blocks with trivial intersection defect groups we show that each block whose defect groups intersect pairwise trivially either has cyclic or generalised quaternion defect groups or is morita equivalent to one of a given list of blocks of central extensions of automorphism groups of non abelian simple groups in particular we classify all blocks of automorphism groups of non abelian simple groups whose defect groups are non cyclic and intersect pairwise trivially a consequence is that donovans conjecture holds for blocks whose defect groups intersect pairwise trivially

nilpotent gelfand pairs and spherical transforms of schwartz functions i rank one actions on the centre the spectrum of a gelfand pair of the form k ltimes n k where n is a nilpotent group can be embedded in a euclidean space mathbb r d the identification of the spherical transforms of k invariant schwartz functions on n with the restrictions to the spectrum of schwartz functions on mathbb r d has been proved already when n is a heisenberg group and in the case where n n3 2 is the free two step nilpotent lie group with three generators with k so3 astengo et al in j funct anal 251 772 791 2007 astengo et al in j funct anal 256 1565 1587 2009 fischer and ricci in ann inst fourier gren 59 2143 2168 2009 we prove that the same identification holds for all pairs in which the k orbits in the centre of n are spheres in the appendix we produce bases of k invariant polynomials on the lie algebra mathfrak n of n for all gelfand pairs k ltimes n k in vinberg s list vinberg in trans moscow math soc 64 47 80 2003 yakimova in transform groups 11 305 335 2006

harmonic analysis as spectral theory of laplacians by writing the fourier inversion formula on euclidean space in polar coordinates we obtain f x 0 fgl x dx where fgl x 2fgl x which is the spectral theory of the laplacian how do properties of f relate to properties of the family of eigenfunctions fgl answers are provided for the following spaces l2 s s d in odd dimensions analogous results are obtained for harmonic analysis on hyperbolic space constant curvature semi riemannian spaces the heisenberg group and for differential forms on hyperbolic space for f l2 there is a plancherel formula f 22 limt o 1t bt z f x 2dx d where bt z denotes the ball of radius t about an arbitrary point z which is independent of the dimension and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted for certain semi riemannian symmetric spaces we obtain a paley wiener theorem that explains the role of discrete series in producing functions of compact support and also involve certain non unitary representations that contain the discrete series representations for the heisenberg group the plancherel formula is of a different nature requiring that the eigenfunctions be almost periodic in one variable

on robustness of finite time stability of homogeneous affine nonlinear systems and cascade interconnections this paper investigates robustness of finite time stability property for a homogeneous nonlinear dynamical system with sufficiently small affine inputs in addition robust stability conditions are presented for the systems admitting homogeneous approximations at the origin or at infinity the effects of additional stable unmodeled dynamics in the input channel on robust stability are investigated the utility of the obtained results is illustrated via robustness analysis of homogeneous observer with time varying gains

vers une decomposition de jordan des blocs des groupes reductifs finis abstract let g be a reductive algebraic group over an algebraic closure of a prime field f p defined over f q with frobenius endomorphism f let g f be the subgroup of rational points let l be a prime number assume that l is different from p if g f is in duality with g f then by a theorem of m broue and j michel m broue j michel blocs et series de lusztig dans un groupe reductif fini j reine angew math 395 1989 56 67 for any l bloc b of g f there exists a unique g f conjugacy class s of l semi simple elements such that some irreducible representation of b is in the rational lusztig s series associated to s if s 1 b is said to be unipotent if g is not connected with identity component g define the unipotent l blocs of g f as the l blocks that cover some unipotent l bloc of g f assuming l is good for g we construct from g f and s some reductive algebraic group g s f defined over f q and a one to one map from the set of l blocks of g f with associated class s onto the set of unipotent l blocks of g s f such that if b corresponds to b then there is a significant height preserving one to one map from the set of irreducible representations irr b onto the set irr b the respective defect groups of b and b are isomorphic the associated brauer categories are equivalent one do not assume that the center of g is connected the class of s may be isolated in g

decomposition numbers of so7 q and sp6 q abstract we complete the l modular decomposition numbers of the unipotent characters in the principal block of the special orthogonal groups so 7 q and the symplectic groups sp 6 q for all prime powers q and all odd primes l different from the defining characteristic

