some small unipotent representations of indefinite orthogonal groups abstract we study a family of small unitary representations of indefinite orthogonal groups these representations arise as analytic continuations of the discrete series and were studied extensively by knapp in k3 we complete knapp s analysis by proving that they are irreducible in order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one moreover we prove that the k type structure of each representation matches up to a shift the k type structure of the ring of functions on the closure a nilpotent k c orbit on p

on the associated cycles of discrete series representations let g k be a classical symmetric pair defined by the involution ong let g k be the corresponding lie algebras given a stable borel subalgebra b we determine a densek orbit ink b k in the case when g k is of type aiii or bdi we use this information to compute the harmonic polynomial determined by the associated cycle of a large discrete series representation

on the variety of a highest weight module 1 2 let u g be the enveloping algebra of g detailed information on the primitive spectrum prim u g of u g was summarized by conjecture 7 4 of 181 here a main question was to relate prim u g with the springer correspondence parts i ii of this conjecture were established in i ii and a partial solution to iii was given in 21 sufficient e g to treat g of type a meanwhile borho and brylinski 5 have established iii for induced ideals while barbasch and vogan 1 2 have established a version of iii in which gr i i e prim u g is replaced by a certain wavefront set known to be contained in y gr 1 this last work involves some case by case analysis and gives little indication as to why the springer correspondence should arise in the study of prim u g

the invariant holonomic system on a semisimple lie algebra on etudie la theorie geometrique des representations du groupe de weyl du point de vue des systemes holonomes sur des algebres de lie

on orbital variety closures in sln iii geometric properties abstract this is the third paper in the series here we define a few combinatorial orders on young tableaux the first order is obtained from the induced duflo order by the extension with the help of vogan t procedure we call it the duflo vogan order the second order is obtained from the generalization of spaltenstein s construction by consideration of an orbital variety as a double chain of nilpotent orbits we call it the chain order again we use vogan s t procedure however this time to restrict the chain order we call it the vogan chain order the order on young tableaux defined by the inclusion of orbital variety closures is called the geometric order and the order on young tableaux defined by inverse inclusion of primitive ideals is called the algebraic order we get the following relations between the orders the duflo vogan order is an extension of the induced duflo order the algebraic order is an extension of the duflo vogan order the geometric order is an extension of the algebraic order the vogan chain order is an extension of the geometric order and finally the chain order is an extension of the vogan chain order the computations show that the duflo vogan and the vogan chain orders coincide on sl n for n 9 and in n 10 there is one case up to t procedure and transposition where the chain vogan order is a proper extension of the duflo vogan order in this only case the algebraic order coincides with the vogan chain order these computations permit us to conjecture that in sl n the algebraic order coincides with the geometric order as well we conjecture that the combinatorics of both the inclusions on primitive ideals and on orbital variety closures is defined by the vogan chain order on young tableaux

primitive ideals and orbital integrals in complex exceptional groups in 2 two related problems were studied for complex classical groups determination of the primitive spectrum of the enveloping algebra as a set and fourier inversion of unipotent orbital integrals the first of these was solved completely extending work of joseph and others in sl n c and small groups although our techniques presumably solve the second as well up to determination of normalizing constants we carried out the calculations only for special unipotent classes see 181 we had of course tried to apply the same methods to the exceptional groups but they seem to be inadequate since that work was done however brylinski and kashiwara and beilinson and bernstein have established the kazhdan lusztig conjecture giving character formulas for irreducible highest weight modules 3 71 because of a conjecture of joseph proved in 24 this determines in principle the primitive ideals with a fixed regular integral infinitesimal character they are in one to one correspondence with what kazhdan and lusztig call left cells in the weyl group however the algorithm given by kazhdan and lusztig to compute these cells is enormously complicated requiring one to compute roughly 1 i polynomials of degrees on the order of half the number of positive roots this is very unsatisfactory however it

non equilibrium critical dynamics of the two dimensional xy model with hamiltonian equations of motion non equilibrium critical dynamics of the two dimensional xy model is investigated with hamiltonian equations of motion critical relaxation starting from both ordered and random states is carefully analyzed and the short time dynamic scaling behavior is revealed logarithmic corrections to scaling are detected for relaxation with a random initial state while power law corrections to scaling are observed for relaxation with an ordered initial state the static exponent and dynamic exponent z are determined around and below the kosterlitz thouless phase transition temperature our results show that the deterministic dynamics described by hamiltonian equations is in the same universality class as the stochastic dynamics described by monte carlo algorithms and langevin equations

