extensions of modules for twisted current algebras abstract twisted current algebras are fixed point subalgebras of current algebras under a finite group action special cases include equivariant map algebras and twisted forms of current algebras their finite dimensional simple modules fall into two categories those which factor through an evaluation map and those which do not we show that there are no nontrivial extensions between finite dimensional simple evaluation and non evaluation modules we then compute extensions between any pair of finite dimensional simple modules for twisted current algebras and use this information to study the block decomposition for the category in the special case of twisted forms this decomposition can be described in terms of maps to the fundamental group of the underlying root system

universal central extensions of twisted forms of split simple lie algebras over rings abstract we give sufficient conditions for the descent construction to be the universal central extension of a twisted form of a split simple lie algebra over a ring in particular the universal central extensions of twisted multiloop lie tori are obtained by the descent construction

quotients of absolute galois groups which determine the entire galois cohomology for a prime power q pd and a field f containing a root of unity of order q we show that the galois cohomology ring h g f mathbb z q is determined by a quotient g f 3 of the absolute galois group gf related to its descending q central sequence conversely we show that g f 3 is determined by the lower cohomology of gf this is used to give new examples of pro p groups which do not occur as absolute galois groups of fields

complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 1 abstract let k be a nonperfect field of characteristic 2 let g be a k split simple algebraic group of type e 6 or g 2 defined over k in this paper we present the first examples of nonabelian non g completely reducible k subgroups of g which are g completely reducible over k our construction is based on that of subgroups of g acting non separably on the unipotent radical of a proper parabolic subgroup of g in our previous work we also present examples with the same property for a non connected reductive group g along the way several general results concerning complete reducibility over nonperfect fields are proved using the recently proved tits center conjecture for spherical buildings in particular we show that under mild conditions a connected k subgroup of g is pseudo reductive if it is g completely reducible over k

magnetic interpretation in three dimensions using euler deconvolution magnetic survey data in grid form may be interpreted rapidly for source positions and depths by deconvolution using euler s homogeneity relation the method employs gradients either measured or calculated data need not be pole reduced so that remanence is not an interfering factor geologic constraints are imposed by use of a structural index model studies show that the method can locate or outline confined sources vertical pipes dikes and contacts with remarkable accuracy a field example using data from an intensively studied area of onshore britain shows that the method works well on real data from structurally complex areas and provides a series of depth labeled euler trends which mark magnetic edges notably faults with good precision

appendix on parahoric subgroups we give the proofs of some simple facts on parahoric subgroups and on iwahori weyl groups used in t haines the base change fundamental lemma for central elements in parahoric hecke algebras preprint 2008 g pappas m rapoport twisted loop groups and their affine flag varieties adv math 219 2008 118 198 m rapoport a guide to the reduction modulo p of shimura varieties asterisque 298 2005 271 318

twisted loop groups and their affine flag varieties we develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field this can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case a consequence of our results is that our construction also includes the flag varieties for kac moody lie algebras of affine type we also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k t and some applications to local models of shimura varieties

the langlands kottwitz method and deformation spaces of p divisible groups we extend the results of kottwitz 17 on points of shimura varieties over finite fields to cases of bad reduction the test function whose twisted orbital integrals appear in the final expression is defined geometrically using deformation spaces of p divisible groups

construction of central elements in the affine hecke algebra via nearby cycles 0 1 overview let g be a connected reductive group over a finite field fq and let g k be the corresponding group over the local field k fq t let g o g k be a maximal compact subgroup of g k here o fq t and let hsph denote the hecke algebra of g k with respect to g o in other words hsph as a vector space consists of compactly supported bi g o invariant functions g k q and the product is defined by

differential conformal superalgebras and their forms abstract we introduce the formalism of differential conformal superalgebras which we show leads to the correct automorphism group functor and accompanying descent theory in the conformal setting as an application we classify forms of n 2 and n 4 conformal superalgebras by means of galois cohomology

maximality of galois actions for compatible systems in the sense of serre 19 of course the image of is not generally zariski dense in gln q so the maximality condition must be formulated relative to the zariski closure g of gal k k one might hope that the image of is a maximal compact subgroup of g q but this is too optimistic the center causes problems and we do not even know a priori that g is reductive to formulate a maximality conjecture which avoids such problems we introduce maps g g g

