generic singularities of nilpotent orbit closures according to a theorem of brieskorn and slodowy the intersection of the nilpotent cone of a simple lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity at the opposite extremity of the poset of nilpotent orbits the closure of the minimal nilpotent orbit is also an isolated symplectic singularity called a minimal singularity for classical lie algebras kraft and procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures specifically any such singularity is either a simple surface singularity a minimal singularity or a union of two simple surface singularities of type a 2k 1 in the present paper we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional lie algebras up to normalization in a few cases we summarize the results in some graphs at the end of the paper in most cases we also obtain simple surface singularities or minimal singularities though often with more complicated branching than occurs in the classical types there are however six singularities that do not occur in the classical types three of these are unibranch non normal singularities an sl 2 c variety whose normalization is a 2 an sp 4 c variety whose normalization is a 4 and a two dimensional variety whose normalization is the simple surface singularity a 3 in addition there are three 4 dimensional isolated singularities each appearing once we also study an intrinsic symmetry action on the singularities extending slodowy s work for the singularity of the nilpotent cone at a point in the subregular orbit

on the choice of norms in system identification in this paper we discuss smooth and sensitive norms for prediction error system identification when the disturbances are magnitude bounded formal conditions for sensitive norms which give an order of magnitude faster convergence of the parameter estimate variance are developed however it also is shown that the parameter estimate variance convergence rate of sensitive norms is arbitrarily bad for certain distributions a necessary condition for a norm to be statistically robust with respect to the family f c of distributions with support c c for some arbitrary c 0 is that its second derivative does not vanish on the support a direct consequence of this observation is that the quadratic norm is statistically robust among all l sub p norms p spl les 2 spl infin for f c

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

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

integral constraints on the accuracy of least squares estimation it is common to need to estimate the frequency response of a system from observed input output data this paper uses integral constraints to characterise the undermodelling induced errors involved in solving this problem via parametric least squares methods this is achieved by exploiting the hilbert space structure inherent in the least squares solution in order to provide a geometric interpretation of the nature of frequency domain errors the result is that an intuitive process can be applied in which for a given data collection method and model structure one identifies the sides of a right triangle and then by noting the hypotenuse to be the longest side integral constraints on the magnitude estimation error are obtained by also noting that the triangle sides both lie in a particular plane integral constraints on phase estimation error are derived this geometric approach is in contrast to earlier work in this area which has relied on algebraic manipulation

on designing stable allpass filters using ar modeling by exploiting the minimum phase property of an autoregressive ar modeling technique we propose a simple method for designing allpass filters that are guaranteed to be stable it is assumed that equispaced samples of the desired allpass filter s phase function is given

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

brief finite sample properties of system identification of arx models under mixing conditions the asymptotic convergence properties of system identification methods are well known but comparatively little is known about the practical situation where only a finite number of data points are available in this paper we consider the finite sample properties of prediction error methods for system identification we consider arx models and uniformly bounded criterion functions the problem we pose is how many data points are required in order to guarantee with high probability that the expected value of the identification criterion is close to its empirical mean value the sample sizes are obtained using generalisations of risk minimisation theory to weakly dependent processes we obtain uniform probabilistic bounds on the difference between the expected value of the identification criterion and the empirical value evaluated on the observed data points the bounds are very general in particular no assumption is made about the true system belonging to the model class further analysis shows that in order to maintain a given bound on the difference the number of data points required grows at most at a polynomial rate in the model order and in many cases no faster than quadratically the results obtained here generalise previous results derived for the case where the observed data was independent and identically distributed

homology of the zero set of a nilpotent vector field on a flag manifold 0 1 let x be a linear transformation of a finite dimensional vector space v the configuration of flags in v which are fixed by x has rather remarkable properties when x is unipotent though this case is especially interesting the proper generality in which to study such configurations is in the theory of reductive algebraic groups where their definition can be reformulated in the language of borel subalgebras as follows let g be a connected reductive group over c with lie algebra g and let n e g be a nilpotent element let q be the variety of all borel subalgebras of g and let