universal stabilization of a class of nonlinear systems with homogeneous vector fields abstract under relative degree one and minimum phase assumptions it is well known that the class l of finite dimensional linear single input u single output y systems a b c is universally stabilized by the feedback strategy u y y 2 where is a function of nussbaum type the terminology universal stabilization being used in the sense of rendering s 0 s a global attractor for each member of the underlying class whilst assuring boundedness of the function a natural generalization of this result to a class h k of nonlinear control systems a b c with positively homogeneous of degree k 1 drift vector field a is described specifically under the relative degree one cb 0 and minimum phase hypotheses the latter being interpreted as that of asymptotic stability of the equilibrium of the zero dynamics it is shown that the strategy u vby vb k 1 y dot vby vb k 1 assures h k universal stabilization more generally the strategy u exp vby vb y dot exp vby vb y 2 assures h universal stabilization where h k 1 h k

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

reduced fusion systems over p groups with abelian subgroup of index p ii abstract for an odd prime p we look at simple fusion systems over a finite nonabelian p group s which has an abelian subgroup a of index p when s has more than one such subgroup we reduce this to a case already studied by ruiz and viruel when a is the unique abelian subgroup of index p in s and is not essential equivalently is not radical in the fusion system we give a complete list of all possibilities which can occur this includes several families of exotic fusion systems including some which have proper strongly closed subgroups

zj controlled blocks of quasisimple groups for odd primes abstract we investigated the converses of glauberman s zj theorem and its block theoretic analogue proved by kessar linckelmann and robinson for a finite group g we proved that a block b of g with a defect group d is zj controlled that is n g z j d controls b fusion in g if and only if b is controlled provided that g is quasisimple and the characteristic p of the modular representation is odd this implies by a result of 5 that the converses of glauberman s zj theorem and its block theoretic analogue both hold in these cases as an application we obtain all q d p free quasisimple groups

radical subgroups of the finite exceptional groups of lie type e6 abstract we consider the finite exceptional groups of lie type e 6 1 q e 6 q and e 6 1 q e 6 2 q both the universal versions we classify up to conjugacy the maximal p local subgroups and radical p subgroups of g e 6 e q for p 5 with p q and q e mod p and for p 3 with 3 q and q e mod 3 as an application the essential p rank of the frobenius category f d g is determined where d is a sylow p subgroup of g moreover if p 3 then we show that there is a subgroup h f 4 q of g containing d such that f d g f d h that is h controls 3 fusion in g

on homogeneous finite time control for linear evolution equation in hilbert space based on the notion of generalized homogeneity a new algorithm of feedback control design is developed for a plant modeled by a linear evolution equation in a hilbert space with a possibly unbounded operator the designed control law steers any solution of the closed loop system to zero in a finite time a method of homogeneous extension is presented in order to make the developed control design principles to be applicable for evolution systems with nonhomogeneous operators the design scheme is demonstrated for the heat equation with the control input distributed on the segment 0 1

brauer s height zero conjecture for quasi simple groups abstract we show that brauer s height zero conjecture holds for blocks of finite quasi simple groups this result is used in navarro spath s reduction of this conjecture for general groups to the inductive alperin mckay condition for simple groups

finite time observers application to secure communication in this paper control theory is used to formalize finite time chaos synchronization as a nonlinear finite time observer design issue this paper introduces a finite time observer for nonlinear systems that can be put into a linear canonical form up to an output injection the finite time convergence relies on the homogeneity properties of nonlinear systems the observer is then applied to the problem of secure data transmission based on the finite time chaos synchronization and the two channel transmission method

on the inductive alperin mckay and alperin weight conjecture for groups with abelian sylow subgroups abstract we study the inductive alperin mckay conjecture the inductive isaacs navarro refinement and the inductive blockwise alperin weight conjecture for groups of lie type in the generic case of abelian sylow l subgroups we also show that the alternating groups the suzuki groups and the ree groups satisfy the inductive condition necessary for spath s reduction of the blockwise alperin weight conjecture to the case of simple groups

