singular solutions in nonlinear parabolic equations with anisotropic nonstandard growth conditions in this paper we study extinction and blow up properties of weak solutions for the doubly nonlinear parabolic problem involving p x t laplacian operator subject to homogeneous dirichlet boundary conditions we first prove the existence and uniqueness of weak solutions in some anisotropic orlicz sobolev spaces and also establish suitable regularity of weak solutions with respect to the introduced control functional second we use the energy estimate and the comparison principle of the ordinary differential equation to study the extinction or non extinction of solutions third we combine some new energy functions with the embedding inequalities to show blow up criteria of solutions moreover the singular rates of weak solutions are determined it is interesting that blow up or extinction phenomena depend sensitively on the sign of the precise relationship among the different anisotropic variable exponents which discovers a key clue to the different effects of diffusion and reaction of the equations on the singularities of weak solutions in this paper we study extinction and blow up properties of weak solutions for the doubly nonlinear parabolic problem involving p x t laplacian operator subject to homogeneous dirichlet boundary conditions we first prove the existence and uniqueness of weak solutions in some anisotropic orlicz sobolev spaces and also establish suitable regularity of weak solutions with respect to the introduced control functional second we use the energy estimate and the comparison principle of the ordinary differential equation to study the extinction or non extinction of solutions third we combine some new energy functions with the embedding inequalities to show blow up criteria of solutions moreover the singular rates of weak solutions are determined it is interesting that blow up or extinction phenomena depend sensitively on the sign of the precise relationship among the different anisotropic variable exponents which discovers a key clue to the different effects of diffusion and reaction of the equations

quenching of solutions of parabolic equations with nonlinear boundary conditions in several dimensions on considere le probleme suivant a n dimensions u t au t 0 x u x t 0 t 0 x u x t u x t t 0 x u x 0 0 x ou est un domaine de r n n 2 a est un operateur elliptique lineaire independant du temps est une derivee oblique sur et 1 0 est continument differentiable et croissante et 1

mixed boundary value problems for elliptic and parabolic differential equations of second order on considere les problemes aux valeurs limites mixtes suivants lu f 1 dans mu f 2 sur u f 3 sur avec l operateur elliptique et pu f 1 dans 0 t mu f 2 sur 0 t u f 3 sur 0 t 0 pour des operateurs p paraboliques pour des operateurs obliques generaux m et pour des domaines ou et se rencontrent a des angles arbitraires on montre l existence de solutions classiques quand les angles sont non nuls et f 3 est continue de holder

cohomology of infinitesimal and discrete groups an infinitesimal algebraic group over a field k is a connected algebraic group scheme whose coordinate ring is a finite dimension al k algebra if one begins with a reductive algebraic group g defined over fp then one has a close relationship between the representation theory of g that of its infinitesimal subgroups g defined as the kernels of the frobenius endomorphisms rr g g and that of its finite chevalley subgroups g fq of fq rational points q pd a basic theorem of cline et al i 9 theorem 6 6 relates the cohomology of g in a rational representation to that of the finite groups g fq related work of cline et al 6 and of the authors 1 16 relates the cohomology of g to that of its infinitesimal subgroups a determination of the cohomology of the first infinitesimal subgroup g 1 of g with trivial coefficients has enabled the authors in 14 to make certain explicit cohomology calculations for the finite groups g fq with certain nontrivial coefficients in this paper we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge in particular we provide a computation which applies in a range of degrees to chevalley groups over the prime field we also investigate the cohomology of infinitesimal subgroups of nilpotent groups enabling us for example to determine the cohomology of their lie algebras in certain cases we apply this lie algebra cohomology computation to determine the integral cohomology of various torsion free nilpotent groups modulo small prime torsion thereby extending a theorem of dwyer 12 in more detail the paper begins in sect 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup g which was previously studied in 14 from this spectral sequence it easily follows that h g1 m is a noetherian h g1 k module for each finite dimensional rational gl module m we then apply the work of andersen and jantzen 1 on the gl cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup gz our analysis also permits a determination of the generic

the multivariate meixner polynomials as matrix elements of so d 1 representations on oscillator states the multivariate meixner polynomials are shown to arise as matrix elements of unitary representations of the so d 1 group on oscillator states these polynomials depend on d discrete variables and are orthogonal with respect to the negative multinomial distribution the emphasis is put on the bivariate case for which the so 2 1 connection is used to derive the main properties of the polynomials orthogonality relation raising lowering relations generating function recurrence relations and difference equations as well as explicit expressions in terms of standard univariate krawtchouk and meixner polynomials it is explained how these results generalize directly to d variables

