unipotent elements and generalized exponential maps abstract let g be a simple and simply connected algebraic group over an algebraically closed field k of characteristic p 0 assume that p is good for the root system of g and that the covering map g s c g is separable in previous work we proved the existence of a not necessarily unique springer isomorphism for g that behaved like the exponential map on the restricted nullcone of g in the present paper we give a formal definition of these maps which we call generalized exponential maps we provide an explicit and uniform construction of such maps for all root systems demonstrate their existence over z p and give a complete parameterization of all such maps one application is that this gives a uniform approach to dealing with the saturation problem for a unipotent element u in g providing a new proof of the known result that u lies inside a subgroup of c g u that is isomorphic to a truncated witt group we also develop a number of other explicit and new computations for g and for g this paper grew out of an attempt to answer a series of questions posed to us by p deligne who also contributed several of the new ideas that appear here

on the geometry of four qubit invariants the geometry of four qubit entanglement is investigated we replace some of the polynomial invariants for four qubits introduced recently by new ones of direct geometrical meaning it is shown that these invariants describe four points six lines and four planes in complex projective space cp3 for the generic entanglement class of stochastic local operations and classical communication they take a very simple form related to the elementary symmetric polynomials in four complex variables moreover suitable powers of their magnitudes are entanglement monotones that fit nicely into the geometric set of n qubit ones related to grassmannians of l planes found recently we also show that in terms of these invariants the hyperdeterminant of order 24 in the four qubit amplitudes takes a more instructive form than the previously published expressions available in the literature finally in order to understand two three and four qubit entanglement in geometric terms we propose a unified setting based on cp3 furnished with a fixed quadric

component groups of unipotent centralizers in good characteristic abstract let g be a connected reductive group over an algebraically closed field of good characteristic for u g unipotent we describe the conjugacy classes in the component group a u of the centralizer of u our results extend work of the second author done for simple adjoint g over the complex numbers when g is simple and adjoint the previous work of the second author makes our description combinatorial and explicit moreover it turns out that knowledge of the conjugacy classes suffices to determine the group structure of a u thus we obtain the result previously known through case checking that the structure of the component group a u is independent of good characteristic

on ad n x2 k series of universal quantum dimensions we present a universal in vogel s sense expression for the quantum dimension of the cartan product of arbitrary powers of the adjoint and x2 representations of simple lie algebras the same formula mysteriously yields quantum dimensions of some other representations of the same lie algebra for permuted universal parameters provided linear resolvability of singularities is applied we list these representations for the exceptional algebras and their stable versions for the classical algebras when the rank of the classical algebra is sufficiently large with regard to the powers of representations universal formulas may have singularities at the points in vogel s plane corresponding to some simple lie algebras we prove that our formula is linearly resolvable at all those singular points i e yields finite answers when restricted either on the classical or the exceptional lines and make a conjecture that these answers coincide with quantum dimensions of some irreducible representations in a number of cases particularly in the case of so 8 algebra which belongs both to the orthogonal and the exceptional lines it is confirmed that both resolutions yield relevant answers we note that an irreducible representation may have several universal formulas for its quantum dimension and discuss the impact of this phenomenon to the method presented by cohen and de man c r acad sci ser 1 322 5 427 432 1996 for derivation of universal formulas

moduli spaces and castelnuovo mumford regularity of sheaves on surfaces we generalize donaldson gieseker li and o grady s results on generic smoothness of the moduli spaces of sheaves with fixed determinant and large discriminant to positive characteristic we also show optimal bounds on the castelnuovo mumford regularity of sheaves on surfaces and we use it to give the first general effective results on irreducibility of the moduli spaces

restrictions of representations of a surface group to a pair of free subgroups let be the fundamental group of a compact orientable genus m surface and let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero define two free rank m subgroups of by a a 1 a m and b b 1 b m where a 1 a m b 1 b m j 1 m a j b j is the standard presentation of we consider representations of of a and of b into g restriction of representations induces a morphism from c g the variety of closed conjugacy classes of representations of to c a g c b g we prove that if m is greater than the semisimple rank of g then this morphism is dominant and almost all fibers are finite

