absorbing boundary in one dimensional anomalous transport in this paper we study the space time probability distribution q x t of a random walk subject to an absorbing boundary at the origin x 0 for motion controlled by l evy flights and l evy walks characterized by the exponent ensuremath gamma we find that the method of images usually applicable to brownian motion may break down for l evy processes we calculate the distribution q x t to be at xg0 after time tg0 having started at the origin assuming that the boundary is effective at time tg0 we show that q x t depends on the details of the underlying process q x t ensuremath sim mathit x ensuremath gamma 2 mathit t 1 1 ensuremath gamma 1 ensuremath le ensuremath gamma ensuremath le 2 for small x while total survival is independent of the spatial realization of motion and displays a universal behavior we also discuss the related smoluchowski boundary condition problem

trace functions in the ring of fractions of polycyclic group rings let kg be the group ring of a polycyclic by finite group g over a field k of characteristic zero r be the goldie ring of fractions of kg s be an arbitrary subring of r n n we prove that the intersection of the commutator subring s s with the center z s is nilpotent this implies the existence of a nontrivial trace function in r n n

a fixed point formula and harish chandra s character formula the main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators for proper actions by connected semisimple lie groups on possibly noncompact manifolds with compact quotients for compact groups and manifolds this reduces to the atiyah segal singer fixed point formula other special cases include an index theorem by connes and moscovici for homogeneous spaces and an earlier index theorem by the second author both in cases where the group acting is connected and semisimple as an application of this fixed point formula we give a new proof of harish chandra s character formula for discrete series representations

invariant lattices and modular decomposition of irreducible representations let p be a prime let h be a finite group and let c be an absolutely irreducible character of h any thorough study of the p modular decom position of c requires passage to some integral representation if s is a finite group acting on h and leaving invariant such an integral represen tation will not be s invariant in general it is important to have invariant integral representations for purposes of clifford theory in fact modular decomposition often gives new insights in such classical questions like extendability of characters for this reason we consider the more general situation that h is a normal subgroup of some group g and let s g h s still acts on the characters of h and the isomorphism types of h modules we also fix a p modular system k r k where

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

levy brownian motion on finite intervals mean first passage time analysis we present the analysis of the first passage time problem on a finite interval for the generalized wiener process that is driven by levy stable noises the complexity of the first passage time statistics mean first passage time cumulative first passage time distribution is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non gaussian noises the validity of the method is tested numerically and compared against analytical formulas when the stability index alpha approaches 2 recovering in this limit the standard results for the fokker planck dynamics driven by gaussian white noise

on the first passage time and leapover properties of levy motions we investigate two coupled properties of levy stable random motions the first passage times fpts and the first passage leapovers fpls while in general the fpt problem has been studied quite extensively the fpl problem has hardly attracted any attention considering a particle that starts at the origin and performs random jumps with independent increments chosen from a levy stable probability law x the fpt measures how long it takes the particle to arrive at or cross a target the fpl addresses a different question given that the first passage jump crosses the target then how far does it get beyond the target these two properties are investigated for three subclasses of levy stable motions i symmetric levy motions characterized by levy index 0 2 and skewness parameter 0 ii one sided levy motions with 0 1 1 and iii two sided skewed levy motions the extreme case 1 2 1

a transition to sharp timing in stochastic leaky integrate and fire neurons driven by frozen noisy input the firing activity of intracellularly stimulated neurons in cortical slices has been demonstrated to be profoundly affected by the temporal structure of the injected current mainen sejnowski 1995 this suggests that the timing features of the neural response may be controlled as much by its own biophysical characteristics as by how a neuron is wired within a circuit modeling studies have shown that the interplay between internal noise and the fluctuations of the driving input controls the reliability and the precision of neuronal spiking cecchi et al 2000 tiesinga 2002 fellous rudolph destexhe sejnowski 2003 in order to investigate this interplay we focus on the stochastic leaky integrate and fire neuron and identify the holder exponent h of the integrated input as the key mathematical property dictating the regime of firing of a single unit neuron we have recently provided numerical evidence taillefumier magnasco 2013 for the existence of a phase transition when becomes less than the statistical holder exponent associated with internal gaussian white noise h 1 2 here we describe the theoretical and numerical framework devised for the study of a neuron that is periodically driven by frozen noisy inputs with exponent h 0 in doing so we account for the existence of a transition between two regimes of firing when h 1 2 and we show that spiking times have a continuous density when the holder exponent satisfies h 1 2 the transition at h 1 2 formally separates rate codes for which the neural firing probability varies smoothly from temporal codes for which the neuron fires at sharply defined times regardless of the intensity of internal noise