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

controlled 2 blocks of quasisimple groups abstract controlled p blocks introduced by alperin and broue in 1979 were classified for quasisimple groups g for odd primes in 10 we classify the controlled 2 blocks of quasisimple groups g the results imply that every nilpotent 2 block of g has abelian defect groups this agrees with one of the main results proved in 15 we also give an explicit characterization of non controlled 2 blocks of all quasisimple groups this implies that the p 2 version of the block theoretic analogue of glauberman s zj theorem proved by kessar linckelmann and robinson in 27 holds for quasisimple groups g

homogeneous approximation recursive observer design and output feedback we introduce two new tools that can be useful in nonlinear observer and output feedback design the first one is a simple extension of the notion of homogeneous approximation to make it valid both at the origin and at infinity homogeneity in the bi limit exploiting this extension we give several results concerning stability and robustness for a homogeneous in the bi limit vector field the second tool is a new recursive observer design procedure for a chain of integrator combining these two tools we propose a new global asymptotic stabilization result by output feedback for feedback and feedforward systems

generalized harish chandra theory for unipotent characters of finite classical groups the harish chandra theory for a finite group g of lie type states the following if p is an irreducible character of g then there is a parabolic subgroup p of g a levi decomposition lb of p and a cuspidal character ii of the levi subgroup such that p is a constituent of the induced character indg of the pullback of to p furthermore the pair l is uniquely determined by p up to conjugacy in g in the case where g is a classical group that is a general linear unitary symplectic or orthogonal group over cfq we generalize this theory relative to an odd prime r different from the defining characteristic there is an integer e and a polynomial 4 x of the form x 1 or y 1 with r dividing c q such that the following holds to each unipotent character p of g corresponds a pair l ic where l is a regular subgroup of g of the form a product of a classical group and cyclic tori of order 4 q ic is a unipotent character of l of degree divisible by the full power of r dividing i l z l l and p is a constituent of the virtual character rf ic moreover the pair l is determined by p up to conjugacy in g the harish chandra theory is the case e i this work arose in an investigation by the authors of the brauer r blocks of g a similar generalization of the harish chandra theory has been proposed by r boyce

local ergodic theorems for k spherical averages on the heisenberg group given a gelfand pair h n k where h n is the heisenberg group and k is a compact subgroup of the unitary group u n we consider the sphere and ball averages of certain k invariant measures on h n we prove local ergodic theorems for these measures when n geq 3 we also consider averages over annuli in the case of reduced heisenberg group and show that when the functions have zero mean value the maximal function associated to the annulus averages behave better than the spherical maximal function we use square function arguments which require several properties of the k spherical functions

on the global finite time stabilization of bilinear systems by homogeneous feedback laws applications to some pde s abstract in this paper the problem of global finite time stabilization of bilinear control systems by means of homogeneous feedback law is investigated we prove under some reasonable assumptions on the operators a and b that continuous bounded and discontinuous unbounded feedbacks stabilize globally in finite time the closed loop system for illustrative the examples of heat schrodinger and transport equations are considered where homogeneous stabilizing feedback is built for these systems

finite time stabilization of 2 times2 hyperbolic systems on tree shaped networks we investigate the finite time boundary stabilization of a one dimensional first order quasilinear hyperbolic system of diagonal form on 0 1 the dynamics of both boundary controls are governed by a finite time stable ode the solutions of the closed loop system issuing from small initial data in lip 0 1 are shown to exist for all times and to reach the null equilibrium state in finite time when only one boundary feedback law is available a finite time stabilization is shown to occur roughly in a twice longer time the above feedback strategy is then applied to the saint venant system for the regulation of water flows in a network of canals

on gelfand pairs associated with solvable lie groups let g be a locally compact group and let k be a compact subgroup of aut g the group of automorphisms of g there is a natural action of k on the convolution algebra l g and we denote by lk g the subalgebra of those elements in l g that are invariant under this action the pair k g is called a gelfand pair if li g is commutative in this paper we consider the case where g is a connected simply connected solvable lie group and k c aut g is a compact connected group we characterize such gelfand pairs k g and determine a moduli space for the associated k spherical functions