reductive subgroups of reductive groups in nonzero characteristic abstract let g be a possibly nonconnected reductive linear algebraic group over an algebraically closed field k and let n n the group g acts on gn by simultaneous conjugation let h be a reductive subgroup of g we prove that if k has nonzero characteristic then the natural map of quotient varieties hn h gn g is a finite morphism we use methods introduced by vinberg who proved the same result in characteristic zero as an application we show that if is a finite group then the character variety c g of closed conjugacy classes of representations from to g is finite

explicit orbit classification of reducible jordan algebras and freudenthal triple systems we determine explicit orbit representatives of reducible jordan algebras and their corresponding freudenthal triple systems this work has direct application to the classification of extremal black hole solutions of mathcal n 2 4 locally supersymmetric theories of gravity coupled to an arbitrary number of abelian vector multiplets in d 4 5 space time dimensions

steady states invariant measures and response theory using the method of computer simulation we test the predictions of nonlinear response theory for classical systems subject to dissipative external fields we provide convincing numerical evidence that kawasaki methods agree with both the transient time correlation function predictions and with a direct measurement of the nonlinear response furthermore this numerical agreement is observed over a time scale which is sufficiently long for the response to have relaxed to within ensuremath sim 1 of its nonequilibrium steady state value this is in spite of the fact that in the steady state the n particle distribution function ultimately becomes fractal we discuss the normalization of the kawasaki distribution and derive a lagrangian form of the kawasaki response function and show that it is consistent with predictions that are obtained using a natural invariant measure for nonequilibrium steady states c 1995 the american physical society

causality response theory and the second law of thermodynamics we show that there is a close connection between the assumption of causality and the second law of thermodynamics we also show that for a class of classical reversible deterministic systems it is overwhelmingly improbable either to find causal steady states that violate the second law or anticausal states that satisfy the second law these arguments indicate that the existence of and the sign associated with the second law of thermodynamics is ultimately determined by causality our discussion employs a green kubo relation that we derive for an anticausal linear transport coefficient

the orbit structure of dynkin curves let g be a simple algebraic group over an algebraically closed field k assume that char k is zero or good for g let mathcal b be the variety of borel subgroups of g and let e in rm lie g be nilpotent there is a natural action of the centralizer cg e of e in g on the springer fibre mathcal b e b in mathcal b mid e in rm lie b associated to e in this paper we consider the case where e lies in the subregular nilpotent orbit in this case mathcal b e is a dynkin curve we give a complete description of the cg e orbits in mathcal b e in particular we classify the irreducible components of mathcal b e on which cg e acts with finitely many orbits in an application we obtain a classification of all subregular orbital varieties admitting a finite number of b orbits for b a fixed borel subgroup of g

poincare families of g bundles on a curve let g be a reductive group over an algebraically closed field k consider the moduli space of stable principal g bundles on a smooth projective curve c over k we give necessary and sufficient conditions for the existence of poincare bundles over open subsets of this moduli space and compute the orders of the corresponding obstruction classes this generalizes the previous results of newstead ramanan and balaji biswas nagaraj newstead to all reductive groups to all topological types of bundles and also to all characteristics

cocharacter closure and the rational hilbert mumford theorem for a field k let g be a reductive k group and v an affine k variety on which g acts using the notion of cocharacter closed g k orbits in v we prove a rational version of the celebrated hilbert mumford theorem from geometric invariant theory we initiate a study of applications stemming from this rationality tool a number of examples are discussed to illustrate the concept of cocharacter closure and to highlight how it differs from the usual zariski closure

the fluctuation theorem and lyapunov weights the fluctuation theorem ft is a generalisation of the second law of thermodynamics that applies to small systems observed for short times for thermostatted systems it gives the probability ratio that entropy will be consumed rather than produced in this paper we derive the transient and steady state fts using lyapunov weights rather than the usual gibbs weights at long times the fts so derived are identical to those derived using the more standard gibbs weights

