explicit determination of the picard group of moduli spaces of semistable g bundles on curves let g be a connected simply connected simple affine algebraic group and cg be a smooth irreducible projective curve of any genus g 1 over c denote by mcg g the moduli space of semistable principal g bundles on cg let pic mcg g be the picard group of mcg g and let x be the infinite grassmannian of the affine kac moody group associated to g it is known that pic x z and is generated by a homogenous line bundle l 0 also as proved by kumarnarasimhan kn there exists a canonical injective group homomorphism

some conjectures and results concerning the homology of nilpotent lie algebras in this paper we consider the lie algebra homology of for l a complex lie algebra our goal is to express the homology of l in terms of the homology of l this problem comes up in previous work by the author on the macdonald root system conjectures we present a number of conjectures related to this problem the simplest of these conjectures asserts that h l h l l i

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

symplectic surgery and the spinc dirac operator abstract letgbe a compact connected lie group and m a compact hamiltoniang space with moment mapj m g under the assumption that these data are pre quantizable one can construct an associated spinc dirac operator formula whose equivariant index yields a virtual representation ofg we prove a conjecture of guillemin and sternberg that if 0 is a regular value ofj the multiplicityn 0 of the trivial representation in the index space formula is equal to the index of the spinc dirac operator for the symplectic quotientm0 j 1 0 g this generalizes previous results for the case thatg tis abelian i e a torus

infinite grassmannians and moduli spaces of g bundles let c be a smooth projective irreducible algebraic curve over c of any genus and g a connected simply connected simple affine algebraic group over c in this paper we elucidate the relationship between 1 the space of vacua conformal blocks defined in conformal field theory using an integrable highest weight representation of the affine kac moody algebra associated to g and 2 the space of regular sections generalised theta functions of a line bundle on the moduli space ta of semistable principal g bundles on c fix a point p in c and let p resp kp be the completion of the local ring p of c at p resp the quotient field of c p let g kp the kp rational points of the algebraic group g be the loop group of g and let g p be the standard maximal parahoric subgroup of then the generalised flag variety x n is an inductive limit of projective varieties in fact of generalised schubert varieties one has a natural r line bundle s on x cf sect 2 2 and the picard group pic x is isomorphic to z which is generated by s proposition 2 3 where f is the universal central extension of by the multiplicative group c cf sect 2 2 by an analogue of the borel weil theorem proved in the kac moody setting by kumar and also by mathieu the space h 162 s of the regular sections of the line bundle y dxo y xo for any d 0 is canonically isomorphic with the full vector space dual v dxo of the integrable highest weight irreducible module v dxo of the affine kacmoody group or of the associated affine kac moody algebra which is a certain one dimensional central extension of the loop algebra lie g with highest weight dxo cf sect 6 1

critical attractive spin systems we study a class of attractive spin systems we prove that for these processes the system dies out when the parameters are on the critical surface we also prove that the supercritical process survives with positive probability in a sufficiently thick space time slab

two point ag codes on the gk maximal curves we determine the weierstrass semigroup of a pair of certain rational points on the giulietti korchmaros maximal curves we use this semigroup to obtain two point algebraic geometric ag codes with better parameters than comparable one point ag codes arising from these curves these parameters are new records in the mint s tables

a new family of maximal curves over a finite field it has been known for a long time that the deligne lusztig curves associated to the algebraic groups of type 2a 2 2b 2 and 2g 2 defined over the finite field mathbb f n all have the maximum number of mathbb f n rational points allowed by the weil explicit formulas and that these curves are mathbb f q 2 maximal curves over infinitely many algebraic extensions mathbb f q 2 of mathbb f n serre showed that an mathbb f q 2 rational curve which is mathbb f q 2 covered by an mathbb f q 2 maximal curve is also mathbb f q 2 maximal this has posed the problem of the existence of mathbb f q 2 maximal curves other than the deligne lusztig curves and their mathbb f q 2 subcovers see for instance garcia on curves with many rational points over finite fields in finite fields with applications to coding theory cryptography and related areas pp 152 163 springer berlin 2002 and garcia and stichtenoth a maximal curve which is not a galois subcover of the hermitan curve bull braz math soc n s 37 139 152 2006 in this paper a positive answer to this problem is obtained for every q n 3 with n p r 2 p 2 prime we give a simple explicit construction of an mathbb f q 2 maximal curve mathcal x that is not mathbb f q 2 covered by any mathbb f q 2 maximal deligne lusztig curve furthermore the mathbb f q 2 automorphism group aut mathcal x has size n 3 n 3 1 n 2 1 n 2 n 1 interestingly mathcal x has a very large mathbb f q 2 automorphism group with respect to its genus g frac 1 2 n 3 1 n 2 2 1