first passage and arrival time densities for levy flights and the failure of the method of images we discuss the first passage time problem in the semi infinite interval for homogeneous stochastic markov processes with levy stable jump length distributions x l x 1 x l namely levy flights lfs in particular we demonstrate that the method of images leads to a result which violates a theorem due to sparre andersen according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density fptd governed by the universal long time decay t 3 2 conversely we show that for lfs the direct definition known from gaussian processes in fact defines the probability density of first arrival which for lfs differs from the fptd our findings are corroborated by numerical results

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

levy flights on the half line we study the probability distribution function pdf of the position of a levy flight of index 0 2 in the presence of an absorbing wall at the origin the solution of the associated fractional fokker planck equation can be constructed using a perturbation scheme around the brownian solution corresponding to 2 as an expansion in e 2 we obtain an explicit analytical solution exact at the first order in e which allows us to conjecture the precise asymptotic behavior of this pdf including the first subleading corrections for any careful numerical simulations as well as an exact computation for 1 confirm our conjecture

representation groups for the schur index fix an algebraic number field k and an irreducible complex character x of some finite group g assume k k x contains the values of x so that the simple component a 4 x k of the group algebra kg determined by 1 is central over k it is a basic consequence of the brauer witt theorem that a is a so called cyclotomic algebra see ls we exhibit here a slightly different point of view namely that of constructing a representation group for a similar to the concept used in clifford theory from class field theory it is known that the order of the class a of a in the brauer group br k equals its index m x it suffices to study the primary components of a so fix a prime p and let a denote the pcomponent of a in br k let k be the field obtained from k by adjoining the nth roots of unity if n exp g then by brauer k is a splitting field for a and g we shall see that there is a field l s k with p power degree over k such that

extending irreducible representations of normal subgroups let h be a normal subgroup of some finite group g and let v be an irreducible right fh module where f is an arbitrary field it is one of the primary objectives of clifford theory to find necessary and sufhcient conditions so that v can be extended to an fg module an obvious necessary condition is that v has to be g invariant stable sufficiency relies upon various properties of the group extension h g w s where s g h and the module itself the theory is well developed in case v is absolutely irreducible recently isaacs 12 and dade 4 proved generalizing a theorem by gallagher that v can be always extended to an fg module provided h is a hall subgroup of g and v is g invariant the present paper resulted from an effort to understand this remarkable and somewhat surprising result assume in the sequel that v is g invariant let d end v be the opposite endomorphism ring of k basic clifford theory yields a group extension d 52 p p s such that there exists an fg module extending v if and only if 52 52 v splits over the multiplicative group d of the division algebra d see theorem 1 below in general d is noncommutative and so the usual cohomological methods seem to fail dade even remarked in 4 sceptically that none of the nice theorems about extensions of v such as those of gallagher which depend upon calculations in the cohomology groups of s with coefficients in a commutative group can be proved this way in the following situation however one can argue as usual

dirac operator and its cohomology for the quantum group uq x1d598 x1d591 x1d7d0 we introduce a dirac operator d for the quantum group uq x1d530 x1d529 2 as an element of the tensor product of uq x1d530 x1d529 2 with the clifford algebra on two generators we study the properties of d including an analogue of vogan s conjecture we compute the cohomology of d acting on various uq x1d530 x1d529 2 modules

using g algebras for schur index computation abstract conditions for the existence of a bijective character correspondence that preserves schur indices are given based on the theory of central simple g algebras over fields of characteristic 0 it is then shown how this applies to schur index computation in the case of finite solvable groups

steady state levy flights in a confined domain we derive the generalized fokker planck equation associated with a langevin equation driven by arbitrary additive white noise we apply our result to study the distribution of symmetric and asymmetric levy flights in an infinitely deep potential well the fractional fokker planck equation for levy flights is derived and solved analytically in the steady state it is shown that levy flights are distributed according to the beta distribution whose probability density becomes singular at the boundaries of the well the origin of the preferred concentration of flying objects near the boundaries in nonequilibrium systems is clarified

blocks with cyclic defect groups and clifford extensions abstract we prove that the residual clifford extension of any non exceptional character belonging to a p block with cyclic defect groups of a twisted group algebra is naturally isomorphic to the clifford extension of any simple module belonging to that block we also show that dade s inductive conjecture holds for blocks with cyclic defect groups

