yang s system of particles and hecke algebras the graded hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions this operator algebra arises from the study of the root system analogue of yang s system of n particles on the real line with delta function potential it turns out that the spectral problem for this generalization of yang s system is related to the problem of finding the spherical tempered representations of the graded hecke algebra this observation turns out to be very useful for both these problems application of our technique to affine hecke algebras yields a simple formula for the formal degree of the generic iwahori spherical discrete series representations

types and unitary representations of reductive p adic groups we prove that for every bushnell kutzko type that satisfies a certain rigidity assumption the equivalence of categories between the corresponding bernstein component and the category of modules for the hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories moreover we show that every irreducible smooth g representation contains a rigid type this is a generalization of the unitarity criterion of barbasch and moy for representations with iwahori fixed vectors

duality for ext groups and extensions of discrete series for graded hecke algebras in this paper we study extensions of graded affine hecke algebra modules in particular based on an explicit projective resolution on graded affine hecke algebra modules we prove a duality result for ext groups this duality result with an ind res resolution gives an algebraic proof of the fact that all higher ext groups between a tempered module and a discrete series vanish

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

the based ring of the lowest two sided cell of an affine weyl group we show that lusztig s homomorphism from an affine hecke algebra to the direct summand of its asymptotic hecke algebra corresponding to the lowest two sided cell is related to the homomorphism constructed by chriss and ginzburg using equivariant k theory by a matrix over the representation ring of the associated algebraic group

hecke algebras of classical groups over p adic fields and supercuspidal representations let k be a p adic field with involution 0 and let k 0 be its fixed subfield of k let v be a finite dimensional vector space defined over k equipped with e hermitian form v k 0 is large

the geometry of generalized steinberg varieties for a reductive algebraic group g the steinberg variety of g is the set of all triples consisting of a unipotent element u in g and two borel subgroups of g that contain u we define generalized steinberg varieties that depend on four parameters and analyze in detail two special cases that turn out to be related to distinguished double coset representatives in the weyl group using one of the two special cases we define a parabolic version of a map from the weyl group to a set of nilpotent orbits of g in lie g defined by joseph and study some of its properties

the saito kurokawa lifting and functoriality certain nontempered liftings from pgl 2 pgl 2 to pgsp 4 are constructed using the theory of local and global theta lifts the resulting representations on pgsp 4 are the saito kurokawa representations the lifting is shown to be functorial under certain reasonable assumptions on the local langlands correspondence for pgsp 4

local langlands correspondence for classical groups and affine hecke algebras using the results of colette moeglin on the representations of p adic classical groups based on methods of james arthur and its relation with representations of affine hecke algebras established by the author we show that the category of smooth complex representations of a quasi split p adic classical group and its pure inner forms is naturally decomposed into subcategories which are equivalent to a tensor product of categories of unipotent representations of classical groups a statement of this kind had been conjectured by g lusztig all classical groups general linear orthogonal symplectic and unitary groups appear in this context we get also parameterizations of representations of affine hecke algebras which seem not all to be in the literature yet all this sheds some light on what is known as the stable bernstein center

generalized springer correspondence and green functions for type b c graded hecke algebras this paper presents a combinatorial approach to the tempered representation theory of a graded hecke algebra h of classical type b or c with arbitrary parameters we present various combinatorial results which together give a uniform combinatorial description of what becomes the springer correspondence in the classical situation of equal parameters more precisely by using a general version of lusztig s symbols which describe the classical springer correspondence we associate to a discrete series representation of h with central character w0c a set w0c of w0 characters where w0 is the weyl group this set w0c is shown to parametrize the central characters of the generic algebra which specialize into w0c using the parabolic classification of the central characters of h t r on the one hand and a truncated induction of weyl group characters on the other hand we define a set w0c for any central character w0c of h t r and show that this property is preserved we show that in the equal parameter situation we retrieve the classical springer correspondence by considering a set u of partitions which replaces the unipotent classes of so2n 1 c and sp2n c and a bijection between u and the central characters of h t r we end with a conjecture which basically states that our generalized springer correspondence determines h t r exactly as the classical springer correspondence does in the equal label case in particular we conjecture that w0c indexes the modules in h t r with central character w0c in the following way a module m in h t r has a natural grading for the action of w0 and the w0 representation m in its top degree is irreducible when m runs through the modules in h t r with central character w0c m runs through w0c moreover still in analogy with the equal parameter case we conjecture that the w0 structure of the modules in h t r can be computed using generalized green functions