cyclic homology and the macdonald conjectures leta k denote the ring t t k 1 and letg be a reductive complex lie algebra with exponentsm 1 m n this paper concerns the lie algebra cohomology ofg a k considered as a bigraded algebra here one of the gradings is homological degree and the other which we callweight is inherited from the obvious grading ofg a k we conjecture that this lie algebra cohomology is an exterior algebra withk 1 generators of homological degree 2m s 1 fors 1 2 n of thesek 1 generators of degree 2m s 1 one has weight 0 and the others have weights k 1 m s t fort 1 2 k it is shown that this conjecture about the lie algebra cohomology of a k implies the macdonald root system conjectures next we consider the case thatg is a classical lie algebra with root systema n b n c n ord n it is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofa k lastly we discuss the relevance of this limiting case to the case of finiten in this situation

two point codes for the generalized gk curve we improve previously known lower bounds for the minimum distance of certain two point ag codes constructed using a generalized giulietti korchmaros curve ggk castellanos and tizziotti recently described such bounds for two point codes coming from the giulietti korchmaros curve our results completely cover and in many cases improve on their results using different techniques while also supporting any ggk curve our method builds on the order bound for ag codes to enable this we study certain weierstrass semigroups this allows an efficient algorithm for computing our improved bounds we find several new improvements upon the mint minimum distance tables

on the motivic class of the stack of bundles let g be a split connected semisimple group over a field we give a conjectural formula for the motivic class of the stack of g bundles over a curve c in terms of special values of the motivic zeta function of c the formula is true if or g sln if upon applying the poincare or called the serre characteristic by some authors the formula reduces to results of teleman and atiyah bott on the gauge group if upon applying the counting measure it reduces to the fact that the tamagawa number of g over the function field of c is p1 g

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

the positive rates problem for attractive nearest neighbor spin systems on ℤ it is shown that a spin system on has only one invariant probability measure if it has attractive or repulsive nearest neighbor flip rates which are strictly positive and periodic under translation along

a stability criterion for attractive nearest neighbor spin systems on mathbb z we study attractive nearest neighbor spin systems on the integers having all 0 s and all 1 s as traps a necessary and sufficient condition for stability under small one sided perturbations of the flip rates is established

quantization of branching coefficients for classical lie groups abstract we study natural quantizations k of branching coefficients corresponding to the restrictions of the classical lie groups to the levi subgroups of their standard parabolic subgroups the polynomials obtained can be regarded as generalizations of lusztig q analogues of weight multiplicities for gl n they coincide with poincare polynomials previously studied by shimozono and weyman they also appear in the hilbert series of the euler characteristic of certain graded virtual g modules and by a result of broer admit nonnegative coefficients providing that restrictive conditions are verified when the levi subgroup considered is isomorphic to a direct product of linear groups we prove that these polynomials admit a stable limit k which decomposes as nonnegative integer combination of poincare polynomials for a general levi subgroup it is conjectured that the polynomials k have nonnegative coefficients when they are parametrized by two partitions when g gl n the polynomials k can be interpreted as quantizations of the littlewood richardson coefficients we show that there also exists a duality between tensor product coefficients for types b c d defined as the analogues of the littlewood richardson coefficients and branching coefficients corresponding to the restriction of so 2 n to subgroups defined from orthogonal decompositions of the root system d n which are not levi subgroups these coefficients can also be quantified but the q analogues obtained may have negative coefficients given a tensor product of irreducible gl n modules we then introduce for each classical group g so n or sp n some q analogues d of the multiplicities obtained by decomposing into its g irreducible components we establish a duality between the polynomials d and k according to a conjecture by shimozono the stable one dimensional sums for nonexceptional affine crystals are expected to coincide with the polynomials d associated to a sequence of rectangular partitions of decreasing heights