the 2 blocks of the covering groups of the symmetric groups abstract let n be a double cover of the finite symmetric group s n of degree n i e n has a central involution z such that n z s n an irreducible character of n is called ordinary or spin according to whether it has z in its kernel or not the purpose of this paper is to determine the distribution of the spin characters of n into 2 blocks the methods applied here are essentially different from those applied to previous questions of this type we also discuss some consequences of our main result for the decomposition numbers an analogue of james well known result for the decomposition numbers of the symmetric groups is proved providing also a generalization of a theorem of benson ben theorem 1 2 in section 1 we present the background for our results and give some preliminaries in section 2 we give an explicit formula for the number of spin characters in a 2 block we also prove a result about the weight of a block containing a given non self associate spin character which will be important for the proof of our theorem on the 2 block distribution of spin characters section 3 presents some fundamental combinatorial concepts used in sections 4 and 5 the theorem concerning the spin characters in a given 2 block is proved in section 4 and in section 5 we present our results on the decomposition numbers

isolated blocks in finite classical groups ii the l blocks of finite classical groups where l is a prime not equal to the defining characteristic of the groups are classified by l semisimple classes in the dual groups the blocks corresponding to semisimple classes of elements with eigenvalues 1 in the natural representations of the dual groups are called isolated blocks these blocks were described in an earlier paper of the author for the groups sp 2n q and so 2n q q odd l odd via lusztig induction in this paper a combinatorial description of the isolated blocks of sp 2n q and o 2n q q odd l odd in terms of lusztig symbols is given for large q using some results of waldspurger

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

verification of iss iiss and ioss properties applying weighted homogeneity several conditions are proposed to check different robustness properties iss iiss ioss and oss for generic nonlinear systems applying the weighted homogeneity concept global or local the advantages of this result are that under some mild conditions the system robustness can be established as a function of the degree of homogeneity

nilpotent approximations of control systems and distributions a constructive method is given to approximate the vector fields in a nonlinear control system which is linear in the controls by a system of similar form and on the same state space with the describing vector fields of the approximating system generating a lie algebra which has a certain relevant sub algebra nilpotent given a k dimensional distribution on r n say locally defined near zero the method leads to an approximating k distribution which has nilpotent basis agrees with the original distribution at zero and each derived distribution of the approximating distribution and original distribution also agree at zero

tracking control of the uncertain heat and wave equation via power fractional and sliding mode techniques in the present paper preliminary results towards the generalization to the infinite dimensional setting of some well known robust finite dimensional control algorithms are illustrated more specifically we deal with the tracking problem for some classes of linear uncertain infinite dimensional systems evolving in hilbert spaces we design distributed variable structure stabilizers that are shown to be effective in the presence of external disturbances the main focus of the present paper is on the rejection of nonvanishing external disturbances the generalization to the infinite dimensional setting of the well known finite dimensional controllers namely the power fractional controller s p bhat and d s bernstein siam j control optim 38 2000 pp 751 766 and two second order sliding mode control algorithms the twisting and supertwisting algorithms l fridman and a levant higher order sliding modes as a natural phenomenon in control theory in robust control via variable structure and lyapunov techniques springer verlag berlin 1996 pp 107 133 a levant internat j control 58 1993 pp 1247 1263 is the main contribution of the present investigation first the distributed twisting control algorithm is developed to address the asymptotic state tracking of the perturbed wave equation next the finite time state tracking of the unperturbed heat equation is provided by means of a distributed power fractional controller finally the distributed supertwisting controller is suggested to address the asymptotic state tracking of the heat equation in spite of the presence of persistent disturbances constructive proofs of stability are developed via the lyapunov functional technique which leads to simple tuning rules for the controller parameters simulation results are discussed to verify the effectiveness of the proposed schemes