scattering symmetry and bias analysis of source position estimates in euler deconvolution and its practical implications despite being widely used euler deconvolution has received little attention in its theoretical aspects we obtain analytical estimators x 0 and z 0 for the horizontal x0 and the vertical z0 source positions which are the parameters to be estimated when the structural index is assigned a tentative value by analyzing the estimators properties we show two things 1 scattering of the alternative solutions associated with different data window positions has two components data noise and the use of a wrong tentative value for the structural index 2 the differences x 0 x0 and z 0 z0 as a function of the data window position exhibit antisymmetric and symmetric behaviors respectively about x0 when the magnetization and the geomagnetic field inclinations are 90 0 or 45 for intermediate inclinations there is a moderate departure from the symmetry properties this analysis shows that the criterion for determining the structural index as the tentative value producing the smallest solution scat

polynomial identity rings as rings of functions ii abstract in characteristic zero zinovy reichstein and the author generalized the usual relationship between irreducible zariski closed subsets of the affine space their defining ideals coordinate rings and function fields to a non commutative setting where varieties carry a pgl n action regular and rational functions on them are matrix valued coordinate rings are prime polynomial identity algebras and function fields are central simple algebras of degree n in the present paper much of this is extended to prime characteristic in addition a mistake in the earlier paper is corrected one of the results is that the finitely generated prime pi algebras of degree n are precisely the rings that arise as coordinate rings of n varieties in this setting for n 1 the definitions and results reduce to those of classical affine algebraic geometry

essential dimension of algebraic tori the essential dimension is a numerical invariant of an algebraic group g which may be thought of as a measure of complexity of g torsors over fields a recent theorem of n karpenko and a merkurjev gives a simple formula for the essential dimension of a finite p group we obtain similar formulas for the essential p dimension of a broad class of groups which includes all algebraic tori

irreducible cuspidal representations with prescribed local behavior let g be a simple algebraic group defined over the global field k in this paper we use the simple trace formula to determine the sum of the multiplicities of the irreducible representations in the cuspidal spectrum of g with specified local behavior at a finite set of places of k and unramified elsewhere this sum is expressed as the product of the values of modified artin l functions at negative integers

lattices of minimum covolume in chevalley groups over local fields of positive characteristic in this article we show that if g is a simply connected chevalley group of either classical type of rank bigger than 1 or type e6 and q 9 is a power of a prime number p 5 then g g fq t 1 up to an automorphism has a unique lattice of minimum covolume which is g fq t 1

on conjugacy of cartan subalgebras in extended affine lie algebras abstract that finite dimensional simple lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as coxeter dynkin diagrams and indecomposable irreducible root systems is arguably one of the most elegant results in mathematics the definition of the root system is done by fixing a cartan subalgebra of the given lie algebra the remarkable fact is that up to isomorphism this construction is independent of the choice of the cartan subalgebra the modern way of establishing this fact is by showing that all cartan subalgebras are conjugate for symmetrizable kac moody lie algebras with the appropriate definition of cartan subalgebra conjugacy has been established by peterson and kac an immediate consequence of this result is that the root systems and generalized cartan matrices are invariants of the kac moody lie algebras the purpose of this paper is to establish conjugacy of cartan subalgebras for extended affine lie algebras a natural class of lie algebras that generalizes the finite dimensional simple lie algebra and affine kac moody lie algebras

the kernel of the rost invariant serre s conjecture ii and the hasse principle for quasi split groups 3 6d4 e6 e7 we prove that for a simple simply connected quasi split group of type 3 6d4 e6 e7 defined over a perfect field f of characteristic 2 3 the rost invariant has trivial kernel in certain cases we give a formula for the rost invariant it follows immediately from the result above that if cd f 2 resp vcd f 2 then serre s conjecture ii resp the hasse principle holds for such a group for a c2 field in particular x y we prove the stronger result that serre s conjecture ii holds for all not necessary quasi split exceptional groups of type 3 6d4 e6 e7

topological flatness of orthogonal local models in the split even case i local models are schemes defined in terms of linear algebra that were introduced by rapoport and zink to study the etale local structure of integral models of certain pel shimura varieties over p adic fields a basic requirement for the integral models or equivalently for the local models is that they be flat in the case of local models for even orthogonal groups genestier observed that the original definition of the local model does not yield a flat scheme in a recent article pappas and rapoport introduced a new condition to the moduli problem defining the local model the so called spin condition and conjectured that the resulting spin local model is flat we prove a preliminary form of their conjecture in the split iwahori case namely that the spin local model is topologically flat an essential combinatorial ingredient is the equivalence of admissibility and permissibility for two minuscule cocharacters in root systems of type d