hilbert series of nearly holomorphic sections on generalized flag manifolds abstract let x g p be a complex flag manifold and e x be a g homogeneous holomorphic vector bundle fix a u invariant kahler metric on x with u g maximal compact we study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in e coincides with the space of u finite smooth sections in e the degree of nearly holomorphic sections defines a u invariant filtration on this space using sheaf cohomology we determine in suitable cases the corresponding hilbert series it turns out that this is given in terms of lusztig s q analog of kostant s weight multiplicity formula and hence gives a new representation theoretic interpretation of these formulas

one point ag codes on the gk maximal curves in this paper algebraic geometric ag codes associated to a recently discovered class of maximal curves are investigated as a result some linear codes with better parameters with respect to the previously known ones are discovered and 70 improvements on mint s tables tables of optimal parameters for linear codes university of salzburg salzburg austria are obtained

the yang mills heat flow on the moduli space of framed bundles on a surface we study the analog of the yang mills heat flow on the moduli space of framed bundles on a cut surface existence and convergence of the heat flow give a stratification of morse type invariant under the action of the loop group we use the stratification to prove versions of kahler quantization commutes with reduction and kirwan surjectivity

the strong macdonald conjecture and hodge theory on the loop grassmannian we prove the strong macdonald conjecture of hanlon and feigin for reductive groups g in a geometric reformulation we show that the dolbeault cohomology hq x op of the loop grassmannian x is freely generated by de rham s forms on the disk coupled to the indecomposables of h bg equating the two euler characteristics gives an identity independently known to macdonald m which generalises ramanujan s 1 1 sum for simply laced root systems at level 1 we also find a strong form of bailey s 4 4 sum failure of hodge decomposition implies the singularity of x and of the algebraic loop groups some of our results were announced in t2

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

highest weight theory for truncated current lie algebras abstract in this paper a highest weight theory for the truncated current lie algebra g g g k k t t n 1 k t is developed when the underlying lie algebra g possesses a triangular decomposition the principal result is the reducibility criterion for the verma modules of g for a wide class of lie algebras g including the symmetrizable kac moody lie algebras the heisenberg algebra and the virasoro algebra this is achieved through a study of the shapovalov form

on the decomposition of the tensor algebra of the classical lie algebras abstract let g n be the lie algebra gln c let s g n be the symmetric algebra of g n and let t g n be the tensor algebra of g n in a recent paper r k gupta studied certain sequences of representations r rn n 1 where rn is a representation of g n these sequences have the property that every irreducible representation occurring in s g n is in exactly one of these sequences fixing f she considers s r f which is the limit on n of the multiplicity of rn in sf g n the fth graded piece of s g n she and r p stanley independently showed that the limit s r f exists and is given by an amazingly elegant formula they call s r f the stable multiplicity of rn in sf g n in this paper an entirely different approach is used to extend the above result in several directions appropriately defined sequences r for all of the classical lie algebras g n are studied and a simple formula for the stable multiplicity m r f g of rn in the isotypic component of tf g n where is any irreducible character of the symmetric group tsf is obtained as in the work of gupta and stanley the expressions for m r f g are amazingly simple special cases include the stable decomposition of the tensor algebra the symmetric algebra and the exterior algebra of g n as a byproduct of our proof a stable decomposition of every isotypic component of t g n is obtained this combinatorial decomposition is in some sense a generalization of kostant s decomposition of s g n into direct sum of the harmonics and the ideal generated by the invariants of positive degree to be precise for f g n projects onto kostant s decomposition of sf g n