berry s phase as the asymptotic limit of an exact evolution an example the authors exhibit for arbitrary time variations of the parameters of the generalised harmonic oscillator hamiltonian a canonical transformation which facilitates an exact analysis of quantal phase and classical angle formulae for berry s phase and hannay s angle are readily obtained by taking the adiabatic limit of the exact theory

angle variable holonomy in adiabatic excursion of an integrable hamiltonian if an integrable classical hamiltonian h describing bound motion depends on parameters which are changed very slowly then the adiabatic theorem states that the action variables i of the motion are conserved here the fate of the angle variables is analysed because of the unavoidable arbitrariness in their definition angle variables belonging to distinct initial and final hamiltonians cannot generally be compared however they can be compared if the hamiltonian is taken on a closed excursion in parameter space so that initial and final hamiltonians are the same the result shows that the angle variable change arising from such an excursion is not merely the time integral of the instantaneous frequency omega dh di but differs from it by a definite extra angle which depends only on the circuit in parameter space not on the duration of the process the 2 form which describes this angle variable holonomy is calculated

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

a metric for arma processes autoregressive moving average arma models seek to express a system function of a discretely sampled process as a rational function in the z domain treating an arma model as a complex rational function we discuss a metric defined on the set of complex rational functions we give a natural measure of the distance between two arma processes the paper concentrates on the mathematics behind the problem and shows that the various algebraic structures endow the choice of metric with some interesting and remarkable properties which we discuss we suggest that the metric can be used in at least two circumstances i in which we have signals arising from various models that are unknown so we construct the distance matrix and perform cluster analysis and ii where there are several possible models m sub i all of which are known and we wish to find which of these is closest to an observed data sequence modeled as m

on geometric phases and dynamical invariants a formalism is developed for calculating berry phases for nonadiabatic time periodic quantum systems when a dynamical invariant is known it is found that when the invariant is periodic and has a nondegenerate spectrum this method allows a convenient way to obtain generalized berry phases and the proper cyclic initial states the method is applied to the generalized harmonic oscillator and the two level system where the invariant operators are explicitly constructed formulae for the conventional berry s phases are readily obtained by taking the adiabatic limit of the exact results

a remark on the orthogonality relations for green functions let g be a connected linear algebraic group defined over the finite field f of characteristic p we denote by k an algebraic closure of f and by f the frobenius automorphism q h ay of k over if then f operates on g and its group g of fixed points is the finite group of 5 rational points of g now assume g to be reductive and let ti be the if variety of unipotent elements of g the green functions of gf are class functions q7 c defined on the set j of unipotent elements of g parametrized by a maximal torus t of g defined over f and depending only on the g iconjugacy class of t they are of importance for the representation theory of g in his fundamental paper 7 on the character theory of the finite groups gl f i a green first introduced such functions in a combinatorial guise subsequently various definitions of green functions for general reductive groups were given in 131 elaborated in 141 in 6 and most recently in 9 conjecturally all definitions should be equivalent at least if the characteristic p is good in the sense of 141 as far as i know this has not yet been established and the equivalence of the various definitions is known only under more severe restrictions on p a basic property of the green functions are their orthogonality relations which can be written as

subspace based identification of infinite dimensional multivariable systems from frequency response data a new identification algorithm which identifies low complexity models of infinite dimensional systems from equidistant frequency response data is presented the new algorithm is a combination of the fourier transform technique with the recent subspace techniques given noisefree data finite dimensional systems are exactly retrieved by the algorithm when noise is present it is shown that identified models strongly converge to the balanced truncation of the identified system if the measurement errors are covariance bounded several conditions are derived on consistency illustrating the trade offs in the selection of certain parameters of the algorithm two examples are presented which clearly illustrate the good performance of the algorithm