covering algebras ii isomorphism of loop algebras this paper studies the loop algebras that arise from pairs consisting of a symmetrizable kac moody lie algebra g and a finite order automorphism of g we obtain necessary and sufficient conditions for two such loop algebras to be isomorphic

lie algebroids arising from simple group schemes abstract a classical construction of atiyah assigns to any real or complex lie group g manifold m and principal homogeneous g space p over m a lie algebroid over m 1 the spirit behind our work is to put this work within an algebraic context replace m by a scheme x and g by a simple reductive group scheme g over x in the sense of demazure grothendieck that arise naturally with an attached torsor which plays the role of p in the study of extended affine lie algebras see 9 for an overview lie algebroids in an algebraic sense were also considered by beilinson and bernstein we will explain how the present work relates to theirs

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

derivations of skew polynomial rings abstract let f be a field of characteristic 0 and let i j f for 1 i j n define r f x 1 x 2 x n to be the skew polynomial ring with x i x j i j x j x i and let s f x 1 x 2 x n x 11 x 12 x 1n be the corresponding laurent polynomial ring in a recent paper kirkman procesi and small considered these two rings under the assumption that s is simple and showed for example that the lie ring of inner derivations of s is simple furthermore when n 2 they determined the automorphisms of s related its ring of inner derivations to a certain block algebra and proved that every derivation of r is the sum of an inner derivation and a derivation which sends each xi to a scalar multiple of itself in this paper we extended these results to a more general situation specifically we study twisted group algebras ft g where g is a commutative group and f is a field of any characteristic furthermore we consider certain subalgebras ft h where h is a subsemigroup of g which generates g as a group finally if e g g f is a skew symmetric bilinear form then we study the lie algebra fe g associated with e and we consider its relationship to the lie structure defined on various twisted group algebras ft g

zero cycles of degree one on principal homogeneous spaces abstract let k be a field of characteristic different from 2 let g be an absolutely simple simply connected or adjoint semisimple algebraic k group of classical type we show that if a principal homogeneous space under g over k admits a zero cycle of degree 1 then it has a k rational point

galois cohomology and forms of algebras over laurent polynomial rings 3 loop torsors 8 3 1 the witt tits index of a loop torsor 9 3 2 reducibility 12 3 3 almost commutative subgroups 14 3 4 almost commuting families of elements of finite order 16 3 5 almost commuting pairs and their invariants 17 3 6 failure in the anisotropic case 24

lower bounds for essential dimensions via orthogonal representations let us first recall what is the essential dimension of a functor cf br 97 and r 00 let k be a field and let f be a functor from the category of field extensions of k into the category of sets let f k be an extension and let be an element of f f if e is a field with k e f we say that comes from e if it belongs to the image of f e f f the essential dimension ed of is the minimum of the transcendence degrees e k for all e with k e f such that comes from e one has ed tr deg f if there is equality we say that is incompressible the essential dimension ed f of f is ed f max ed the maximum being taken over all pairs f with k f and f f

3d euler deconvolution theoretical basis for automatically selecting good solutions we derive the analytical estimators for the horizontal and vertical source positions in 3d euler deconvolution as a function of the x y and z derivatives of the magnetic anomaly within a data window from these expressions we show that in the case of noise corrupted data the x y and z coordinate estimates computed at the anomaly borders are biased toward the respective horizontal coordinate of the data window center regardless of the true or presumed structural indices and regardless of the magnetization inclination and declination on the other hand in the central part of the anomaly the x and y coordinate estimates are very close to the respective source horizontal coordinates regardless of the true or presumed structural indices and regardless of the magnetization inclination and declination this contrasting behavior of the horizontal coordinate estimates may be used to automatically delineate the region associated with the best solutions applying the euler deconvolution operator inside this region would decrease the dispersion of all position estimates improving source location precision