pursuing the double affine grassmannian i transversal slices via instantons on a_k singularities this article is the first in a series that describes a conjectural analog of the geometric satake isomorphism for an affine kac moody group for simplicity we only consider the untwisted and simply connected case here the usual geometric satake isomorphism for a reductive group g identifies the tensor category rep g of finite dimensional representations of the langlands dual group g with the tensor category pervg o grg of g o equivariant perverse sheaves on the affine grassmannian grg g k g o of g here k c t and o c t as a by product one gets a description of the irreducible g o equivariant intersection cohomology ic sheaves of the closures of g o orbits in grg in terms of q analogs of the weight multiplicity for finite dimensional representations of g the purpose of this article is to try to generalize the above results to the case when g is replaced by the corresponding affine kac moody group gaff we refer to the not yet constructed affine grassmannian of gaff as the double affine grassmannian more precisely in this article we construct certain varieties that should be thought of as transversal slices to various gaff o orbits inside the closure of another gaff o orbit in grgaff we present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding q analog of the weight multiplicity for the langlands dual affine group gaff and we check this conjecture in a number of cases some further constructions such as convolution of the corresponding perverse sheaves analog of the beilinson drinfeld grassmannian and so forth will be addressed in another publication

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

the index formula for the moduli of g bundles on a curve we prove the formulae conjectured by the first author for the index of k theory classes over the moduli stack of algebraic g bundles on a smooth projective curve the formulae generalize e verlinde s for line bundles and have witten s integrals over the moduli space of stable bundles as their large level limits as an application we prove the newstead ramanan conjecture on the vanishing of high chern classes of certain moduli spaces of semi stable g bundles

a wonderful embedding of the loop group i describe the wonderful compactification of loop groups these compactifications are obtained by adding normal crossing boundary divisors to the group lg of loops in a reductive group g or more accurately to the semi direct product c lg in a manner equivariant for the left and right c lg actions the analogue for a torus group t is the theory of toric varieties for an adjoint group g this is the wonderful compactification of de concini and procesi the loop group analogue is suggested by work of faltings in relation to the compactification of moduli of g bundles over nodal curves

borel weil bott theory on the moduli stack of g bundles over a curve let g be a semi simple group and m the moduli stack of g bundles over a smooth complex projective curve using representation theoretic methods i prove the pure dimensionality of sheaf cohomology for certain evaluation vector bundles over m twisted by powers of the fundamental line bundle this result is used to prove a borel weil bott theorem conjectured by g segal for certain generalized flag varieties of loop groups along the way the homotopy type of the group of algebraic maps from an affine curve to g and the homotopy type the hodge theory and the picard group of m are described one auxiliary result in appendix a is the alexander cohomology theorem conjectured in gro2 a self contained account of the uniformization theorem of ls for the stack m is given including a proof of a key result of drinfeld and simpson in characteristic 0 a basic survey of the simplicial theory of stacks is outlined in appendix b

a brylinski filtration for affine kac moody algebras abstract braverman and finkelberg have recently proposed a conjectural analogue of the geometric satake isomorphism for untwisted affine kac moody groups as part of their model they conjecture that at dominant weights lusztig s q analog of weight multiplicity is equal to the poincare series of the principal nilpotent filtration of the weight space as occurs in the finite dimensional case we show that the conjectured equality holds for all affine kac moody algebras if the principal nilpotent filtration is replaced by the principal heisenberg filtration the main body of the proof is a lie algebra cohomology vanishing result we also give an example to show that the poincare series of the principal nilpotent filtration is not always equal to the q analog of weight multiplicity finally we give some partial results for indefinite kac moody algebras

first layer formulas for characters of sl n c some problems concerning the decomposition of certain characters of sl n c are studied from a combinatorial point of view the specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the euler characteristic of hanlon s so called macdonald complex a general recursion is given for computing the irreducible decomposition of these characters the recursion is explicitly solved for the first layer representations which are the irreducible representations corresponding to partitions of n in the case of the exterior algebra this settles a conjecture of gupta and hanlon a further application of the recursion is used to give a family of formal laurent series identities that generalize the equal parameter q dyson theorem