orbits and invariants associated with a pair of commuting involutions let be commuting involutions of the connected reductive algebraic group g where and g are defined over a usually algebraically closed field k char k 2 we have fixed point groups h g and k g and an action h k g g where h k g hgk 1 h h k k g g let g h k denote spec o g h k the categorical quotient let a be maximal among subtori s of g such that s s s 1 for all s s there is the associated weyl group w wh k a we show the inclusion a g induces an isomorphism a w g h k in particular the closed h k orbits are precisely those which intersect a the fibers of g g h k are the same as those occurring in certain associated symmetric varieties in particular the fibers consist of finitely many orbits we investigate the structure of w and its relation to other naturally occurring weyl groups and to the action of on the a weight spaces of the relation of the orbit type stratifications of a w and g h k along the way we simplify some of richardson s proofs for the symmetric case and at the end we quickly recover results of berger flensted jensen hoogenboom and matsuki ber57 fj78 hoo84 mat97 for the case k

relative springer isomorphisms let g be a simple algebraic group over the algebraically closed field k a slightly strengthened version of a theorem of t a springer says that under some mild restrictions on g and k there exists a g equivariant isomorphism of varieties u n where u denotes the unipotent variety of g and n denotes the nilpotent variety of g lieg such is called a springer isomorphism let b be a borel subgroup of g u the unipotent radical of b and u the lie algebra of u in this note we show that a springer isomorphism induces a b equivariant isomorphism u m u m where m is any unipotent normal subgroup of b and m liem we call such a map a relative springer isomorphism we also use relative springer isomorphisms to describe the geometry of u orbits in u

three qutrit entanglement and simple singularities in this paper we use singularity theory to study the entanglement nature of pure three qutrit systems we first consider the algebraic variety x of separable three qutrit states within the projective hilbert space given a quantum pure state we define the x hypersuface by cutting x with a hyperplane h defined by the linear form the x hypersurface of x is we prove that when ranges over the stochastic local operation and classical communication entanglement classes the worst possible singular x hypersuface with isolated singularities has a unique singular point of type d 4

the spherical maximal operator on radial functions abstract we use simple one dimensional operators to bound pointwise the spherical maximal operator acting on radial functions with this bounds we obtain weighted inequalities which are sharp for power weights we also discuss boundedness results on morrey spaces and give a fefferman stein type inequality the bounds of the two dimensional case can be use for the universal maximal operator as well

casimir eigenvalues for universal lie algebra for two different natural definitions of casimir operators for simple lie algebras we show that their eigenvalues in the adjoint representation can be expressed polynomially in the universal vogel s parameters and give explicit formulae for the generating functions of these eigenvalues

singularity of type d_4 arising from four qubit systems an intriguing correspondence between four qubit systems and simple singularity of type d4 is established we first consider the algebraic variety x of separable states within the projective hilbert space then cutting x with a specific hyperplane h we prove that the x hypersurface defined from the section x h x has an isolated singularity of type d4 it is also shown that this is the worst possible isolated singularity that one can obtain by this construction moreover it is demonstrated that this correspondence admits a dual version by proving that the equation of the dual variety of x which is nothing but the cayley hyperdeterminant of type 2 2 2 2 can be expressed in terms of the stochastic local operation and classical communication invariant polynomials as the discriminant of the miniversal deformation of the d4 singularity as a consequence of this correspondence one obtains a finer grained classification of entanglement classes of four qubit systems

complete reducibility and separability let g be a reductive linear algebraic group over an algebraically closed field of characteristic p 0 a subgroup of g is said to be separable in g if its global and infinitesimal centralizers have the same dimension we study the interaction between the notion of separability and serre s concept of g complete reducibility for subgroups of g the separability hypothesis appears in many general theorems concerning g complete reducibility we demonstrate that many of these results fail without this hypothesis on the other hand we prove that if g is a connected reductive group and p is very good for g then any subgroup of g is separable we deduce that under these hypotheses on g a subgroup h of g is g completely reducible provided the lie algebra of g is semisimple as an h module recently guralnick has proved that if h is a reductive subgroup of g and c is a conjugacy class of g then the intersection of c and g is a finite union of h conjugacy classes for generic p when certain extra hypotheses hold including separability this follows from a well known tangent space argument due to richardson but in general it rests on lusztig s deep result that a connected reductive group has only finitely many unipotent conjugacy classes we show that the analogue of guralnick s result is false if one considers conjugacy classes of n tuples of elements from h for n 1