multiloop realization of extended affine lie algebras and lie tori an important theorem in the theory of infinite dimensional lie algebras states that any affine kac moody algebra can be realized that is to say constructed explicitly using loop algebras in this paper we consider the corresponding problem for a class of lie algebras called extended affine lie algebras ealas that generalize affine algebras ealas occur in families that are constructed from centreless lie tori so the realization problem for ealas reduces to the realization problem for centreless lie tori we show that all but one family of centreless lie tori can be realized using multiloop algebras in place of loop algebras we also obtain necessary and sufficient conditions for two centreless lie tori realized in this way to be isotopic a relation that corresponds to isomorphism of the corresponding families of ealas

the jordan holder series for nearby cycles on some shimura varieties and affine flag varieties we study the jordan holder series for nearby cycles on certain shimura varieties and rapoport zink local models and on finite dimensional pieces of beilinson s deformation of the affine grassmannian to the affine flag variety and their p adic analogues we give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups and we also provide an algorithm to compute multiplicities in terms of the affine hecke algebra msc classification 14m15 14g35 20c08

the arason invariant and mod 2 algebraic cycles introduction 73 1 review of the arason invariant 75 2 the special clifford group 76 3 k cohomology of split reductive algebraic groups 78 4 k cohomology of bg 86 5 gl n and cliff n n 92 6 two invariants for clifford bundles 95 7 snaking a bloch ogus differential 100 8 proof of theorem 1 101 9 application to quadratic forms 102 appendix a toral descent 104 appendix b the rost invariant 108 appendix c an amusing example 113 acknowledgements 116 references 116

when is the range of a multiplier on a banach algebra closed in this paper we prove the theorem let a be a banach algebra with a bounded approximate identity bai such that every proper closed ideal of a is contained in a proper closed ideal with a bai then a multiplier t a a has a closed range iff t factors as a product of an idempotent multiplier and an invertible multiplier

hermitian analogue of a theorem of springer with involutions and in fact in a much more general setupse for algebraic groups the weak form of springer s theorem forhermitian forms was proved by bayer fluckiger and lenstra in bl andthe analogue of springer s theorem for hermitian forms was raised as anopen question in bst the aim of this paper is to settle this question inthe affirmative for hermitian forms over quaternion algebras with respectto involutions of the first kind we prove that if

a characterization of the closed unital ideals of the fourier stieltjes algebra b g of a locally compact amenable group g abstract let g be a locally compact amenable group b g its fourier stieltjes algebra and i be a closed ideal of it in this paper we prove the following result the ideal i has a unit element iff it is principal this is the noncommutative version of the glicksberg host parreau theorem the paper also contains an abstract version of this theorem

shells of twisted flag varieties and the rost invariant we introduce two new general methods to compute the chow motives of twisted flag varieties and settle a 20 year old conjecture of markus rost about the rost invariant for groups of type e 7

the structural index in gravity and magnetic interpretation errors uses and abuses abstractthe structural index si is based on the concept of euler homogeneity a description of scaling behavior it has found wide use in potential field depth estimation and is a constant integer for simple sources with single singularities points lines thin bed faults sheet edges infinite contacts for these cases the si is identical to the index of a simple power law field fall off with distance the simple euler formulation is only strictly correct for such simple sources and integer si values the widespread use of the simple euler method on more complex structures using fractional si values is likely to produce misleading results because the si is no longer a constant for any given source we examine a recently published example that used an arbitrary si to estimate depth to the base of the crust for africa and produced misleading results extension to more complex sources such as tabular bodies or thick steps requires one of several more generalized approaches which recognize all variabl

an abstract form of a theorem of helson and applications to sets of synthesis and sets of uniqueness let e be a compact perfect subset of the real line r such that the restriction of the fourier transform a a e from l1 r into c e is onto helson proved that then for m e lim y y 0 is possible only if 0 in this paper we present an abstract version of this theorem of helson and provide some applications of it to the study of sets of spectral synthesis and sets of uniqueness