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

derived categories and deligne lusztig varieties ii this paper is a continuation and a completion of boro1 we extend the jordan decomposition of blocks we show that blocks of finite groups of lie type in non describing characteristic are morita equivalent to blocks of subgroups associated to isolated elements of the dual group the key new result is the invariance of the part of the cohomology in a given modular series of deligne lusztig varieties associated to a given levi subgroup under certain variations of parabolic subgroups we also show that the equivalence arises from a splendid rickard equivalence even in the setting of boro1 the finer homotopy equivalence was unknown as a consequence the equivalence preserves defect groups and categories of subpairs we finally determine when deligne lusztig induced representations of tori generate the derived category of representations

a classification of multiplicity free actions abstract the action of a complex algebraic groupgon an affine varietyvis said to be multiplicity free if each irreducible representation ofgoccurs at most once in the ring c v of regular functions onv this paper concerns the classical setting wherevis a vector space andgacts linearly onv the irreducible linear multiplicity free actions have been classified by v kac the focus here is on nonirreducible linear multiplicity free actions we exhibit many such actions which do not decompose as direct products of irreducible actions our main results provide a classification for all linear multiplicity free actions on a vector space

lp harmonic analysis and radon transforms on the heisenberg group in this paper harmonic analysis in the heisenberg group heisn cn r with group law z t z t z z t t 12im z z is taken to mean the joint spectral theory of the operators l heisenberg laplacian and t t the spectral decomposition of e l2 is given explicitly as e 0 e d where e 1 k 0 1 2 and e z t n 2 n 1 n 2k n 1 exp ie tn 2k exp z 24 n 2k lkn 1 z 22 n 2k where lkn 1 denotes the laguerre polynomial the eigenvalues are it e e n 2k e and l k e k e this decomposition is essentially well known but has not previously been described in this light it is also the decomposition into irreducibles of the representation of the heisenberg motion group semi direct product of heisn and u n on l2 heisn the main result is a summability in lp of the decomposition for e lp 1 p limr 1 rk 0 ed for p 1 only a weaker substitute is available and for p the decomposition is false there is also a plancherel formula for l2 2 2 k e n 2k 0 k e z 0 2dzd a number of explicit examples of the decomposition are computed including part of the schrodinger propagator eisl the heisenberg radon transform is defined by r z t n z t w 0 dw this operator was studied by geller and stein it is shown to be bounded from lp to lq if and only if p 2n 2 2n 1 and q 2n 2 some generalizations of these results to the free step two nilpotent lie groups are given but the results are not as complete

abelian subgroups nilpotent subgroups and the largest character degree of a finite group abstract let h be an abelian subgroup of a finite group g and the set of prime divisors of h we prove that h o g o g is bounded above by the largest character degree of g a similar result is obtained when h is nilpotent

homogeneity approach to high order sliding mode design it is shown that a general uncertain single input single output regulation problem is solvable only by means of discontinuous control laws giving rise to the so called high order sliding modes the homogeneity properties of the corresponding controllers yield a number of practically important features in particular finite time convergence is proved and asymptotic accuracy is calculated in a very general way in the presence of input noises discrete measurements and switching delays a robust homogeneous differentiator is included in the control structure thus yielding robust output feedback controllers with finite time convergence it is demonstrated that homogeneity features significantly simplify the design and investigation of a new family of high order sliding mode controllers finally simulation results are presented

bounded k spherical functions on heisenberg groups abstract let hn be the 2n 1 dimensional heisenberg group and let k be a compact subgroup of aut hn the group of automorphisms of hn the pair k hn is called a gelfand pair if lk1 hn the subalgebra of elements of l1 hn that are invariant under the action of k is commutative in this case the continuous homomorphisms on lk1 hn are given by integrating against certain k invariant functions on hn these functions are the k spherical functions associated to the gelfand pair k hn in this paper we show how to compute the bounded k spherical functions on hn

the blocks and weights of finite special linear and unitary groups abstract this paper has two main parts first we give a classification of the l blocks of finite special linear and unitary groups sl n q in the non defining characteristic l 3 second we describe how the l weights of sl n q can be obtained from the l weights of gl n q when l gcd n q and verify the alperin weight conjecture for sl n q under the condition l gcd n q as a step to establish the alperin weight conjecture for all finite groups we prove the inductive blockwise alperin weight condition for any unipotent l block of sl n q if l gcd n q