the multivariate krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states an algebraic interpretation of the bivariate krawtchouk polynomials is provided in the framework of the three dimensional isotropic harmonic oscillator model these polynomials in two discrete variables are shown to arise as matrix elements of unitary reducible representations of the rotation group in three dimensions many of their properties are derived by exploiting the group theoretic setting the bivariate tratnik polynomials of krawtchouk type are seen to be special cases of the general polynomials that correspond to particular rotations involving only two parameters it is explained how the approach generalizes naturally to d 1 dimensions and allows us to interpret multivariate krawtchouk polynomials as matrix elements of so d 1 unitary representations indications are given on the connection with other algebraic models for these polynomials

remarks on quenching on considere le probleme aux limites parabolique non lineaire suivant u t u f u t 0 x u 0 t 0 n u 0 x u 0 x 0 x on demontre un phenomene de trempe pour des non linearites non convexes dans un espace de dimension arbitraire

second cohomology groups for frobenius kernels and related structures abstract let g be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p let b be a borel subgroup of g and u be its unipotent radical in this paper we determine the second cohomology groups of b and its frobenius kernels for all simple b modules we also consider the standard induced modules obtained by inducing a simple b module to g and compute all second cohomology groups of the frobenius kernels of g for these induced modules also included is a calculation of the second ordinary lie algebra cohomology group of lie u with coefficients in k

generic cohomology for twisted groups let g be a simple algebraic group defined and split over ko fp and let a be a surjective endomorphism of g with finite fixed point set go we give conditions under which cohomology groups of g are isomorphic to cohomology groups of go let g be a simple algebraic group defined and split over k fp and for q pm let g q be the subgroup of fq rational points for a finite dimensional rational g module v and a nonnegative integer e let v e be the g module obtained by twisting the original g action on v by the frobenius endomorphism x x1p of g cline parshall scott and van der kallen proved in 2 that for sufficiently large q and e depending on v and n the restriction map induces an isomorphism from the rational cohomology group h g v e to h g q v e this implies that as q increases the groups h g q v e have a stable or generic value h en g v in this paper we prove an analogous theorem with g q replaced by go for a surjective endomorphism a of g having finite fixed point set the first section of the paper summarizes the basic results on endomorphisms of algebraic groups required for the proof some arithmetic facts are established in the second section and the main theorem is proved in the third section the author is grateful to leonard l scott for several helpful conversations and to the referee for comments on an earlier version of this paper 1 endomorphisms of algebraic groups we briefly review here some results on endomorphisms of algebraic groups which will be needed later we refer the reader to 4 and 5 for more details let k be the algebraic closure of the prime field ko fp and let g be a simple algebraic group defined and split over ko if a is a surjective rational endomorphism of g having finite fixed point set g0 a stabilizes a borel subgroup b and a maximal torus t ua is a t equivariant isomorphism then axa u x cauq a for some received by the editors may 5 1980 and in revised form november 12 1980 1980 mathematics subject classification primary 20g10

parabolic problems with nonlinear absorptions and releases at the boundaries let a p and q be positive constants t d 0 a d be the closure of d w dx 0 t and hu u t u x x this article studies the following heat conduction problem with nonlinear boundary conditions hu 0in w u x 0 u 0 x ond u x 0 t 1 u 0 t p u x a t 1 u a t q 0

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

interbasis expansions for the isotropic 3d harmonic oscillator and bivariate krawtchouk polynomials an explicit expression for the general bivariate krawtchouk polynomials is obtained in terms of the standard krawtchouk and dual hahn polynomials the bivariate krawtchouk polynomials occur as matrix elements of the unitary reducible representations of so 3 on the energy eigenspaces of the three dimensional isotropic harmonic oscillator and the explicit formula is obtained from the decomposition of these representations into their irreducible components the decomposition entails expanding the cartesian basis states in the spherical bases that span irreducible so 3 representations the overlap coefficients are obtained from the clebsch gordan problem for the lie algebra