ein neuer zusammenhang zwischen einfachen gruppen und einfachen singularitaten we describe a new construction to obtain a simple hypersurface singularity from the corresponding simple complex lie groupg letx be the closed orbit in the projective space attached to the lie algebra mathfrak g ofg consider a regular nilpotent element y 0 in mathfrak g and denote byh y 0 the hyperplane orthogonal toy 0 with respect to the killing form then the hyperplane sectionx h y 0 has exactly one singularity which is simple of desired type by variation of the pointy 0 we obtain a versal deformation the construction generalizes with minor modifications to any characteristicp of the basefield even in bad characteristic we recover at least the positive part of the semiuniversal deformation we prove that forp 2 a simple quasihomogeneous singularity of type a7 resp d8 is adjacent to e7 resp e8 provided its dimension is even furthermore a8 is adjacent to e8 forp 3

the local structure of compactified jacobians this paper studies the local geometry of compactified jacobians the main result is a presentation of the completed local ring of the compactified jacobian of a nodal curve as an explicit ring of invariants described in terms of the dual graph of the curve the authors have investigated the geometric and combinatorial properties of these rings in previous work and consequences for compactified jacobians are presented in this paper similar results are given for the local structure of the universal compactified jacobian over the moduli space of stable curves

nilpotent commuting varieties of reductive lie algebras let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 and lie g in positive characteristic suppose in addition that p is good for g and the derived subgroup of g is simply connected let denote the nilpotent variety of and nil x y x y 0 the nilpotent commuting variety of our main goal in this paper is to show that the variety nil is equidimensional in characteristic 0 this confirms a conjecture of vladimir baranovsky see 2 when applied to gl n our result in conjunction with an observation in 2 shows that the punctual local hilbert scheme n hilb n 2 is irreducible over any algebraically closed field

geometric descriptions of entangled states by auxiliary varieties the aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry in the context of this paper geometric means each stratum of the hilbert space corresponding to an entangled state is an open subset of an algebraic variety built by classical geometric constructions tangent lines secant lines from the set of separable states in this setting we describe well known classifications of multipartite entanglement such as 2 2 n 1 for n 1 quantum systems and a new description with the 2 3 3 quantum system our results complete the approach of miyake and make stronger connections with recent work of algebraic geometers moreover for the quantum systems detailed in this paper we propose an algorithm based on the classical theory of invariants to decide to which subvariety of the hilbert space a given state belongs

triality exceptional lie algebras and deligne dimension formulas abstract we give a computer free proof of the deligne cohen and de man formulas for the dimensions of the irreducible g modules appearing in g k k 4 where g ranges over the exceptional complex simple lie algebras we give additional dimension formulas for the exceptional series as well as uniform dimension formulas for other representations distinguished by freudenthal along the rows of his magic chart our proofs use the triality model of the magic square which we review and present a simplified proof of its validity we conclude with some general remarks about obtaining series of lie algebras in the spirit of deligne and vogel

frobenius splitting of cotangent bundles of flag varieties let g be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic p 0 let u be the unipotent radical of a borel subgroup b g and u the lie algebra of u springer 16 has shown for good primes that there is a b equivariant isomorphism u u where b acts through conjugation on u and through the adjoint action on u for g sln one has the well known equivariant isomorphism a 7 a i between unipotent and nilpotent upper triangular matrices let p be a good prime for g then there is an isomorphism of homogeneous bundles x g b u g b u where the latter can be identified with the cotangent bundle t g b of g b motivated in part by 12 we establish a link between the g invariant form on the steinberg module st h0 g b p 1 cf 1 8 and frobenius splittings 15 of the cotangent bundle of g b the representation h0 g b 2 p 1 is a quotient of the space of functions h x ox on x here h0 g b m denotes the g module induced from the b module m and half the sum of the roots r opposite to the roots