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

differential operators on homogeneous spaces iii characteristic varieties of harish chandra modules and of primitive ideals letg be a semi simple complex algebraic group with lie algebra and flag varietyx g b for each primitive idealj with trivial central character in the enveloping algebrau we define a characteristic variety in the cotangent bundle ofx which projects under the springer resolution mapt x onto the closure of a nilpotent orbit we prove that this characteristic variety is theg saturation of the characteristic variety of a highest weight module with annihilatorj we conjecture that it is irreducible forg sl n our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of weyl group representations as achieved by a joseph the presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space and to study more generally characteristic varieties of harish chandra modules

orbital varieties in slnand the smith conjecture abstract let o be a nilpotent orbit in the lie algebra s l n c that is a class of nilpotent elements for conjugation bysln c let v be an orbital variety contained in o andpbe the largest parabolic subgroup ofsln c stabilizing v the smith conjecture asserts that v contains a denseporbit this is shown to fail in general and further those nilpotent orbits for which such a dense orbit exists are determined

characteristic varieties and vanishing cycles on presente une etude systematique des varietes caracteristiques des systemes holonomes a singularites regulieres ce sont des sous varietes lagrangiennes coniques dans le fibre cotangent t x

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

nonholonomic modules over enveloping algebras of semisimple lie algebras abstract we prove that the enveloping algebras of all semisimple lie algebras over the complex numbers which are not of type a 1 admit nonholonomic simple modules

on varieties in an orbital variety closure in semisimple lie algebras abstract in this note we discuss the closure of an orbital variety as a union of varieties we show that if semisimple lie algebra g contains factors not of type a n then there are orbital varieties whose closure contains components which are not lagrangian we show that the argument does not work if all the factors are of type a n and provide the facts supporting the conjecture claiming that if all the factors of g are of type a n then the closure of an orbital variety is a union of orbital varieties

on differential operators on complete symmetric varieties of type a1 and a2 abstract in this paper we consider the algebra of global differential operators d x on wonderful compactifications x of symmetric spaces g h of type a 1 and a 2 we will first construct a global differential operator on these varieties that does not come from the infinitesimal action of the lie algebra of g we will then focus on type a 2 where we will show that d x is an algebra of finite type and that for any invertible sheaf l on x h 0 x l is either 0 or a simple left d x l module finally we will show with the help of local cohomology that this still holds for higher cohomology groups h i x l

robust sampled data control robust stabilization of a class of linear systems using sampled data feedback control is studied it is shown that a linear time invariant system subject to additive perturbations can be stabilized using high gain periodic feedback from sampled values of the state provided that the perturbations satisfy certain structural conditions this result is also extended to decentralized sampled data control of interconnected systems

direct singular perturbation analysis of high gain and cheap control problems a new and direct approach to the analysis and design of high gain feedback and cheap control problems is developed transformation of such singular problems to singular perturbation models of conventional type early at the onset of analysis or design allows a direct application of the readily available singular perturbation literature the approach permits a simple characterization of a multivariable root loci under a high gain feedback and b asymptotic behavior of optimal closed loop poles state and control trajectories performance index and optimal transfer function as the control cost coefficient in the performance index goes to zero

picard lefschetz theory and characters of a semisimple lie group the paper applies picard lefschetz theory to the distribution characters of infinite dimensional representations of semisimple lie groups and analyzes their asymptotic behaviour at the identity

universelle versus relative einhullende eine geometrische untersuchung von quotienten von universellen einhullenden halbeinfacher lie algebren

differential operators on rational projective curves fix a singular projective rational curve 3 over an algebraically closed field k of characteristic zero with normalisation map a p 3 and write s 5 r 9 for its ring of global differential operators we prove theorem 1 the ring g a is a noerherian domain finirely generated as a k algebra with a unique minimal non zero ideal j z moreover f y 9 j f is finite dimensional over k in the case when n is injective we give complete description of the structure of 2 x

adaptive robust sampled data control of a class of systems under structured perturbations robust adaptive sampled data control of a class of linear systems under structured perturbations is considered the controller is a time varying state feedback law having a fixed structure containing an adjustable parameter and operating on sampled values the sampling period and the controller parameter are adjusted with simple adaptation rules the resulting closed loop system is shown to be stable for a class of unknown perturbations the same result is also shown to be applicable to decentralized control of interconnected system

pattern avoidance and smoothness of closures for orbits of a symmetric subgroup in the flag variety abstract we give a pattern avoidance criterion to classify the orbits of sp p c sp q c resp gl n c on the flag variety of type c p q resp d n with rationally smooth closure we show that all such orbit closures fiber with smooth fiber over a smaller flag variety and hence are in fact smooth in addition we prove that the classification is insensitive to isogeny