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

generalized berry phase for the most general time dependent damped harmonic oscillator the most general time dependence hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum it describes in general a harmonic oscillator with a time dependent mass a time dependent friction or antifriction constant and a time dependent spring constant acted upon by a time dependent force the energy whose time derivative is the power is in general different from the hamiltonian the generalized berry phase for a given energy eigenstate has a state dependent part which vanishes if there is no damping and an arbitrary state independent part if the hamiltonian is identified as the energy both the energy eigenvalues and the generalized berry phase are different in the adiabatic limit both approaches give the same total phase which is the sum of the dynamical and berry phases

frequency domain analysis of tracking and noise performance of adaptive algorithms in this paper an analysis of the tracking and noise performance of several adaptive algorithms is carried out for the case of model structures with fixed pole positions such structures have previously been proposed as an efficient generalization of the common fir model structure the focus of this work is to analyze the associated tradeoff between noise sensitivity and tracking ability in the frequency domain by illustrating how it is influenced by such things as input and noise spectral densities step size and in particular the choice of the fixed pole locations the latter influence is not described by preexisting analysis but is shown here to be amenable to attack by a particular class of orthonormal bases

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

an exact invariant for a class of time dependent anharmonic oscillators with cubic anharmonicity an exact invariant is constructed for a class of time dependent anharmonic oscillators using the method of the lie theory of extended groups the presence of the anharmonic term imposes a constraint on the nature of the time dependence for a subclass it is possible to obtain an energy like integral and a condition under which the motion is bounded

base sizes for simple groups and a conjecture of cameron let g be a permutation group on a finite set a base for g is a subset b with pointwise stabilizer in g that is trivial we write b g for the smallest size of a base for g in this paper we prove that b g 6 if g is an almost simple group of exceptional lie type and is a primitive faithful g set an important consequence of this result when combined with other recent work is that b g 7 for any almost simple group g in a non standard action proving a conjecture of cameron the proof is probabilistic and uses bounds on fixed point ratios

on the irreducibility of associated varieties of w algebras abstract we investigate the irreducibility of the nilpotent slodowy slices that appear as the associated variety of w algebras furthermore we provide new examples of vertex algebras whose associated variety has finitely many symplectic leaves

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

a variant of the induction theorem for springer representations abstract let g be a simple algebraic group over c with the weyl group w for a unipotent element u g let b u be the variety of borel subgroups of g containing u let l be a levi subgroup of a parabolic subgroup of g with the weyl subgroup w l of w assume that u l and let b u l be a similar variety as b u for l for a certain choice of l u l and e 1 we describe the w modules n k mod e h 2 n b u for k 0 e 1 in terms of the w l module h b u l with some additional data which is a refinement of the induction theorem due to lusztig as an application we give an explicit formula for the values of green functions at root of unity in the case where u is a regular unipotent element in l

an effective method to compute closure ordering for nilpotent orbits of θ representations we develop an algorithm for computing the closure of a given nilpotent g0 orbit in g1 where g1 and g0 are coming from a z or a z mz grading g gi of a simple complex lie algebra g

on the values of unipotent characters of finite chevalley groups of type e6 in characteristic 3 abstract let g be a finite chevalley group we are concerned with computing the values of the unipotent characters of g by making use of lusztig s theory of character sheaves in this framework one has to find the transformation between several bases for the class functions on g in principle this has been achieved by lusztig and shoji but the underlying process involves some scalars which are still unknown in many cases we shall determine these scalars in the specific case where g is one of the groups e 6 q e 6 2 q and q is a power of the bad prime p 3 for e 6 by exploiting known facts about the representation theory of the hecke algebra associated with g