hochschild homology of affine hecke algebras abstract let h h r q be an affine hecke algebra with complex possibly unequal parameters q which are not roots of unity we compute the hochschild and the cyclic homology of h it turns out that these are independent of q and that they admit an easy description in terms of the extended quotient of a torus by a weyl group both of which are canonically associated to the root datum r for positive q we also prove that the representations of the family of algebras h r q c come in families which depend analytically on analogous results are obtained for graded hecke algebras and for schwartz completions of affine hecke algebras

affine hecke algebras and generalized standard young tableaux abstract this paper introduces calibrated representations for affine hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations the primary technique is to provide indexing sets for controlling the weight space structure of finite dimensional modules for the affine hecke algebra using these indexing sets we show that 1 irreducible calibrated representations are indexed by skew local regions 2 the dimension of an irreducible calibrated representation is the number of chambers in the local region 3 each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine hecke algebra on a specific basis in the representation space the indexing sets for weight spaces are generalizations of standard young tableaux and the construction of the irreducible calibrated affine hecke algebra modules is a generalization of a young s seminormal construction of the irreducible representations of the symmetric group in this sense young s construction has been generalized to arbitrary lie type

residue construction of hecke algebras abstract hecke algebras are usually defined algebraically via generators and relations we give a new algebra geometric construction of affine and double affine hecke algebras the former is known as the iwahori hecke algebra and the latter was introduced by cherednik in terms of residues more generally to any generalized cartan matrixaand a pointqin a one dimensional complex algebraic group c we associate an associate algebrahq ifais of finite type and c c the algebrahqis theaffinehecke algebra of the corresponding finite root system ifais of affine type and c c thenhqis essentially the cherednik algebra the case c c corresponds to degenerate counterparts of the above objects considered by drinfeld and lusztig finally taking c to be an elliptic curve one gets some new elliptic analogues of the affine hecke algebra

lagrangian construction for representations of hecke algebras 1 let k be a non archimedean local field with the residue field consisting of q elements according to langlands philosophy irreducible complex representations of a reductive k group g should roughly speaking correspond to homomorphisms of the weil deligne group into the socalled l group let zc g be the iwahori subgroup the category of admissible gkmodules generated by z fixed vectors is known to be equivalent to the category of finite dimensional representations of the hecke algebra h z g z langlands philosophy suggests via that equivalence the following classification of irreducible hecke algebra modules stated in purely complex terms and not involving any local held

affine cellular algebras abstract graham and lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras many classes of finite dimensional algebras including various hecke algebras and diagram algebras have been shown to be cellular and the theory due to graham and lehrer successfully has been applied to these algebras we will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field affine hecke algebras of type a and infinite dimensional diagram algebras like the affine temperley lieb algebras are shown to be examples of our definition the isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties in this way representation theory of non commutative algebras is linked with commutative algebra moreover conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification these conditions are shown to be satisfied for the affine hecke algebra of type a if the quantum parameter is not a root of the poincare polynomial

the modular branching rule for affine hecke algebras of type a for the affine hecke algebra of type a at roots of unity we make explicit the correspondence between geometrically constructed simple modules and combinatorially constructed simple modules and prove the modular branching rule the latter generalizes work by vazirani

on the signature character of representations of p adic general linear groups in this article we calculate the signature character of certain hermitian representations of gln f for a p adic field f we further give a conjectural description for the signature character of unramified representations in terms of kostka numbers

level raising and symmetric power functoriality iii we apply automorphy lifting techniques to establish new cases of symmetric power functoriality for hilbert modular forms of regular algebraic weight the proof is based on a novel application of an automorphy lifting theorem for residually reducible galois representations

