oracle minimum variance estimator

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

on unipotent and nilpotent pieces for classical groups abstract we show that the definition of unipotent resp nilpotent pieces for classical groups given by lusztig resp lusztig and the author coincides with the combinatorial definition using closure relations on unipotent classes resp nilpotent orbits moreover we give a closed formula for a map from the set of unipotent classes resp nilpotent orbits in characteristic 2 to the set of unipotent classes in characteristic 0 such that the fibers are the unipotent resp nilpotent pieces

generalized green functions associated to complex reflection groups abstract in this paper we consider the set of r symbols in a full generality we construct hall littlewood functions and kostka functions associated to those r symbols we also discuss a multi parameter version of those functions we show that there exists a general algorithm of computing the multi parameter kostka functions as an application we show that the generalized green functions of symplectic groups can be described combinatorially in terms of our one parameter kostka functions

nilpotent orbits in classical lie algebras over f2n and the springer correspondence we give the number of nilpotent orbits in the lie algebras of orthogonal groups under the adjoint action of the groups over f2n let g be an adjoint algebraic group of type b c or d defined over an algebraically closed field of characteristic 2 we construct the springer correspondence for the nilpotent variety in the lie algebra of g

characteristic cycles for the loop grassmannian and nilpotent orbits x which is a linear combination of closures of conormal bundles to submanifolds of x intuitively the microlocal multiplicities c me asurethesingularity of at in settings related to representation theory a group g acts on x is g equivariant and the microlocal multiplicities play a significant but only partially understood role in representation theory see ro sv abv and kasa e g in this paper we compute microlocal multiplicities for certain cases of interest in representation theory let g be a connected reductive group with loop group lg and let p bethesubgroup of lg consisting of loops with positive fourier coefficients then p orbits on theloop grassmannian lg p correspond to the irreducible representations l of thedual re ductivegroup g for dominant weights and of a torus of

modular generalized springer correspondence iii exceptional groups we complete the construction of the modular generalized springer correspondence for an arbitrary connected reductive group with a uniform proof of the disjointness of induction series that avoids the case by case arguments for classical groups used in previous papers in the series we show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the sylow subgroups of the weyl group under some assumptions we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character we also provide tables and other information on the modular generalized springer correspondence for quasi simple groups of exceptional type including a complete classification of cuspidal pairs in the case of good characteristic and a full determination of the correspondence in type g 2

target location estimation in sensor networks with quantized data a signal intensity based maximum likelihood ml target location estimator that uses quantized data is proposed for wireless sensor networks wsns the signal intensity received at local sensors is assumed to be inversely proportional to the square of the distance from the target the ml estimator and its corresponding crameacuter rao lower bound crlb are derived simulation results show that this estimator is much more accurate than the heuristic weighted average methods and it can reach the crlb even with a relatively small amount of data in addition the optimal design method for quantization thresholds as well as two heuristic design methods are presented the heuristic design methods which require minimum prior information about the system prove to be very robust under various situations

computational proof of the mackey formula for q 2 let g be a connected reductive group defined over a finite field with q elements we prove that the mackey formula for the lusztig induction and restriction holds in g whenever q 2 or g does not have a component of type e

produit scalaire elliptique abstract suppose that w is a weyl group let c w be a space of functions on w with complex values invariant under conjugation we can define an elliptic scalar product on c w it is a natural ingredient to the representation theory of p adic reductive groups let g be a reductive group over the algebraic closure of a finite field the generalized springer correspondence gives a bijection between two sets the set of pairs u e where u is an unipotent orbit of g and e is a g equivariant irreducible local system on u the disjoint union of the sets of irreducible representations of certain weyl groups related to g using kazhdan lusztig polynomials we modify the generalized springer correspondence by the modified correspondence a pair u e as above maps to a representation of a certain weyl group and this representation is in general reducible there is no simple formula that relates the elliptic scalar product and the generalized springer correspondence but a simple formula does exist and we prove it that relates the elliptic scalar product and the modified generalized springer correspondence our result is in fact a corollary of a theorem of lusztig on the restriction of character sheaves to the unipotent variety

a vanishing theorem for dirac cohomology of standard modules abstract this paper studies the dirac cohomology of standard modules in the setting of graded hecke algebras with geometric parameters we prove that the dirac cohomology of a standard module vanishes if and only if the module is not twisted elliptic tempered the proof makes use of two deep results one is some structural information from the generalized springer correspondence obtained by s kato and lusztig another one is a computation of the dirac cohomology of tempered modules by barbasch ciubotaru trapa and ciubotaru we apply our result to compute the dirac cohomology of ladder representations for type a n for each of such representations with non zero dirac cohomology we associate to a canonical weyl group representation we use the dirac cohomology to conclude that such representations appear with multiplicity one