generalized sample hold functions frequency domain analysis of robustness sensitivity and intersample difficulties over the past 5 years there has been substantial interest in the use of generalized sample hold functions for control in this correspondence the authors use a tool which is novel in this context namely amplitude modulation theory the authors employ this tool to analyze the quantitative and qualitative features of the intersample behavior in a frequency domain setting this offers new theoretical and practical insights into the method the authors conclusion is that the perceived benefits come at substantial cost which makes its practical use questionable

characteristic cycles of holomorphic discrete series the characteristic cycles of standard modules arising from closed at orbits in a flag variety are studied and those of holomorphic discrete series are determined explicitly also the asymptotic expansion of the distribution characters of holomorphic discrete series are computed the result verifies a conjecture of d vogan in this special case suppose gr is a linear semisimple lie group and ar a maximal compact subgroup of gr in case the symmetric space gr ar carries a hermitian structure gr has a family of holomorphic discrete series representations let be the underlying harish chandra module of an arbitrary holomorphic dis crete series representation and o be the corresponding distribution character on the group the associated cycle 2 7 of is an algebraic cycle defined al gebraically on n the dual to the complexification of the lie algebra or of gr on the other hand according to barbasch vogan the fourier transform of the leading term of the asymptotics of 8 near the identity is a linear combination of canonical invariant measures on coadjoint gr orbits in g r it therefore de termines a cycle on gm in this paper we determine both invariants attached to the holomorphic discrete series it then verifies in this special case a conjecture of d vogan which asserts that the major part of both invariants coincide under the orbit correspondence of sekiguchi the complexified tangent space of gr a r at the identity coset is naturally identified to a quotient of q therefore via the invariant nondegenerate killing form the holomorphic tangent space is identified to a subspace in g here the holomorphic structure is the one chosen to define then the associated cycle 2 7 is the holomorphic tangent space with multiplicity the dimension of the lowest a r type of i 2 13 the computation is made through the corre sponding 2 module on the flat variety x of 9 and the moment map from the cotangent bundle t x to g generally modules giving rise to the discrete series are supported on closed a orbits on x k is the complexification of ar the normal derivatives then induce a good filtration on these modules this then computes the associated graded modules as sheaves of coherent cf modules on t x although in general the moment map is quite complicated on the support of the associated graded module in the situation where holo

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

robust adaptive sampled data control of a class of systems under structured nonlinear perturbations a robust adaptive sampled data feedback stabilization scheme is presented for a class of systems with nonlinear additive perturbations the proposed controller generates a control input by using high gain static or dynamic feedback from nonuniform sampled values of the output a simple adaptation rule adjusts the gain and the sampling period of the controller

multivariable feedback sensitivity and decentralized control in this paper the problem of sensitivity reduction by feedback is studied and related to a problem of decentralized control a plant will be represented by an n times n matrix of frequency responses which may be unstable or irrational the object will be to find conditions on p s under which a diagonal feedback f s can make the sensitivity parallel i p s f s 1 parallel arbitrarily small over some specified frequency interval j omega 0 j omega 0 without violating a global sensitivity bound parallel i p s f s 1 parallel leq m m geq some const 1 for re s geq 0 it will be shown that such a diagonal feedback of the high gain type can be constructed whenever p 1 s is analytic in re s geq 0 p s satisfies an attenuation condition near s infty and p s approaches diagonal dominance at high frequencies it will also be shown that these conditions on the plant can be interpreted as conditions for the existence of a decentralized wide band control scheme

rational cherednik algebras and hilbert schemes ii representations and sheaves let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring gr u c mathbb c mathfrac h mathfrac h w where is the n th symmetric group using the z algebra construction from gs it is also possible to associate to a filtered h c or u c module hat m on the hilbert scheme hilb n using this technique we study the representation theory of u c and h c and we relate it to hilb n and to the resolution of singularities hilb n mathfrac h mathfrac h w for example we prove the following if c 1 n so that l c triv is the unique one dimensional simple h c module then hat e l c triv mathcal o z n where z n 1 0 is the punctual hilbert scheme if c 1 n k for k in mathbb n then under a canonical filtration on the finite dimensional module l c triv gr e l c triv has a natural bigraded structure that coincides with that on h 0 z n mathscr l k where mathscr l mathcal o hilb n 1 this confirms conjectures of berest etingof and ginzburg beg2 conjectures 7 2 7 3 under mild restrictions on c the characteristic cycle of hat e c equals sum k z where k are kostka numbers and the z are known irreducible components of 1 mathfrak h w