nilpotent orbits over ground fields of good characteristic let x be an f rational nilpotent element in the lie algebra of a connected and reductive group g defined over the ground field f suppose that the lie algebra has a non degenerate invariant bilinear form we show that the unipotent radical of the centralizer of x is f split this property has several consequences when f is complete with respect to a discrete valuation with either finite or algebraically closed residue field we deduce a uniform proof that g f has finitely many nilpotent orbits in f when the residue field is finite we obtain a proof that nilpotent orbital integrals converge under some further fairly mild assumptions on g we prove convergence for arbitrary orbital integrals on the lie algebra and on the group the convergence of orbital integrals in the case where f has characteristic 0 was obtained by deligne and ranga rao 1972

involutions of reductive lie algebras in positive characteristic abstract let g be a reductive group over a field k of characteristic 2 let g lie g let be an involutive automorphism of g and let g k p be the associated symmetric space decomposition for the case of a ground field of characteristic zero the action of the isotropy group g on p is well understood since the well known paper of kostant and rallis b kostant s rallis orbits and representations associated with symmetric spaces amer j math 93 1971 753 809 such a theory in positive characteristic has proved more difficult to develop here we use an approach based on some tools from geometric invariant theory to establish corresponding results in good positive characteristic among other results we prove that the variety n of nilpotent elements of p has a dense open orbit and that the same is true for every fibre of the quotient map p p g however we show that the corresponding statement for g conjectured by richardson is not true we provide a new mostly calculation free proof of the number of irreducible components of n extending a result of sekiguchi for k c finally we apply a theorem of skryabin to describe the infinitesimal invariants k p k

the moduli space of three qutrit states we study the invariant theory of trilinear forms over a three dimensional complex vector space and apply it to investigate the behavior of pure entangled three partite qutrit states and their normal forms under local filtering operations slocc we describe the orbit space of the slocc group sl 3 c 3 both in its affine and projective versions in terms of a very symmetric normal form parametrized by three complex numbers the parameters of the possible normal forms of a given state are roots of an algebraic equation which is proved to be solvable by radicals the structure of the sets of equivalent normal forms is related to the geometry of certain regular complex polytopes

pointwise ergodic theorems for radial averages on the heisenberg group leth hn cn r denote the heisenberg group and let rdenote the normalized lebesgue measure on the sphere z 0 z r let x b m be a standard borel probability space on whichhacts measurably and ergodically by measure preserving transformations and let r denote the operator canonically associated with ronlp x we prove maximal and pointwise ergodic theorems inlp for radial averages ron the heisenberg grouphn n 1 the results are best possible for actions of the reduced heisenberg group the method of proof is to use the spectral theory of the banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods in particular littlewood paley stein square functions and analytic interpolation

a geometric approach to complete reducibility let g be a connected reductive linear algebraic group we use geometric methods to investigate g completely reducible subgroups of g giving new criteria for g complete reducibility we show that a subgroup of g is g completely reducible if and only if it is strongly reductive in g this allows us to use ideas of r w richardson and hilbert mumford kempf from geometric invariant theory we deduce that a normal subgroup of a g completely reducible subgroup of g is again g completely reducible thereby providing an affirmative answer to a question posed by j p serre and conversely we prove that the normalizer of a g completely reducible subgroup of g is again g completely reducible some rationality questions and applications to the spherical building of g are considered many of our results extend to the case of non connected g

singularities of duals of grassmannians abstract let x p n be a smooth irreducible nondegenerate projective variety and let x p n denote its dual variety the locus of bitangent hyperplanes i e hyperplanes tangent to at least two points of x is a component of the singular locus of x in this paper we provide a sufficient condition for this component to be of maximal dimension and show how it can be used to determine which dual varieties of grassmannians are normal that last part may be compared to what has been done for hyperdeterminants by j weyman and a zelevinsky 1996 in 23