extremal part of the pbw filtration and nonsymmetric macdonald polynomials abstract given a reduced irreducible root system the corresponding nil daha is used to calculate the extremal coefficients of nonsymmetric macdonald polynomials in the limit t and for antidominant weights which is an important ingredient of the new theory of nonsymmetric q whittaker function these coefficients are pure q powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so called pbw filtration in the corresponding finite dimensional irreducible representations of the simple lie algebras for any root systems this is a particular case of a general conjecture in terms of the level one demazure modules we prove this coincidence for all lie algebras of classical type and for g 2 and also establish the relations of our extremal degrees to minimal q degrees of the extremal terms of the kostant q partition function they coincide with the latter only for some root systems

scaling of conformal blocks and generalized theta functions over overline mathcal m _ g n by way of intersection theory on overline mathcal m g n we show that geometric interpretations for conformal blocks as sections of ample line bundles over projective varieties do not have to hold at points on the boundary we show such a translation would imply certain recursion relations for first chern classes of these bundles while recursions can fail geometric interpretations are shown to hold under certain conditions

two point coordinate rings for gk curves giulietti and korchmaros presented new curves with the maximal number of points over a field of size q6 garcia guneri and stichtenoth extended the construction to curves that are maximal over fields of size q2n for odd n 3 the generalized gk curves have affine equations xq x yq 1 and yq2 y zr for r qn 1 q 1 we give a new proof for the maximality of the generalized gk curves and we outline methods to efficiently obtain their two point coordinate ring

counts of maps to grassmannians and intersections on the moduli space of bundles we show that intersection numbers on the moduli space of stable bundles of coprime rank and degree over a smooth complex curve can be recovered as highestdegree asymptotics in formulas of vafa intriligator type in particular we explicitly evaluate all intersection numbers appearing in the verlinde formula our results are in agreement with previous computations of witten jeffrey kirwan and liu moreover we prove the vanishing of certain intersections on a suitable quot scheme which can be interpreted as giving equations between counts of maps to the grassmannian

a positive combinatorial formula for symplectic kostka foulkes polynomials i rows abstract we prove a conjecture of lecouvey which proposes a closed positive combinatorial formula for symplectic kostka foulkes polynomials in the case of rows of arbitrary weight to show this we construct a new algorithm for computing cocyclage in terms of which the conjecture is described our algorithm is free of local constraints which were the main obstacle in lecouvey s original construction in particular we show that our model is governed by the situation in type a this approach works for arbitrary weight and we expect it to lead to a proof of the conjecture in full generality

an analytic proof of the geometric quantization conjecture of guillemin sternberg we present a direct analytic approach to the guillemin sternberg conjecture gs that geometric quantization commutes with symplectic reduction which was proved recently by meinrenken m1 m2 and vergne v1 v2 et al besides providing a new proof of this conjecture our methods also lead immediately to further extensions in various contexts

the survival of branching annihilating random walk branching annihilating random walk is an interacting particle system on as time evolves particles execute random walks and branch and disappear when they meet other particles it is shown here that starting from a finite number of particles the system will survive with positive probability if the random walk rate is low enough relative to the branching rate but will die out with probability one if the random walk rate is high since the branching annihilating random walk is non attractive standard techniques usually employed for interacting particle systems are not applicable instead a modification of a contour argument by gray and griffeath is used

bounds for the critical line of the theta contact processes with 1 or theta or 2 we study a family of the one dimensional contact processes introduced by durrett and griffeath 1983 which is parametrized by theta for each theta or 1 there is a unique critical value lambda c theta so that any process becomes extinct with probability 1 for lambda lambda c theta in this paper we give rigorous upper and lower bounds for the critical line lambda lambda c theta for 1 or theta or 2 in order to obtain the upper bound we extend the holley liggett argument which was originally given for the case theta 2 the basic contact process we construct a new class of attractive renewal measures with positive densities and the upper bound of lambda c theta is given as the largest root of a cubic equation theta lambda 3 3 theta 2 lambda 2 3 2 theta lambda 2 theta 0 recently liggett 1991 reported an upper bound of the critical value for the case theta 1 the threshold contact process by a modified version of the holley liggett argument our result includes these previous results as special cases