lifting modular representations of p solvable groups let p be a prime the celebrated fong swan theorem states that for psolvable groups every absolutely irreducible p modular character can be lifted to characteristic 0 e g serre 7 theorem 381 recently isaacs 5 sharpened the conclusion by showing that one can choose p rational characters as liftings including a uniqueness statement for odd p the objective of this note is to demonstrate that it is unnecessary to assume splitting fields throughout we fix a finite group g and a p modular system k r k where k is a finite extension of the p adic numbers ujp and k r n is its residue class field a kg module v is said to lift a kg module w if there exists an rg lattice u such that v k or u and w u nu assume w is irreducible here then v is irreducible as well and the isomorphism type of u is uniquely determined by v further k end w then is a finite cyclic extension of k wedderburn and k end v is an unramified extension field of k the residue extension being a subextension of k k see lemma 2 of course this subextension will be proper unless r end u maps onto k

properties of levy flights on an interval with absorbing boundaries we consider a l evy yer of orderthat starts from a point x0 on an interval 0 l with absorbing boundaries we 7nd a closed form expression for an arbitrary average quantity char acterizing the trajectory of the yer such as mean 7rst passage time average total path length probability to be absorbed by one of the boundaries using fractional di erential equations with a riesz kernel we 7nd exact analytical expressions for these quantities in the continuous limit we 7nd numerically the eigenfunctions and the eigenvalues of these equations we study how the results of monte carlo simulations of the l evy ights with di erent ight length distribu tions converge to the continuous approximations we show that if x0 is placed in the vicinity of absorbing boundaries the average total path length has a minimum near 1 corresponding to the cauchy distribution we discuss the relevance of these results to the problem of biological foraging and transmission of light through cloudy atmosphere c 2001 elsevier science b v all rights reserved

jacquet modules and dirac cohomology abstract we show that dirac cohomology of the jacquet module of a harish chandra module is a harish chandra module for the corresponding levi subgroup we obtain an explicit formula of dirac cohomology of the jacquet module for most of the principal series based on our determination of dirac cohomology of irreducible generalized verma modules with regular infinitesimal characters

dirac series for some real exceptional lie groups abstract up to equivalence this paper classifies all the irreducible unitary representations with non zero dirac cohomology for the following simple real exceptional lie groups ei e 6 6 eiv e 6 26 fi f 4 4 fii f 4 20 along the way we find an irreducible unitary representation of f 4 4 whose dirac index vanishes while its dirac cohomology is non zero this disproves a conjecture raised in 2015 asserting that there should be no cancellation between the even part and the odd part of the dirac cohomology

holomorphic discrete models of semisimple lie groups and their symplectic constructions abstract let g be a connected real semisimple lie group which contains a compact cartan subgroup such that it has non empty discrete series a holomorphic discrete model of g is a unitary g representation consisting of all its holomorphic discrete series with multiplicity one we perform geometric quantization to a class of g invariant pseudo kahler manifolds and construct a holomorphic discrete model the construction of discrete series which are not holomorphic is also discussed

to hit or to pass it over remarkable transient behavior of first arrivals and passages for levy flights in finite domains the term levy flights was coined by benoit mandelbrot who thus poeticized stable levy random motion a markovian process with stationary independent increments distributed according to the stable levy probability law contrary to the brownian motion the trajectories of the stable levy motion are discontinous that is exhibit jumps this feature implies that the process of first passage through the boundary of a given space domain or the first escape is different from the process of first arrival hit at the boundary here we investigate the properties of first escapes and first arrivals for levy flights and explore how the asymptotic behavior of the corresponding passage and hit probabilities is sensitive to the size of the domain in particular we find that the survival probability to stay in a large enough finite domain has a universal sparre andersen temporal scaling which is transient and changes to an exponential non universal decay at longer times also the probability to arrive at a finite domain possesses a similar transient sparre andersen universality that turns into a non universal and slower power law decay in course of time finally we demonstrate that the probability density of the leapover length l over the boundary related to overshooting events has an intermediate asymptotics which is inherent for the escape from a semi infinite domain however for larger leapovers the probability density decays faster according to the law thus we find that the laws derived for the stable processes on the semi infinite domain manifest themselves as transients for levy flights on the finite domain