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

stability analysis and improvement of structural index estimation in euler deconvolution euler deconvolution has been widely used in automatic aeromagnetic interpretations because it requires no prior knowledge of the source magnetization direction and assumes no particular interpretation model provided the structural index defining the anomaly falloff rate related to the nature of the magnetic source is determined in advance estimating the correct structural index and electing optimum criteria for selecting candidate solutions are two fundamental requirements for a successful application of this method we present a new criterion for determining the structural index this criterion is based on the correlation between the total field anomaly and the estimates of an unknown base level these estimates are obtained for each position of a moving data window along the observed profile and for several tentative values for the structural index the tentative value for the structural index producing the smallest correlation is the best estimate of the correct structural index we also propose a n

local models of shimura varieties and a conjecture of kottwitz we give a group theoretic definition of local models as sought after in the theory of shimura varieties these are projective schemes over the integers of a p adic local field that are expected to model the singularities of integral models of shimura varieties with parahoric level structure our local models are certain mixed characteristic degenerations of grassmannian varieties they are obtained by extending constructions of beilinson drinfeld gaitsgory and the second named author to mixed characteristics and to the case of general tamely ramified reductive groups we study the singularities of local models and hence also of the corresponding integral models of shimura varieties in particular we study the monodromy inertia action and show a commutativity property for the sheaves of nearby cycles as a result we prove a conjecture of kottwitz which asserts that the semi simple trace of frobenius on the nearby cycles gives a function which is central in the parahoric hecke algebra

torsors over the punctured affine line we provide a classification of g torsors over the punctured affine line rm spec k t pm 1 where g is a reductive algebraic group defined over a field k of good characteristic our classification is in terms of the galois cohomology of the complete field k t with values in g

quadratic spaces over laurent extensions of dedekind domains let abea dedekind domain in which 2 is invertible we show in this paper that any isotropic quadratic space over r t 7 1 is isometric to n ii q q hip with rank p n where if p denotes the hyperbolic space 1 some preliminary results in this section we prove some lemmas which will be used in the later sections we assume throughout the paper that in all the rings considered 2 is invertible if a q2 are quadratic spaces over r a j tq2 denotes the quadratic space a r r t t x l t a2 r r t t x over r t t x

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

estimating the nature and the horizontal and vertical positions of 3d magnetic sources using euler deconvolution abstractwe have developed a new method that drastically reduces the number of the source location estimates in euler deconvolution to only one per anomaly our method employs the analytical estimators of the base level and of the horizontal and vertical source positions in euler deconvolution as a function of the x and y coordinates of the observations by assuming any tentative structural index defining the geometry of the sources our method automatically locates plateaus on the maps of the horizontal coordinate estimates indicating consistent estimates that are very close to the true corresponding coordinates these plateaus are located in the neighborhood of the highest values of the anomaly and show a contrasting behavior with those estimates that form inclined planes at the anomaly borders the plateaus are automatically located on the maps of the horizontal coordinate estimates by fitting a first degree polynomial to these estimates in a moving window scheme spanning all estimates the positio

derivations of certain algebras defined by etale descent we give an explicit description of the lie algebra of derivations for a class of infinite dimensional algebras which are given by etale descent the algebras under consideration are twisted forms of central algebras over rings and include the multiloop algebras that appear in the construction of extended affine lie algebras

resolving g torsors by abelian base extensions abstract let g be a linear algebraic group defined over a field k we prove that under mild assumptions on k and g there exists a finite k subgroup s of g such that the natural map h 1 k s h 1 k g is surjective for every field extension k k we give several applications of this result in the case where k an algebraically closed field of characteristic zero and k k is finitely generated in particular we prove that for every h 1 k g there exists an abelian field extension l k such that l h 1 l g is represented by a g torsor over a projective variety from this we deduce that l has trivial fixed point obstruction we also show that a strong variant of the algebraic form of hilbert s 13th problem implies that the maximal abelian extension of k has cohomological dimension 1 the last assertion if true would prove conjectures of bogomolov and konigsmann answer a question of tits and establish an important case of serre s conjecture ii for the group e 8

a non split torsor with trivial fixed point obstruction abstract let g be a linear algebraic group and x be an irreducible algebraic variety with a generically free g action all defined over an algebraically closed base field of characteristic zero it is well known that x can be viewed as a g torsor representing a class x in h 1 k g where k is the field of g invariant rational functions on x we have previously shown that if x has a smooth h fixed point for some non toral diagonalizable subgroup h of g then x 1 it is natural to ask if the converse is true assuming g is connected and x is projective and smooth in this note we show that the answer is no

automorphisms of toroidal lie superalgebras abstract we give a description of the algebraic group aut g of automorphisms of a simple finite dimensional lie superalgebra g over an algebraically closed field k of characteristic 0 which is obtained by viewing g as a module over a levi subalgebra of its even part as an application we give a detailed description of the group of automorphism of the k lie superalgebra g k r for a large class of commutative rings r