on characters and formal degrees of discrete series of affine hecke algebras of classical types we address two fundamental questions in the representation theory of affine hecke algebras of classical types one is an inductive algorithm to compute characters of tempered modules and the other is the determination of the constants in the formal degrees of discrete series in the form conjectured by reeder j reine angew math 520 37 93 2000 the former is completely different from the lusztig shoji algorithm shoji in invent math 74 239 267 1983 lusztig in ann math 131 355 408 1990 and it is more effective in a number of cases the main idea in our proof is to introduce a new family of representations which behave like tempered modules but for which it is easier to analyze the effect of parameter specializations our proof also requires a comparison of the c theoretic results of opdam delorme slooten solleveld j inst math jussieu 3 531 648 2004 arxiv 0909 1227 int math res not 2008 adv math 220 1549 1601 2009 acta math 205 105 187 2010 and the geometric construction from kato duke math j 148 305 371 2009 am j math 133 518 553 2011 ciubotaru and kato adv math 226 1538 1590 2011

higgs bundles gauge theories and quantum groups the appearance of the bethe ansatz equation for the nonlinear schrodinger equation in the equivariant integration over the moduli space of higgs bundles is revisited we argue that the wave functions of the corresponding two dimensional topological u n gauge theory reproduce quantum wave functions of the nonlinear schrodinger equation in the n particle sector this implies the full equivalence between the above gauge theory and the n particle sub sector of the quantum theory of the nonlinear schrodinger equation this also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine hecke algebra we propose a similar construction based on the g g gauged wzw model leading to the representation theory of the double affine hecke algebra

on classical saito kurokawa liftings there exist two different generalizations of the classical saito kurokawa lifting to modular forms with square free level one lifting produces modular forms with respect to 0 m the other one with respect to the paramodular group m we shall give an alternative and unified construction of both liftings using group theoretic methods the construction shows that a single elliptic modular form may in fact have many saito kurokawa liftings we also obtain precise information about the spin l function of the resulting siegel modular forms

unipotent representations of complex semisimple groups algebraic group over r or c in this paper we restrict attention to c we generalize arthur s definition slightly or perhaps simply make it more precise all of the resulting representations except for a finite set are then unitarily induced from representations of the same kind on proper parabolic subgroups we call the finite set remaining special unipotent representations a precise definition will be given later definition 1 17 our main result theorem iii of this introduction is a character formula for any special unipotent representation of course such a formula can be deduced from the kazhdan lusztig conjecture cf v3 the advantages of theorem iii are that it is in closed form and that it lends itself to verification of some conjectures of arthur in a so let g be a connected complex semisimple lie group and q its lie algebra choose

isomorphisms between affine hecke algebras of type a2 abstract we prove that two affine hecke algebras h p and h q of type a 2 with nonzero parameters p q c are not isomorphic whenever p q and p q 1 theorem 3 1 however they are isomorphic locally more precisely we prove that the quotient algebras h s q and h t p are isomorphic provided that n s q n t p for type a 2 theorem 2 4

on the non unitary unramified dual for classical p adic groups in this paper we give a zelevinsky type classification of unramified irreducible representations of split classical groups

discrete series characters for affine hecke algebras and their formal degrees we introduce the generic central character of an irreducible discrete series representation of an affine hecke algebra using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine hecke algebras except for the types e n 1 n 6 7 8 with arbitrary positive parameters and we prove an explicit product formula for their formal degrees in all cases

unipotent representations and unipotent classes in sl n to achieve the stabilization of the trace formula for a reductive group 18 there are two related problems that must first be solved one is the matching of the sorbital integrals of smooth functions on a reductive group g over a p adic field of characteristic zero with stable orbital integrals on its endoscopic groups h igusa theory clarifies the nature of this first problem 11 12 13 19 22 23 the second problem is the fundamental lemma which asserts the compatibility of the matching of smooth functions with the hecke algebras on g and h buildings have been used to study this problem 25 the statements of the problems are closely related and the main results of this paper give a method of reducing one problem to the other if the first problem matching smooth functions is solved for sl n then the second problem the fundamental lemma for the hecke algebras follows as a consequence at least for the identity of the algebra the solutions of the two problems are linked by making a detailed study of the unipotent representations and the unipotent classes of sl n representations with iwahori fixed vectors on sl n f are called unipotent in this paper to emphasize the close relationship between unipotent conjugacy classes and these representations not long after this paper was written waldspurger refined reformulated and greatly extended the results of this paper this paper made a modest contribution to this subsequent work of his he eliminated the hypothesis that functions on sl n have matching smooth functions by making clever use of a strengthened version of kazhdan s lemma the opinion of the experts is still divided will the solution of the fundamental lemma for other reductive groups come from the matching of smooth functions or will it come from even stronger versions of kazhdan s lemma this much is clear the fundamental lemma no longer needs to be viewed as an isolated combinatorial problem in buildings howe s conjecture uniform germ expansions homogeneity of germs compact traces unipotent orbits and kazhdan s density theorem give us a framework through which the fundamental lemma may be understood in the final portion of the introduction i would like to state a theorem that has considerable interest beyond its usefulness for the fundamental lemma this theorem will be used in section 1 to produce a uniform germ expansion the