a superintegrable discrete oscillator and two variable meixner polynomials a superintegrable discrete model of the quantum isotropic oscillator in two dimensions is introduced the system is defined on the regular infinite dimensional lattice it is governed by a hamiltonian expressed as a seven point difference operator involving three parameters the exact solutions of the model are given in terms of the two variable meixner polynomials orthogonal with respect to the negative trinomial distribution the constants of motion of the system are constructed using the raising and lowering operators for these polynomials they are shown to generate an invariance algebra the two variable meixner polynomials are seen to support irreducible representations of this algebra in the continuum limit where the lattice constant tends to zero the standard isotropic quantum oscillator in two dimensions is recovered the limit process from the two variable meixner polynomials to a product of two hermite polynomials is carried out by involving the bivariate charlier polynomials

quenching of solutions to a class of semilinear parabolic equations with boundary degeneracy abstract this paper concerns the quenching phenomenon of solutions to a class of semilinear parabolic equations with boundary degeneracy in the case that the degeneracy is not strong it is shown that there exists a critical length which is positive such that the solution exists globally in time if the length of the spatial interval is less than it while quenches in a finite time if the length of the spatial interval is greater than it whereas in the case that the degeneracy is strong enough the solution must be quenching in a finite time no matter how long the spatial interval is furthermore for each quenching solution the set of quenching points is determined and it is proved that its derivative with respect to the time must blow up at the quenching time

carleman estimates for a class of degenerate parabolic operators given alpha in 0 2 and f in l 2 0 t times 0 1 we derive new carleman estimates for the degenerate parabolic problem w t x alpha w x x f where t x in 0 t times 0 1 associated to the boundary conditions w t 1 0 and w t 0 0 if 0 leq alpha 1 or x alpha w x t 0 0 if 1 leq alpha 2 the proof is based on the choice of suitable weighted functions and hardy type inequalities as a consequence for all 0 leq alpha 2 and omega subset subset 0 1 we deduce null controllability results for the degenerate one dimensional heat equation u t x alpha u x x h chi omega with the same boundary conditions as above

complexity for modules over finite chevalley groups and classical lie algebras 1 1 let g be a connected reductive algebraic group defined over a finite prime field fp of p elements for a given rational g module m one can consider the restrictions of m to the restricted lie algebra lie g and to the finite group g fp of fp rational points even though there is no direct functorial connection between the categories of restricted lie g modules and g fp modules the ambient algebraic group g can serve as a bridge linking these two important module categories early results of curtis in the 1960s provided a one to one correspondence between the simple restricted modules over k fp of lie g and those of g fp this correspondence is given by simply restricting certain classes of simple g modules more importantly these results along with work of steinberg allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories for over thirty years this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups lie algebras and finite groups of lie type in the defining characteristic see e g cpsvan jan2 hum3 and jan1 given the previous results in this area a natural course of investigation should be to find relationships between canonical modules like projective modules for lie g and g fp more precisely

quenching phenomena in strongly degenerate diffusion equations with strong absorption abstract we examine the degenerate parabolic equation u t u p u xx u q u 0 with boundary value u 1 in a bounded real interval where p 1 and q 1 p 1 when investigating the evolution of the dead core set u t 0 we find that the two regimes q p and q 1 p show completely different behavior finally we rule out the possibility of quenching in infinite time under certain assumptions on p q and the size of

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

second cohomology groups for algebraic groups and their frobenius kernels let g be a simple simply connected algebraic group scheme defined over an algebraically closed field of characteristic p 0 let t be a maximal split torus in g b t be a borel subgroup of g and u its unipotent radical let f g g be the frobenius morphism for r 1 define the frobenius kernel gr to be the kernel of f iterated with itself r times define ur respectively br to be the kernel of the frobenius map restricted to u respectively b let x t be the integral weight lattice and x t be the dominant integral weights the computations of particular importance are h2 u1 k h2 br for x t h2 gr h0 for x t and h2 b for x t the above cohomology groups for the case when the field has characteristic 2 are computed in this paper these computations complete the picture started by bendel nakano and pillen 2007 5 for p 3 furthermore the computations show h2 gr h0 has a good filtration

some multivariable orthogonal polynomials of the askey tableau discrete families a multivariable generalization is presented for all the discrete families of the askey tableau this significantly extends the multivariable hahn polynomials introduced by karlin and mcgregor the latter are recovered as a limit case from a family of multivariable racah polynomials