gelfand pairs on the heisenberg group and schwartz functions abstract let h n be the 2 n 1 dimensional heisenberg group and k a compact group of automorphisms of h n such that k h n k is a gelfand pair we prove that the gelfand transform is a topological isomorphism between the space of k invariant schwartz functions on h n and the space of schwartz function on a closed subset of r s homeomorphic to the gelfand spectrum of the banach algebra of k invariant integrable functions on h n

null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time varying feedback laws to this end on the one hand we provide a new proof of the well posedness and the optimal bound with respect to damping constants for the solutions of the kernel equations this allows one to deal with variable coefficients even with a weak regularity of these coefficients on the other hand we establish the well posedness and estimates for the heat equations with a nonlocal boundary condition at one side

radical 3 subgroups and essential 3 rank of f4 q abstract we classify maximal 3 local subgroups and radical 3 subgroups of g f 4 q when q odd and then determine the essential 3 rank of the frobenius category f d g where d is a sylow 3 subgroup of g

an equivariant bijection between irreducible brauer characters and weights for sp 2n q abstract the longstanding alperin weight conjecture and its blockwise version have been reduced to simple groups recently by navarro tiep spath and koshitani thus to prove this conjecture it suffices to verify the corresponding inductive condition for all finite simple groups the first is to establish an equivariant bijection between irreducible brauer characters and weights for the universal covering groups of simple groups assume q is a power of some odd prime p we first prove the blockwise alperin weight conjecture for sp 2 n q and odd non defining characteristics if the decomposition matrix of sp 2 n q is unitriangular with respect to an aut sp 2 n q stable basic set this assumption holds for linear primes we can establish an equivariant bijection between the irreducible brauer characters and weights

equivariance and extendibility in finite reductive groups with connected center we show that several character correspondences for finite reductive groups g are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to g has connected center the correspondences we consider are the so called jordan decomposition of characters introduced by lusztig and the generalized harish chandra theory of unipotent characters due to broue malle michel in addition we consider a correspondence giving character extensions due to the second author in order to verify the inductive mckay condition from isaacs malle navarro for the non abelian finite simple groups of lie types 3 mathsf d 4 mathsf e 8 mathsf f 4 2 mathsf f 4 and mathsf g 2

uno s invariant conjecture for chevalley groups g2 q in nondefining characteristics abstract uno s invariant conjecture for the chevalley groups g 2 q has been verified when the characteristic of the modular representation is distinct from the defining characteristic of the groups thus dade s reductive conjecture and the isaacs navarro conjecture both hold for g 2 q in non defining characteristics

h control stabilization and input output stability of nonlinear systems with homogeneous properties new relations are established between h control and stabilization and between input output stability and lyapunov stability of nonlinear systems with homogeneous properties these results can be applied to h controller design and stability analysis

the spherical transform of a schwartz function on the heisenberg group suppose thatk u n is a compact lie group acting on the 2n 1 dimensional heisenberg grouphn we say that k hn is a gelfand pair if the convolution algebral1k hn of integrablek invariant functions onhnis commutative in this case the gelfand space k hn is equipped with the godement plancherel measure and the spherical transform l2k hn l2 k hn is an isometry the main result in this paper provides a complete characterization of the set sk hn f f sk hn of spherical transforms ofk invariant schwartz functions onhn we show that a functionfon k hn belongs to sk hn if and only if the functions obtained fromfvia application of certain derivatives and difference operators satisfy decay conditions we also consider spherical series expansions fork invariant schwartz functions onhnmodulo its center

controlled blocks of the finite quasisimple groups for odd primes abstract we classify controlled blocks introduced by alperin and broue in 1979 for all quasisimple groups g for odd primes the results imply that every nilpotent block of g has abelian defect groups which in turn is one of the main results proved in an and eaton 2011 6 we also give an explicit characterization of non controlled blocks of all quasisimple groups g for odd primes this implies the block theoretic analogue of glauberman s zj theorem for g proved by kessar linckelmann and robinson 2002 18