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

brief new identification approaches for disturbed models recently much research has been conducted in the field of identification of the linear models in general these methods use a time domain estimate or a frequency domain estimate in this paper the time domain estimate and the frequency domain estimate were combined to identify the autoregressive exogenous noise arx interference model the concept of a general prediction error criterion is introduced for the time domain estimate an optimal frequency estimation is introduced for the frequency domain estimate a new identification method called the empirical frequency domain optimal parameter estimate is proposed for disturbed systems it is fully applied and developed for the output error model and a specific case or the arx model the algorithm theoretically provides the globally optimum frequency domain estimate of the model some simulations are included to illustrate the new identification method

springer s weyl group representations through characteristic classes of cone bundles on realise le groupe symetrique s n comme le groupe des permutations des valeurs propres des matrices diagonales pour l identifier au groupe de weyl w de g sl n c

generalized gelfand graev representations of exceptional simple algebraic groups over a finite field i introduction let ffi be an adjoint simple algebraic group of type e n 6 7 8 f 4 or g2 over an algebraically closed field k of characteristic p 0 we assume that 5 has a fixed fq rational structure for a finite subfield fq of k the purpose of this paper is to determine explicitly the values of the irreducible characters of g ffi fq at the unipotent elements when p is good 38 i 4 3 in this part i we shall do this only for large p and the full result will be proved in the forthcoming part ii after submitting the first version of the present paper the author was informed by lusztig that he solved analogous problems for classical groups using his theory of character sheaves and that he can also recover the result of this paper mentioned above our method depends upon 1 a parametrization of the irreducible characters of g and the determination of their multiplicities in deligne lusztig virtual representations lusztig 25 2 the computation of the green functions of g in large characteristic springer 36 7 1 shoji 33 beynon and spaltenstein 3 4 3 the determination of the values of generalized gelfand graev characters 20 of g in good characteristic see section 3 by 1 and 2 the uniform parts of the irreducible characters are already known explicitly in large characteristic this implies that in large characteristic our problem for groups of type e 6 o r e 7 has already been solved it should be mentioned here that the whole character table of the g2 group is also known by chang and ree 8 hence in the g2 case this paper gives just another approach to a part of the result of chang and ree if g is of type g2 f4 or e 8 the restriction of an irreducible character r of g to the set of unipotent elements can be written as a sum of its uniform part with a function of the form

rational points on generalized flag varieties and unipotent conjugacy in finite groups of lie type let g be a connected reductive algebraic group defined over the finite field fq where q is a power of a good prime for g we write f for the frobenius morphism of g corresponding to the fq structure so that gf is a finite group of lie type let p be an f stable parabolic subgroup of g and u the unipotent radical of p in this paper we prove that the number of uf conjugacy classes in gf is given by a polynomial in q under the assumption that the centre of g is connected this answers a question of j alperin in 1 in order to prove the result mentioned above we consider for unipotent u gf the variety p0u of g conjugates of p whose unipotent radical contains u we prove that the number of fq rational points of p0u is given by a polynomial in q with integer coefficients moreover in case g is split over fq and u is split in the sense of 22 5 the coefficients of this polynomial are given by the betti numbers of p0u we also prove the analogous results for the variety pu consisting of conjugates of p that contain u

perverse coherent sheaves and the geometry of special pieces in the unipotent variety let x be a scheme of finite type over a noetherian base scheme s admitting a dualizing complex and let ux be an open set whose complement has codimension at least 2 we extend the deligne bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension or intersection cohomology functor from perverse sheaves on u to perverse sheaves on x may be defined for a much broader class of perversities than has previously been known we also introduce a derived category version of the coherent intermediate extension functor under suitable hypotheses we introduce a construction called s2 extension in terms of perverse coherent sheaves of algebras on x that takes a finite morphism to u and extends it in a canonical way to a finite morphism to x in particular this construction gives a canonical s2 ification of appropriate x the construction also has applications to the macaulayfication problem and it is particularly well behaved when x is gorenstein our main goal however is to address a conjecture of lusztig on the geometry of special pieces certain subvarieties of the unipotent variety of a reductive algebraic group the conjecture asserts in part that each special piece is the quotient of some variety previously unknown for the exceptional groups and in positive characteristic by the action of a certain finite group we use s2 extension to give a uniform construction of the desired variety