dirac cohomology and geometric quantization let g be a connected real semisimple lie group having a finite center and a compact cartan subgroup t with lie algebra t0 let be a g t invariant symplectic form on xdg t0 we incorporate dirac cohomology into the geometric quantization of x and study the resulting multiplicity free unitary g t representation on a hilbert space h x we also perform symplectic reduction of x and show that our quantiza tion method satisfies the principle quantization commutes with reduction as an application we construct various models of discrete series

a phase transition in the first passage of a brownian process through a fluctuating boundary with implications for neural coding finding the first time a fluctuating quantity reaches a given boundary is a deceptively simple looking problem of vast practical importance in physics biology chemistry neuroscience economics and industrial engineering problems in which the bound to be traversed is itself a fluctuating function of time include widely studied problems in neural coding such as neuronal integrators with irregular inputs and internal noise we show that the probability p t that a gauss markov process will first exceed the boundary at time t suffers a phase transition as a function of the roughness of the boundary as measured by its holder exponent h the critical value occurs when the roughness of the boundary equals the roughness of the process so for diffusive processes the critical value is hc 1 2 for smoother boundaries h 1 2 the probability density is a continuous function of time for rougher boundaries h 1 2 the probability is concentrated on a cantor like set of zero measure the probability density becomes divergent almost everywhere either zero or infinity the critical point hc 1 2 corresponds to a widely studied case in the theory of neural coding in which the external input integrated by a model neuron is a white noise process as in the case of uncorrelated but precisely balanced excitatory and inhibitory inputs we argue that this transition corresponds to a sharp boundary between rate codes in which the neural firing probability varies smoothly and temporal codes in which the neuron fires at sharply defined times regardless of the intensity of internal noise

dirac cohomology unitary representations and a proof of a conjecture of vogan u radu je dokazana slutnja d vogana koja odredjuje infinitesimalni karakter ireducibilnog g k modula koji ima diracovu kohomologiju razlicitu od nula infinitesimalni karakter je dan kao rho translat najvece tezine k tipa u diracovoj kohomologiji diracova kohomologija se definira kao ker d im d cap ker d gdje d oznacava diracov operator na zadanom g k modulu tenzoriranom sa spin modulom

representation groups of characters the subject we are concerned with has its roots in the work of schur who introduced the notion of a representation group darstellungs gruppe these groups have the property to lift linearize all projective representations of some finite group over an algebraically closed field 16 fong 8 j noticed that schur s method applies in context with clif ford s work exhibiting some single invariant stable character or module the familiar proof for the famous fong swan theorem gives the model for this kind of representation groups see serre 17 or feit 7 throughout we fix a finite group g with normal subgroup h and quotient group s g h we also fix an absolutely irreducible frobenius character x of h and a field f containing the values of x we assume that x is g invariant as passage from the inertia group to the whole group usually is easy to handle this is not too restrictive it is a necessary con dition for x to be extendible to g

morita equivalences related to the glauberman correspondence abstract starting with p interior algebras where p is a finite p group we prove two theorems establishing certain group graded morita equivalences these apply in particular to the case of blocks with normal defect groups and defect zero blocks of normal subgroups respectively

principal series representations and harmonic spinors abstract let g be a real reductive lie group and g h a reductive homogeneous space we consider kostant s cubic dirac operator d on g h twisted with a finite dimensional representation of h under the assumption that g and h have the same complex rank we construct a nonzero intertwining operator from principal series representations of g into the kernel of d the langlands parameters of these principal series are described explicitly in particular we obtain an explicit integral formula for certain solutions of the cubic dirac equation d 0 on g h

fractional laplacians in bounded domains killed reflected censored and taboo levy flights the fractional laplacian 2 0 2 has many equivalent albeit formally different realizations as a nonlocal generator of a family of stable stochastic processes in r n on the other hand if the process is to be restricted to a bounded domain there are many inequivalent proposals for what a boundary data respecting fractional laplacian should actually be this ambiguity not only holds true for each specific choice of the process behavior at the boundary e g absorbtion reflection conditioning or boundary taboos but extends as well to its particular technical implementation dirichlet neumann etc problems the inferred jump type processes are inequivalent as well differing in their spectral and statistical characteristics which may strongly influence the ability of the formalism if uncritically adopted to provide an unambiguous description of real geometrically confined physical systems with disorder specifically that refers to their relaxation properties and the near equilibrium asymptotic behavior in the present paper we focus on levy flight induced jump type processes which are constrained to stay forever inside a finite domain this refers to a concept of taboo processes imported from brownian to levy stable contexts to so called censored processes and to reflected levy flights whose status still remains to be unequivocally settled as a by product of our fractional spectral analysis with reference to neumann boundary conditions we discuss disordered semiconducting heterojunctions as the bounded domain problem