adding an integrator for the stabilization problem abstract we study the relationship between the following two properties p1 the system x f x y y v is locally asymptotically stabilizable and p2 the system x f x u is locally asymptotically stabilizable where x r n y r dayawansa martin and knowles have proved that these properties are equivalent if the dimension n 1 here using the so called control lyapunov function approach a we propose another more constructive and somewhat simpler proof of dayawansa martin and knowles s result b we show that in general p1 does not imply p2 for dimensions n larger than 1 c we prove that p2 implies p1 if some extra assumptions are added like homogeneity of the system by using the latter result recursively we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form

on rouquier blocks for finite classical groups at linear primes abstract w turner has proved that broue s abelian defect group conjecture holds for certain unipotent blocks of the finite general linear group the so called rouquier blocks this together with theorems of a marcus and of j chuang and r rouquier proves that the conjecture holds for all blocks of such groups we prove that other finite classical groups also possess analogues of rouquier blocks at linear primes

gelfand transforms of polyradial schwartz functions on the heisenberg group abstract we prove that the gelfand transform is a topological isomorphism between the space of polyradial schwartz functions on the heisenberg group and the space of schwartz functions on the heisenberg brush we obtain analogous results for radial schwartz functions on heisenberg type groups

on the endomorphism algebra of modular gelfand graev representations abstract we study the endomorphism algebras of a modular gelfand graev representation of a finite reductive group by investigating modular properties of homomorphisms constructed by curtis and curtis shoji

radical 3 subgroups of f4 q with q even abstract in a recent paper an and huang classified the radical 3 subgroups of f 4 q for q odd here we use a similar approach to classify the radical 3 subgroups of f 4 q with q even as an application we determine the essential rank of the frobenius category f d f 4 q for q even where d is a sylow 3 subgroup of f 4 q

on depth zero l packets for classical groups by computing reducibility points of parabolically induced representations we construct to within at most two unramified quadratic characters the langlands parameter of an arbitrary depth zero irreducible cuspidal representation formula presented of a classical group which may be not quasi split over a non archimedean local field of odd residual characteristic from this we can explicitly describe all the irreducible cuspidal representations in the union of one two or four formula presented packets containing formula presented these results generalize the work of debacker reeder in the case of classical groups from regular to arbitrary tame langlands parameters

finite time stability and robust control synthesis of uncertain switched systems stability analysis is developed for uncertain nonlinear switched systems while being asymptotically stable and homogeneous of degree q 0 these systems are shown to approach the equilibrium point in finite time restricted to second order systems this feature is additionally demonstrated to persist regardless of inhomogeneous perturbations based on this fundamental property switched control algorithms are then developed to globally stabilize uncertain minimum phase systems of uniform m vector relative degree 2 2 t the controllers constructed do not rely on the generation of sliding motions while providing robustness features similar to those possessed by their sliding mode counterparts the proposed synthesis procedure is illustrated via application to a friction servo motor

the irreducible brauer characters of the finite special linear groups in non describing characteristics abstract the irreducible brauer characters ofsln q are investigated for primeslnot dividingq they are described in terms of a set of ordinary characters ofsln q whose reductions modulolare a generating set of the additive group of generalized brauer characters and the decomposition numbers of this set these decomposition numbers are described by combinatorial means in terms of the decomposition numbers ofgln q the latter have been investigated in great detail the last 15 years and are known completely forn 10

on the inductive blockwise alperin weight condition for classical groups abstract recently there has been substantial progress on the alperin weight conjecture as a step to establish the alperin weight conjecture for all finite groups we prove the inductive blockwise alperin weight condition for simple groups of classical type under some additional assumption

maximal 2 local subgroups of e7 q abstract let g e 7 q be the finite exceptional group of lie type universal version we classify up to conjugacy all maximal proper 2 local subgroups of g that is all 2 local m g which are maximal with respect to inclusion among all proper subgroups of g which are 2 local for this purpose we first determine up to conjugacy all elementary abelian 2 subgroups of g containing the center z g these classifications are an important first step towards a classification of the 2 radical subgroups of g

convolution kernels versus spectral multipliers for sub laplacians on groups of polynomial growth abstract let l be a sub laplacian on a connected lie group g of polynomial growth it is well known that if f r c is in the schwartz class s r then the convolution kernel k f l of the operator f l is in the schwartz class s g here we prove a sort of converse implication for a class of groups g including all solvable noncompact groups of polynomial growth we also discuss the problem whether integrability of k f l implies continuity of f