weil restriction and support varieties this paper extends the comparison results for cohomological support varieties for chevalley groups of the form g fp established by j carlson z lin and d nakano to chevalley groups of the form g fpd other topics considered are complexity and non maximal support varieties for modules techniques developed by j pevtsova and the author together with the use of weil restriction functors are employed

hat g local systems on smooth projective curves are potentially automorphic dfg grants fg 1920 and spp 1489 erc grant 290766 aamot nsf grant dms 1404769 nsf grant dms 1161671 humboldt research award clay mathematics institute erc grant no 714405 gmlp

entanglement of four qubit systems a geometric atlas with polynomial compass i the finite world we investigate the geometry of the four qubit systems by means of algebraic geometry and invariant theory which allows us to interpret certain entangled states as algebraic varieties more precisely we describe the nullcone i e the set of states annihilated by all invariant polynomials and also the so called third secant variety which can be interpreted as the generalization of ghz states for more than three qubits all our geometric descriptions go along with algorithms which allow us to identify any given state in the nullcone or in the third secant variety as a point of one of the 47 varieties described in the paper these 47 varieties correspond to 47 non equivalent entanglement patterns which reduce to 15 different classes if we allow permutations of the qubits

a classification of nilpotent orbits in infinitesimal symmetric spaces abstract let g be a semisimple algebraic group defined over an algebraically closed field k whose characteristic is very good for g and not equal to 2 suppose is an involution on g we also denote the induced involution on g by let k g g g g and let p be the 1 eigenspace of in g the adjoint action of g on g induces an action of k on p and on the variety n p which consists of the nilpotent elements in p in this paper we give a classification of the k orbits in n p to do so we use the theory of associated cocharacters developed by pommerening

fourier integral operators on noncompact symmetric spaces of real rank one let x g k be a noncompact symmetric space of real rank one the purpose of this paper is to investigate lp boundedness properties of a certain class of radial fourier integral operators on the space x we will prove that if u is the solution at some fixed time of the natural wave equation on x with initial data f and g and 1 p then u lp x cp f lpbp x 1 g lpbp 1 x we will obtain both the precise behavior of the norm cp and the sharp regularity assumptions on the functions f and g i e the exponent bp that make this inequality possible in the second part of the paper we deal with the analog of e m stein s maximal spherical averages and prove exponential decay estimates of a highly non euclidean nature on the lp norm of supt t 1 f d z where d is a normalized spherical measure

sub principal homomorphisms in positive characteristic let g be a reductive group over an algebraically closed field of characteristic p and let u g be a unipotent element of order p suppose that p is a good prime for g we show in this paper that there is a homomorphism sl 2 k g whose image contains u this result was first obtained by d testerman j algebra 1995 using case considerations for each type of simple group and using in some cases computer calculations with explicit representatives for the unipotent orbits the proof we give is free of case considerations except in its dependence on the bala carter theorem our construction of generalizes the construction of a principal homomorphism made by j p serre in invent math 1996 in particular is obtained by reduction modulo from a homomorphism of group schemes over a valuation ring in a number field this permits us to show moreover that the weight spaces of a maximal torus of sl 2 k on lie g are the same as in characteristic 0 the existence of a with this property was previously obtained again using case considerations by lawther and testerman memoirs ams 1999 and has been applied in some recent work of g seitz invent math 2000

representation ring of levi subgroups versus cohomology ring of flag varieties recall the classical result that the cup product structure constants for the singular cohomology with integral coefficients h mathrm mathrm gr r n of the grassmannian of r planes coincide with the littlewood richardson tensor product structure constants for mathrm mathrm gl r specifically the result asserts that there is an explicit surjective ring homomorphism xi mathrm mathrm rep mathrm mathrm poly mathrm mathrm gl r rightarrow h mathrm mathrm gr r n where mathrm mathrm gr r n denotes the grassmannian of r planes in mathbb c n and mathrm mathrm rep mathrm mathrm poly mathrm mathrm gl r denotes the polynomial representation ring of mathrm mathrm gl r this work seeks to achieve one possible generalization of this classical result for mathrm mathrm gl r and the grassmannian mathrm mathrm gr r n to the levi subgroups of any reductive group g and the corresponding flag varieties