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

twisted strong macdonald theorems and adjoint orbits the strong macdonald theorems state that for l reductive and s an odd variable the cohomology algebras h l z z n and h l z s are freely generated and describe the cohomological s and z degrees of the generators the resulting identity for the z weighted euler characteristic is equivalent to macdonald s constant term identity for a finite root system we calculate h mathfrak p z n mathfrak p and h mathfrak p s for mathfrak p a standard parahoric in a twisted loop algebra giving strong macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist in particular we show that h mathfrak p z n mathfrak p contains a parabolic subalgebra of the coinvariant algebra of the fixed point subgroup of the weyl group of l and thus is no longer free we also prove a strong macdonald theorem for h mathfrak b s mathfrak n and h mathfrak b z n mathfrak n when mathfrak b and mathfrak n are iwahori and nilpotent subalgebras respectively of a twisted loop algebra for each strong macdonald theorem proved taking z weighted euler characteristics gives an identity equivalent to macdonald s constant term identity for the corresponding affine root system as part of the proof we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to l proving an analogue of the kostant slice theorem

correction factors for kac moody groups and t deformed root multiplicities we study a correction factor for kac moody root systems which arises in the theory of p adic kac moody groups in affine type this factor is known and its explicit computation is the content of the macdonald constant term conjecture the data of the correction factor can be encoded as a collection of polynomials m lambda in mathbb z t indexed by positive imaginary roots lambda at t 0 these polynomials evaluate to the root multiplicities so we consider m lambda to be a t deformation of mathrm mult lambda we generalize the peterson algorithm and the berman moody formula for root multiplicities to compute m lambda as a consequence we deduce fundamental properties of m lambda

the t analog of the basic string function for twisted affine kac moody algebras abstract we study lusztig s t analog of weight multiplicities associated to level one representations of twisted affine kac moody algebras an explicit closed form expression is obtained for the corresponding t string function using constant term identities of macdonald and cherednik the closed form involves the generalized exponents of the graded pieces of the twisted affine algebra considered as modules for the underlying finite dimensional simple lie algebra this extends previous work on level 1 t string functions for the untwisted simply laced affine kac moody algebras

on the homology of sln tc t and a theorem of stembridge abstract in this paper we apply the garland lepowsky theorem to compute the homology of the lie algebra n sl n t c t by suitably taking euler characteristics we derive a theorem of stembridge which gives a combinatorial decomposition of a virtual character of sl n c in his theorem stembridge indexes the schur functions that appear in his decomposition by an integer vector of length n and a permutation in sn an additional feature of our approach is that we interpret the integer vector and permutation in terms of the affine weyl group of the kac moody lie algebra of type a a second result given in the garland lepowsky paper extending an earlier result of kostant is an explicit formula for the eigenvalues of the laplacian in the koszul complex for computing the homology of n we give an alternative quite different in nature description of those eigenvalues the garland lepowsky formula is algebraic stated in terms of differences of squared norms our description is combinatorial stated in terms of structural features of certain graphs in order to make this combinatorial interpretation seemingly more natural we link it directly with the paper of kostant we work in this part with the lie algebra l t n where t is the span of x 1 with x taken from the strictly upper triangular matrices in sl n c

a simple proof of the stability criterion of gray and griffeath gray and griffeath studied attractive nearest neighbor spin systems on the integers having all 0 s and all 1 s as traps using the contour method they established a necessary and sufficient condition for the stability of the all 1 s equilibrium under small perturbations in this paper we use a renormalized site construction to give a much simpler proof our new approach can be used in many situations as a substitute for the contour method

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

cohomology of flag varieties and the brylinski kostant filtration let g be a reductive algebraic group over c and let n be a g module for any subspace m of n the brylinski kostant filtration on m is defined through the action of a principal nilpotent element in lieg this filtration is related to a q analog of weight multiplicity due to lusztig we generalize this filtration to other nilpotent elements and show that this generalized filtration is related to parabolic versions of lusztig s q analog of weight multiplicity along the way we also generalize results of broer on cohomology vanishing of bundles on cotangent bundles of partial flag varieties we conclude by computing some explicit examples