on the alperin mckay conjecture for simple groups of type a abstract in this paper characters of the normalizer of d split levi subgroups in sl n q and su n q are parametrized with a particular focus on the clifford theory between the levi subgroup and its normalizer these results are applied to verify the alperin mckay conjecture for primes l with l 6 q 2 1 and the alperin weight conjecture for l blocks of those quasi simple groups with abelian defect the inductive alperin mckay condition and inductive alperin weight condition by the second author are verified for certain blocks of sl n q and su n q

on the morita frobenius numbers of blocks of finite reductive groups abstract we show that the morita frobenius number of the blocks of the alternating groups the finite groups of lie type in defining characteristic and the ree and suzuki groups is 1 we also show that the morita frobenius number of almost all of the unipotent blocks of the finite groups of lie type in non defining characteristic is 1 and that in the remaining cases it is at most 2

boundary time varying feedbacks for fixed time stabilization of constant parameter reaction diffusion systems in this paper the problem of fixed time stabilization of constant parameter reaction diffusion partial differential equations by means of continuous boundary time varying feedbacks is considered moreover the time of convergence can be prescribed in the design the design of time varying feedbacks is carried out based on the backstepping approach using a suitable target system with a time varying coefficient one can state that the resulting kernel of the backstepping transformation is time varying and rendering the control feedback to be time varying as well explicit representations of the kernel solution in terms of generalized laguerre polynomials and modified bessel functions are derived the fixed time stability property is then proved a simulation example is presented to illustrate the main results

brauer s height zero conjecture for two primes let p and q be two primes we propose that brauer s height zero conjecture for the principal p blocks of finite groups can naturally be extended from the perspective of q we prove one direction of this new conjecture and show the reverse direction assuming that the inductive alperin mckay condition holds for the finite simple groups

on unipotent blocks and their ordinary characters there is a great deal of evidence suggesting that block theory for finite reductive groups should lead to an enrichment of the harish chandra theory for those groups see s 5 1 fs2 p2 3 4 h 6 1 10 an e generalized harish chandra theory for e a positive integer is introduced in bmm i a the context is as follows let g be a connected reductive group defined over lfq with associated frobenius endomorphism f let e 1 an integer when l is an f stable levi subgroup of g one denotes by r and re the lusztig functor and its adjoint a unipotent e pair l l is defined by an e split levi subgroup l of g and a unipotent character z d l 1 one writes li z1 l2 j 2 if and only if l1 c l 2 and zl l2 l 1 2 l 4 0 see 3 l the notion of unipotent e cuspidal pairs and the resulting partition of g g v 1 are defined as in the classical case corresponding to e 1 if f is a prime not dividing q and e is the order of q mod f it seems natural in connection with blocks to ask if the partition of the unipotent characters 8 g v l resulting from e theory coincides with f blocks the blocks of g v which are defined by unipotent characters are called unipoteut f blocks we prove

unipotent elements of finite groups of lie type and realization fields of their complex representations abstract let p be a prime this paper classifies finite connected reductive groups g in characteristic p with the property that all complex character values of g belong to an unramified above p extension of the field of rational numbers the main application of these results is to the problem of describing the irreducible complex or p adic representations of g that remain absolutely irreducible under the brauer reduction modulo p an efficient approach to solve this problem for p 3 has been developed in our paper proc london math soc 3 84 2002 439 together with proc london math soc 3 84 2002 439 theorem 1 9 of the paper solves this problem for many finite connected reductive groups in characteristic p 3 additionally we show that all complex representations of any finite connected reductive group with no composition factor of type e7 2f e8 2f and e8 5f can be realized over a quadratic extension of an unramified above p extension of q

scopes reduction and morita equivalence classes of blocks in finite classical groups ii abstract we extend our work of j algebra 230 2000 378 423 on unipotent blocks of classical groups to blocks of orthogonal groups containing characters labelled by degenerate symbols we also show that for classical groups in the linear prime case there are only finitely many morita types of unipotent blocks with a given defect group