homogeneous coordinates for algebraic varieties abstract we associate to every divisorial e g smooth variety x with only constant invertible global functions and finitely generated picard group a pic x graded homogeneous coordinate ring this generalizes the usual homogeneous coordinate ring of the projective space and constructions of cox and kajiwara for smooth and divisorial toric varieties we show that the homogeneous coordinate ring defines in fact a fully faithful functor for normal complex varieties x with only constant global functions we even obtain an equivalence of categories finally the homogeneous coordinate ring of a locally factorial complete irreducible variety with free finitely generated picard group turns out to be a krull ring admitting unique factorization

fluctuation theorem for stochastic systems the fluctuation theorem describes the probability ratio of observing trajectories that satisfy or violate the second law of thermodynamics it has been proved in a number of different ways for thermostatted deterministic nonequilibrium systems in the present paper we show that the fluctuation theorem is also valid for a class of stochastic nonequilibrium systems the theorem is therefore not reliant on the reversibility or the determinism of the underlying dynamics numerical tests verify the theoretical result

galois symmetry induced by hecke relations in rational conformal field theory and associated modular tensor categories hecke operators relate characters of rational conformal field theories rcfts with different central charges and extend the previously studied galois symmetry of modular representations and fusion algebras we show that the conductor n of a rcft and the quadratic residues mod n play an important role in the computation and classification of galois permutations we establish a field correspondence in different theories through the picture of effective central charge which combines galois inner automorphisms and the structure of simple currents we then make a first attempt to extend hecke operators to the full data of modular tensor categories the galois symmetry encountered in the modular data transforms the fusion and the braiding matrices as well and yields isomorphic structures in theories related by hecke operators

a universal dimension formula for complex simple lie algebras we present a universal formula for the dimension of the cartan powers of the adjoint representation of a complex simple lie algebra i e a universal formula for the hilbert functions of homogeneous complex contact manifolds as well as several other universal formulas these formulas generalize formulas of vogel and deligne and are given in terms of rational functions where both the numerator and denominator decompose into products of linear factors with integer coefficients we discuss consequences of the formulas including a relation with scorza varieties

on duality and negative dimensions in the theory of lie groups and symmetric spaces we give one more interpretation of the symbolic formulas u n u n and sp 2n so 2n and show that they can be extended to the classical symmetric spaces using macdonald duality for jack and jacobi symmetric functions

complete reducibility and conjugacy classes of tuples in algebraic groups and lie algebras let h be a reductive subgroup of a reductive group g over an algebraically closed field k we consider the action of h on g n the n fold cartesian product of g with itself by simultaneous conjugation we give a purely algebraic characterization of the closed h orbits in g n generalizing work of richardson which treats the case h g this characterization turns out to be a natural generalization of serre s notion of g complete reducibility this concept appears to be new even in characteristic zero we discuss how to extend some key results on g complete reducibility in this framework we also consider some rationality questions

nilpotent orbits in good characteristic and the kempf rousseau theory let g be a connected reductive algebraic group over an algebraically closed field k of characteristic p 0 image and suppose that p is a good prime for the root system of g in this paper we give a fairly short conceptual proof of pommerening s theorem pommerening j algebra 49 1977 525 536 j algebra 65 1980 373 398 which states that any nilpotent element in image is richardson in a distinguished parabolic subalgebra of the lie algebra of a levi subgroup of g as a by product we obtain a short noncomputational proof of the existence theorem for good transverse slices to the nilpotent g orbits in image for earlier proofs of this theorem see kawanaka invent math 84 1986 575 616 premet trans amer math soc 347 1995 2961 2988 spaltenstein j fac sci univ tokyo sect ia math 31 1984 283 286 we extend recent results of sommers internal math res notices 11 1998 539 562 to reductive lie algebras of good characteristic thus providing a satisfactory approach to computing the component groups of the centralisers of nilpotent elements in image and unipotent elements in g earlier computations of these groups in positive characteristics relied mostly on work of mizuno j fac sci univ tokyo sect ia math 24 1977 525 563 tokyo j math 3 1980 391 459 our approach is based on the theory of optimal parabolic subgroups for g unstable vectors also known as the kempf rousseau theory which provides a good substitute for the image theory prominent in the characteristic zero case