k theory of twisted differential operators on flag varieties let mathfrak g be a semisimple lie algebra overk an algebraically closed field of characteristic zero and let mathfrak h subset mathfrak b be a cartan subalgebra inside a borel subalgebra of mathfrak g letu be the enveloping algebra of mathfrak g for mu in mathfrak h letm denote the corresponding verma module and letuu u annm letw be the weyl group and letw 0 be the stabiliser of inw we prove the following theorem which affirms a conjecture of t j hodges

unipotent representations attached to spherical nilpotent orbits a coadjoint nilpotent orbit inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of a complex semisimple lie group g is said to be spherical if it contains an open orbit of a borel subgroup we determine when and how to attach unitary representations to such an orbit for the real orthogonal and symplectic groups our results actually extend to a larger class of nilpotent coadjoint orbits

lie superalgebras clifford algebras induced modules and nilpotent orbits let g be a classical simple lie superalgebra to every nilpotent orbit o in g0 we associate a clifford algebra over the field of rational functions on o we find the rank k o of the bilinear form defining this clifford algebra and deduce a lower bound on the multiplicity of a u g module with o or an orbital subvariety of o as associated variety in some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction the invariant k o is in many cases equal to the odd dimension of the orbit g o where g is a lie supergroup with lie superalgebra g

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

sheets and topology of primitive spectra for semisimple lie algebras abstract let g be a semisimple lie algebra consider the set x of primitive ideals of the enveloping algebra u g given the jacobson topology a basic open problem is to describe x as a countable union of algebraic varieties v with strata v as large as possible towards this aim the notion of a sheet in x is introduced here in analogy with the notion of sheet in the adjoint orbit space g g the sheets in x are classified and given a purely topological description moreover goldie rank is constant and maximal on a dense open subset of a sheet for every positive integer n it is shown that there are only finitely many sheets in x with maximal goldie rank n for n 1 which corresponds to completely prime ideals an extension of the dixmier map from some adjoint orbits to primitive ideals is introduced with the aim of showing that the sheets with n 1 are homeomorphic to sheets in g g and hence explicitly described as algebraic varieties this goal is partly though not fully achieved the proofs of the above results require a detailed knowledge of goldie rank polynomials and even a new difficult result concerning the support of their restrictions to the walls an extension of a theorem of soergel on the topology of x is also required and for this a new approach is given which is both simpler and more general

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

double cells for unitary groups abstract let g r u p q be the unitary group and k the complexification of a maximal compact subgroup of g r we prove that the associated variety of an irreducible harish chandra module for g r with trivial infinitesimal character is the closure of a single nilpotent k orbit

the minimal nilpotent orbit the joseph ideal and differential operators abstract fix a simple complex lie algebra g not of type g 2 f 4 or e 8 let min denote the zariski closure of the minimal non zero nilpotent orbit in g and let g n h n be a triangular decomposition we prove theorem 1 if g is not of type a n then there exists an irreducible component x of min n such that u g j o d x where j o is the joseph ideal and d x denotes the ring of differential operators on x 2 if g is of type a n then for n 2 of the n irreducible components x i of min n there exist distinct maximal ideals j i of u g such that u g j i d x i

on duflo s theorem that minimal primitive ideals are centrally generated abstract let g be a complex semisimple lie algebra duflo s theorem states that each minimal primitive ideal of the enveloping algebra u g is generated by its intersection with the center we prove the following generalization for the relative enveloping algebra a u g i relative to a parabolic subalgebra q of g where i denotes the annihilator of the induced module u g u q q c let z denote the center of a let z denote its normalization and let a az be the slight extension of a obtained by normalizing the center we present a theorem that states that under certain conditions which are always satisfied if g s l n we have that each minimal primitive ideal of a is generated by its intersection with the center z duflo s theorem is the special case where q is a borel subalgebra then i 0 z z and a a u g note that the normalization of the center is really necessary for the theorem the corresponding statement for a and z instead of a and z fails for g s l 3

on irreducible representations of compact p adic analytic groups we prove that the canonical dimension of a coadmissible representation of a semisimple p adic lie group in a p adic banach space is either zero or at least half the dimension of a nonzero coadjoint orbit to do this we establish analogues for p adically completed enveloping algebras of bernstein s inequality for modules over weyl algebras the beilinson bernstein localisation theorem and quillen s lemma about the endomorphism ring of a simple module over an enveloping algebra