first passage leapovers of levy flights and the proper formulation of absorbing boundary conditions an important open problem in the theory of levy flights concerns the analytically tractable formulation of absorbing boundary conditions although the nonlocal approach where the dynamical operators are given restrictions on their domain properties has yielded substantial insights regarding the statistics of first passage the resultant modifications to the dynamical equations hinder the detailed analysis possible in the absence of these nonlocal restrictions in this study it is demonstrated that using the first hit distribution related to the first passage leapover as the absorbing sink preserves the tractability of the dynamical equations for a particle undergoing levy flight in particular knowledge of the first hit distribution is sufficient to fully determine the first passage time and position density of the particle without requiring operator domain restrictions or numerical simulations in addition we report on the first hit and leapover properties of first passages and arrivals for levy flights of arbitrary skew parameter and extend these results to levy flights in a certain ubiquitous class of potentials satisfying an integral condition

on the first passage of one sided levy motions abstract we study the first passage problem for one sided levy motions how does such a motion started at the origin cross a barrier positioned at the point x x 0 since one sided levy motions are pure jump processes they always leap over barriers rather than crossing them continuously we hence explore the following issues i first passage times fpts how long would it take the motion to cross the barrier ii first passage leapovers fpls how far would the motion leap over the barrier and iii what is dependence between the fpts and the fpls formulae for the joint laplace transforms of the fpts and fpls are derived and the scaling limits of the fpts and fpls are computed the scaling limits turn out to display dramatically different behavior when i the underlying levy motion has finite cumulants and when ii the underlying levy motion is heavy tailed special attention is devoted to the investigation of the first passage problem for self similar one sided levy motions

tensor induction of projective representations to helmut wielandt on his 80 birthday there are some standard reduction techniques in representation theory of finite groups mainly based on work by clifford and fong apparently less known is a method suggested by berger l which refers to tensor induction of projective representations it applies in the following situation ti e is a nonabelian normal subgroup

source algebras and source modules the aim of this article is to give a self contained approach in module theoretic terms to two fundamental results in block theory due to puig 6 14 6 first there is an embedding of the source algebra of the brauer correspondent of a block of some finite group into a source algebra of that block and second the source algebras of the brauer correspondence can be described explicitly our proof of the first result theorem 5 and its corollary 6 below is essentially the translation to our terminology of the proof in 1 4 10 the second result describing the source algebras of the brauer correspondent follows also from 3 but the proofs in 3 use both the main result on the structure of nilpotent blocks in 5 as well as techniques from 6 sects 4 6 while our approach from proposition 9 onwards requires only some classical results on the structure of blocks with a central defect group an account of the results in 6 on source algebras of blocks with a normal defect group can also be found in 7 sect 45 except for the explicit description of the central extension e see section 10 below as being defined in terms of the multiplicity algebra of the block which is quoted in 7 45 10 b without proof referring back to 6 at the end of 7 sect 45

average time spent by levy flights and walks on an interval with absorbing boundaries we consider a levy flyer of order alpha that starts from a point x 0 on an interval o l with absorbing boundaries we find a closed form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed these two quantities are equivalent to the mean first passage times for levy flights and levy walks respectively using fractional differential equations with a riesz kernel we find exact analytical expressions for both quantities in the continuous limit we show that numerical solutions for the discrete levy processes converge to the continuous approximations in all cases except the case of alpha 2 and the cases of x 0 0 and x 0 l for alpha 2 when the second moment of the flight length distribution exists our result is replaced by known results of classical diffusion we show that if x 0 is placed in the vicinity of absorbing boundaries the average total length has a minimum at alpha 1 corresponding to the cauchy distribution we discuss the relevance of this result to the problem of foraging which has received recent attention in the statistical physics literature