the best constant in a trace inequality abstract the best possible constant in the trace inequality v l 1 1 0 v 2 v h 1 2 is shown to be given by a quantity in terms of the solution of an elliptic boundary value problem where 1 2 are disjoint subsets of with positive measures and the sobolev space h 1 2 v h 1 v 0 on 2 some examples are presented with either an analytic expression or a numerical value for the best constant as an application the best constant c0 is used to give an upper bound for the first eigenvalue of a modified steklov eigenvalue problem

the multivariate hahn polynomials and the singular oscillator karlin and mcgregor s d variable hahn polynomials are shown to arise in the dimensional singular oscillator model as the overlap coefficients between bases associated with the separation of variables in cartesian and hyperspherical coordinates these polynomials in d discrete variables depend on real parameters and are orthogonal with respect to the multidimensional hypergeometric distribution the focus is put on the d 2 case for which the connection with the three dimensional singular oscillator is used to derive the main properties of the polynomials forward backward shift operators orthogonality relation generating function recurrence relations bispectrality difference equations and explicit expression in terms of the univariate hahn polynomials the extension of these results to an arbitrary number of variables is presented at the end of the paper

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

sheaf cohomology on gb and tensor products of weyl modulus let k be an algebraically closed field it is assumed that char k p 0 unless otherwise indicated g is a simply connected semisimple algebraic group over k t is a fixed maximal torus and b a bore1 subgroup of g containing t let qi g t be the root system of g d the set of all simple roots and and the sets of all positive and negative roots relative to d respectively we assume that the set a is chosen such that b corresponds to 0 let x be the weight lattice of sp which may be considered as the character group of t or b therefore if 1 e x by abuse of notation we also denote by l the one dimensional tor b module determined by a let x be the set of all dominant weights and w the fundamental weight corresponding to a simple root a e a we denote by 6 the half sum of all positive roots and we know that 6 is the sum of all fundamental weights let w be the weyl group of 0 the reflection with respect to a root a is denoted by s the longest element of w by w there are two ways of w acting on x i the ordinary action if 1 e x m e w the image of i under w is denoted by 2 ii dot action w a 5 2 6 s in particular let a w a the dual root system of is denoted by and the dual root of a e by a i e u 2a a u where is the inner product invariant under the ordinary action of w of the euclidean space spanned by for 1 e x a e let a a 1 a all modules of an algebraic group we consider in this paper are assumed to be rational in particular for an infinite dimensional module the term rational module means this module is a direct limit of finite dimensional rational modules let h be a closed subgroup of g containing t such as g b or t m an h module we define for l e x

on quenching for some parabolic problems abstract the universal quenching of a porous medium equation is established we also obtain some universal quenching properties for some semilinear parabolic problems and dichotomy properties for some general one dimensional problems

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

the multivariate charlier polynomials as matrix elements of the euclidean group representation on oscillator states a family of multivariate orthogonal polynomials generalizing the standard univariate charlier polynomials is shown to arise in the matrix elements of the unitary representation of the euclidean group e d on oscillator states these polynomials in d discrete variables are orthogonal on the product of d poisson distributions the accent is put on the d 2 case and the group theoretical setting is used to obtain the main properties of the polynomials orthogonality and recurrence relations difference equation raising lowering relations generating function hypergeometric and integral representations and explicit expression in terms of standard charlier and krawtchouk polynomials the approach is seen to extend straightforwardly to an arbitrary number of variables the contraction of so 3 to e 2 is used to show that the bivariate charlier polynomials correspond to a limit of the bivariate krawtchouk polynomials

evolutionary weighted p laplacian with boundary degeneracy abstract the paper concerns the well posedness problem of an evolutionary weighted p laplacian with boundary degeneracy different from the classical theory for linear equations it is shown that the degenerate portion of the boundary should be decomposed into two parts the strongly degenerate boundary on which the equation exhibits hyperbolic characteristics and the weakly degenerate boundary on which the equation still exhibits parabolic characteristics we formulate reasonably the boundary value condition and establish the existence and uniqueness theorems

no quenching in infinite time for degenerate singular semilinear parabolic equations let q be any nonnegative real number and t for the degenerate singular semilinear parabolic initial boundary value problemx qu t u x x f u for0

modular characteristic classes for representations over finite fields abstract the cohomology of the degree n general linear group over a finite field of characteristic p with coefficients also in characteristic p remains poorly understood for example the lowest degree previously known to contain nontrivial elements is exponential in n in this paper we introduce a new system of characteristic classes for representations over finite fields and use it to construct a wealth of explicit nontrivial elements in these cohomology groups in particular we obtain nontrivial elements in degrees linear in n we also construct nontrivial elements in the mod p homology and cohomology of the automorphism groups of free groups and the general linear groups over the integers these elements reside in the unstable range where the homology and cohomology remain mysterious