representation ring of levi subgroups versus cohomology ring of flag varieties ii abstract for any reductive group g and a parabolic subgroup p with its levi subgroup l the first author in 5 introduced a ring homomorphism p rep poly c l h g p c where rep poly c l is a certain subring of the complexified representation ring of l depending upon the choice of an irreducible representation v of g with highest weight in this paper we study this homomorphism for g sp 2 n and its maximal parabolic subgroups p n k for any 1 k n 1 with the choice of v to be the defining representation v 1 in c 2 n thus we obtain a c algebra homomorphism n k rep 1 poly c sp 2 k h i g n k 2 n c our main result asserts that n k is injective when n tends to keeping k fixed similar results are obtained for the odd orthogonal groups

spherical maximal operator on symmetric spaces of constant curvature we prove an endpoint weak type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension n 2 more explicitly in the lorentz space associated with the natural isometry invariant measure we show that for every radial function f mf n c n f n 1 n the proof uses only geometric arguments and volume estimates and applies uniformly in every dimension

orbit closures and invariants let g be a reductive linear algebraic group h a reductive subgroup of g and x an affine g variety let x h denote the set of fixed points of h in x and n g h the normalizer of h in g in this paper we study the natural map of quotient varieties psi x h x h n g h rightarrow x g induced by the inclusion x h subseteq x we show that given g and h psi x h is a finite morphism for all affine g varieties x if and only if h is a g completely reducible subgroup of g in the sense defined by serre this was proved in characteristic 0 by luna in the 1970s we discuss some applications and give a criterion for psi x h to be an isomorphism we show how to extend some other results in luna s paper to positive characteristic and also prove the following theorem let h and k be reductive subgroups of g then the double coset hgk is closed for generic g in g if and only if h cap gkg 1 is reductive for generic g in g

loci in quotients by finite groups pointwise stabilizers and the buchsbaum property let k v g be the invariant ring of a nite linear group g gl v and let gu be the pointwise stabilizer of a subspace u v we prove that the following numbers associated to the invariant ring do not increase if one passes from k v g to k v gu the minimal number of homogeneous generators the maximal degree of the generators the number of syzygies and other betti numbers the complete intersection defect the dierence between depth and dimension and the type from this theorems of steinberg serre nakajima kac and watanabe and the author follow which say that if k v g is a polynomial ring a hypersurface a complete intersection or cohen macaulay then the same is true for k v gu furthermore k v gu inherits the gorenstein property from k v g we give an algorithm which transforms generators of k v g into generators of k v g u letp be one of the properties mentioned above we consider the locus ofp in v g spec k v g and prove that for x2 spec k v with image x 0 in v g the local ring k v g0 has the property p if and only if p holds for the invariant ring k v g x of the point stabilizer using this we prove that the non cohen macaulay locus in v g is either empty or it has dimension at least one and codimension at least 3 from this we deduce that k v g is buchsbaum if and only if it is cohen macaulay this proves a conjecture of campbell et al

the sextonions and e712 we fill in the hole in the exceptional series of lie algebras that was observed by cvitanovic deligne cohen and deman more precisely we show that the intermediate lie algebra between e7 and e8 satisfies some of the decomposition and dimension formulas of the exceptional simple lie algebras a key role is played by the sextonions a six dimensional algebra between the quaternions and octonions using the sextonions we show similar results hold for the rows of an expanded freudenthal magic chart we also obtain new interpretations of the adjoint variety of the exceptional group g2

moduli spaces for principal bundles in arbitrary characteristic in this article we solve the problem of constructing moduli spaces of semistable principal bundles and singular versions of them over smooth projective varieties over algebraically closed ground fields of positive characteristic

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