hitting distributions of alpha stable processes via path censoring and self similarity after brownian motion stable processes are often considered an exemplary family of processes for which many aspects of the general theory of lproesses can be illus trated in closed form first passage problems which are relatively straightforward to handle in the case of brownian motion become much harder in the setting of a general levy process on account of the inclusion of jumps a collection of articles through the 1960s and early 1970s appealing largely to potential analytic methods for general markov processes were relatively successful in handling a number of first passage prob lems in particular for symmetric stable processes in one or more dimensions see for example 3 24 12 13 27 to name but a few however following this cluster of activity several decades have passed since new results concerning first passage identities for stable processes have appeared the last few years have seen a number of new explicit first passage identities for one dimensional stable processes thanks to a better understanding of the intimate re lationship between the aforesaid processes and positive self similar markov processes see for example 4 6 8 17 22 in this paper we return to the problem of blumenthal et al 3 published in 1961 which gave the law of the position of first entry of a symmetric stable process into the unit ball specifically we are interested in establishing the same law but now for a one dimensional stable process which enjoys two sided jumps and which is not necessarily symmetric our method is modern in the sense that we appeal to the relationship between stable processes and certain positive self similar markov processes however there are two notable additional innovations first we make use of a type of path censoring second we are able to describe in explicit analytical detail a non trivial wiener hopf factorisation of an auxiliary levy process from which the desired solution can be sourced moreover as a consequence of this approach we are able to deliver a number of additional related identities in explicit form for stable processes

translation principle for dirac index let g be a finite cover of a closed connected transpose stable subgroup of rm gl n bbb r with complexified lie algebra frak g let k be a maximal compact subgroup of g and assume that g and k have equal rank we prove a translation principle for the dirac index of virtual frak g k modules as a byproduct to each coherent family of such modules we attach a polynomial on the dual of the compact cartan subalgebra of frak g this index polynomial generates an irreducible representation of the weyl group contained in the coherent continuation representation we show that the index polynomial is the exact analogue on the compact cartan subgroup of king s character polynomial the character polynomial was defined by king on the maximally split cartan subgroup and it was shown to be equal to the goldie rank polynomial up to a scalar multiple in the case of representations of gelfand kirillov dimension at most half the dimension of g k we also conjecture an explicit relationship between our index polynomial and the multiplicities of the irreducible components occurring in the associated cycle of the corresponding coherent family

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

leapover lengths and first passage time statistics for levy flights exact results for the first passage time and leapover statistics of symmetric and one sided levy flights lfs are derived lfs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one sided lfs and surprisingly with an index alpha 2 for symmetric lfs the first passage time distribution scales like a power law with an index 1 2 as required by the sparre andersen theorem for symmetric lfs whereas one sided lfs have a narrow distribution of first passage times the exact analytic results are confirmed by extensive simulations

dirac cohomology k characters and branching laws inspired by work of enright and willenbring ew we prove a generalized littlewood s restriction formula in terms of dirac cohomology our approach is to use a character formula of irreducible unitary lowest weight modules instead of the bernstein gelfand gelfand resolution and the proof is much simpler we also show that our branching formula is equivalent to the formula of enright and willenbring in terms of nilpotent lie algebra cohomology this follows from the close relationship between the dirac cohomology and the corresponding nilpotent lie algebra cohomology for unitary representations of semisimple lie groups of hermitian type which was established in hpr

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

clifford theory of simple modules clifford theory primarily is concerned with the representations of normal subgroups and quotient groups of finite groups the basic contribution being due to a h clifford 3 modern developments founded on the concepts of group graded rings and crossed products have shown close relations with commutative and non commutative galois theory this is most conspicuous in stable clifford theory of simple modules suppose n w h g is an extension of finite groups and v is a simple f n module where f is any field since passage from the inertia group to the whole group usually is easy to handle we may and do assume that v is g invariant or h invariant stable then the annihilator i ann v of v generates a homogeneous ideal z of f h and a f h z is a fully g graded algebra with simple l component r f n z general clifford theory is concerned with such g graded algebras a and their modules cf dade 7 lo assume the l component r a is simple artinian as above then the centre k z r has in a natural way the structure of a g field the main object of study is the endomorphism ring e end v of the a module induced from the simple r module v this is a fully g graded algebra as well generalizing wedderburn s theorem we show that a z m e as g graded algebras where d is the dimension of v over the division algebra d end v theorem 2 1 in particular k z d has the same g structure with respect to e and a and e are morita equivalent in some strong sense now assume that dim r d2m2 is finite and that the schur index m m v is relatively prime to the order of the automorphism group g c k of k induced by g then the graded structure of a e can be described by some distinguished cohomology class w v e h g k more precisely w oa v determines a crossed product sz of k with g