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

nonlinear wave equations on the two dimensional sphere we study a nonlinear wave equation on the two dimensional sphere with a blowing up nonlinearity the existence and uniqueness of a local regular solution are established also the behavior of the solutions is examined we show that a large class of solutions to the initial value problem quench in finite time

quenching behavior of solutions to heat equations with coupled boundary singularities abstract this paper deals with finite time quenching for one dimensional heat equations coupled via negative singular boundary sources in the more general product form of two component powers u p 1 v q 1 and u p 2 v q 2 differently from the generally considered there are two additional singular factors namely u p 1 and v q 2 for the boundary sources of u and v respectively we at first propose a criterion to distinguish simultaneous and non simultaneous quenching and then determine all possible blow up rates in the model three for simultaneous quenching and two for non simultaneous quenching under different nominations this shows real contributions of the two additional singular factors to the quenching behavior of solutions

second cohomology for finite groups of lie type let g be a simple simply connected algebraic group defined over mathbb f p given a power q p r of p let g mathbb f q subset g be the subgroup of mathbb f q rational points let l lambda be the simple rational g module of highest weight lambda in this paper we establish sufficient criteria for the restriction map in second cohomology h 2 g l lambda rightarrow h 2 g mathbb f q l lambda to be an isomorphism in particular the restriction map is an isomorphism under very mild conditions on p and q provided lambda is less than or equal to a fundamental dominant weight even when the restriction map is not an isomorphism we are often able to describe h 2 g mathbb f q l lambda in terms of rational cohomology for g we apply our techniques to compute h 2 g mathbb f q l lambda in a wide range of cases and obtain new examples of nonzero second cohomology for finite groups of lie type

extinction properties of semilinear heat equations with strong absorption consider the initial boundary value problem for ut u uq with 0 0 q 1 the initial data are nonnegative and the boundary data vanish it is well known that the solution becomes extinct in finite time t i e u x t becomes identically zero for t t where t is some positive number in this paper we study the profile of x u x t as t t

limits of weight spaces lusztig s q analogs and fiberings of adjoint orbits let g be a connected complex semisimple algebraic group and t a maximal torus inside a borel subgroup b with g t and b their lie algebras let v be a representation in the category a for g the t decomposition v met va of v into a direct sum of finite dimensional weight spaces is central in the representation theory of g in this paper we introduce on weight spaces a new structure the principal filtration je v where e is a principal nilpotent in g chosen to be compatible with t for example e can be the sum of the simple root vectors relative to t b this filtration is constructed in a very simple way by taking j v to be the space of vectors annihilated by the p 1 th power of e for p 0 our approach is motivated by kostant s fundamental work ki k2 on actions of the principal tds three dimensional subalgebra and coordinate rings of regular adjoint orbits in theorem 3 4 we give a new description in terms of the dimension jumps of the principal filtration of the weight space of lusztig s l q analog mm q of dominant weight multiplicity in a finite dimensional irreducible g representation v the polynomial m q was defined algebraically as an alternating sum over the weyl group through a q analog of kostant s weight multiplicity formula we prove that m q is equal to the jump polynomial

support varieties for modules over chevalley groups and classical lie algebras let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 g 1 be the first frobenius kernel and g f p be the corresponding finite chevalley group let m be a rational g module in this paper we relate the support variety of m over the first frobenius kernel with the support variety of m over the group algebra kg f p this provides an answer to a question of parshall applications of our new techniques are presented which allow us to extend results of alperin mason and janiszczak jantzen and to calculate the dimensions of support varieties for finite chevalley groups

elementary subalgebras of lie algebras abstract we initiate the investigation of the projective varieties e r g of elementary subalgebras of dimension r of a p restricted lie algebra g for various r 1 these varieties e r g are the natural ambient varieties for generalized support varieties for restricted representations of g we identify these varieties in special cases revealing their interesting and varied geometric structures we also introduce invariants for a finite dimensional u g module m the local r j radical rank and local r j socle rank functions which are lower upper semicontinuous on e r g examples are given of u g modules for which some of these rank functions are constant