a new proof for classification of irreducible modules of a hecke algebra of type an 1 abstract in this paper we give a new proof for the classification of irreducible modules of a hecke algebra of type a n 1 which was obtained by dipper and james in 1986

the degenerate analogue of ariki s categorification theorem we explain how to deduce the degenerate analogue of ariki s categorification theorem over the ground field mathbb c as an application of schur weyl duality for higher levels and the kazhdan lusztig conjecture in finite type a we also discuss some supplementary topics including young modules tensoring with sign tilting modules and ringel duality

sparse and distributed analytic hierarchy process abstract the analytic hierarchy process ahp is a de facto standard technique in centralized decision making consider a situation where there is a need to rank a set of elements or alternatives based on their value or utility of which we just know pairwise relative information i e the ratio of their values ahp proved an effective tool to retrieve the value of each element being able to handle also relative information affected by distortions subjective biases and intransitivity a downside of ahp however is that it requires complete information i e knowledge on all pairs in this paper we extend the applicability of the ahp technique to the case of sparse information i e when only a limited amount of information is available and such an information corresponds to an undirected connected graph we complement our sparse framework by developing novel criteria and metrics to evaluate the degree of consistency of the data at hand moreover exploiting the proposed framework we also provide a distributed formulation of ahp in which a set of agents interacting through an undirected graph are able to compute their own values e g for ranking or leader election purposes by only knowing the ratio of their values with respect to their neighbors to this end we develop a novel algorithm to let each agent i compute the dominant eigenvalue and the i th component of the corresponding eigenvector of the sparse ahp matrix we conclude the paper with a simulation campaign that numerically demonstrates the effectiveness of the proposed approach

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

multiplicity matrices for the affine graded hecke algebra in this paper we are looking at the problem of determining the composition factors for the affine graded hecke algebra via the computation of kazhdan lusztig type polynomials we review the algorithms of g lusztig study of perverse sheaves arising from graded lie algebras adv math 112 1995 147 217 g lusztig graded lie algebras and intersection cohomology preprint and use them in particular to compute at every real central character which admits tempered modules the geometric parameterization the kazhdan lusztig polynomials the composition series and the iwahori matsumoto involution for the representations with iwahori fixed vectors of the split p adic groups of type g2 and f4 and by the nature of the algorithms for their levi subgroups

a framework for robust parametric set membership identification this paper proposes a new framework for studying robust parametric set membership identification the authors derive some new results on the fundamental limitations of algorithms in this framework given a particular model structure the new idea is to quantify uncertainty only with respect to the finite dimensional parametric part of the model and not the fixed size unmodeled dynamics thus the measure of uncertainty is different from the measures used in previous robust identification work where system norms are used to quantify uncertainty as an example the results are used to assess the fidelity of a certain approximate robust parametric set membership identification algorithm

the affineq schur algebra abstract we introduce an analogue of the q schur algebra associated to coxeter systems of type a n 1 we give two constructions of this algebra the first construction realizes the algebra as a certain endomorphism algebra arising from an affine hecke algebra of type a r 1 where n r this generalizes the original q schur algebra as defined by dipper and james and the new algebra contains the ordinary q schur algebra and the affine hecke algebra as subalgebras using this approach we can prove a double centralizer property the second construction realizes the affine q schur algebra as the faithful quotient of the action of a quantum group on the tensor power of a certain module analogous to the construction of the ordinary q schur algebra as a quotient of u g l n