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

some small unipotent representations of indefinite orthogonal groups abstract we study a family of small unitary representations of indefinite orthogonal groups these representations arise as analytic continuations of the discrete series and were studied extensively by knapp in k3 we complete knapp s analysis by proving that they are irreducible in order to do so we prove that the representations are unipotent and have irreducible associated cycles in which all multiplicities are exactly one moreover we prove that the k type structure of each representation matches up to a shift the k type structure of the ring of functions on the closure a nilpotent k c orbit on p

a fixed point formula and harish chandra s character formula the main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators for proper actions by connected semisimple lie groups on possibly noncompact manifolds with compact quotients for compact groups and manifolds this reduces to the atiyah segal singer fixed point formula other special cases include an index theorem by connes and moscovici for homogeneous spaces and an earlier index theorem by the second author both in cases where the group acting is connected and semisimple as an application of this fixed point formula we give a new proof of harish chandra s character formula for discrete series representations

one dimensional representations of u p q and the howe correspondence abstract we explicitly determine the theta lifts of all one dimensional representations of u p q in terms of langlands parameters and determine exactly which lifts are unitary moreover we show that such a lift is unitary if and only if it is a weakly fair derived functor module of the form a q finally we show that the correspondence of unitary representations behaves well with respect to associated cycles

unitary representations induced from maximal parabolic subgroups abstract it is known that the problem of classifying the irreducible unitary representations of a linear connected semisimple lie group g comes down to deciding which langlands quotients j man v are infinitesimally unitary here man is any cuspidal parabolic subgroup is any discrete series or nondegenerate limit of discrete series representation of m and v is any complex valued linear functional on the lie algebra of a satisfying certain positivity and symmetry properties the authors determine which langlands quotients are infinitesimally unitary under the conditions that g is simple that dim a 1 and that g is neither split f 4 nor split g 2

dirac operator and its cohomology for the quantum group uq x1d598 x1d591 x1d7d0 we introduce a dirac operator d for the quantum group uq x1d530 x1d529 2 as an element of the tensor product of uq x1d530 x1d529 2 with the clifford algebra on two generators we study the properties of d including an analogue of vogan s conjecture we compute the cohomology of d acting on various uq x1d530 x1d529 2 modules

classification of aq λ modules by their dirac cohomology for type d g2 and sp 2n r abstract let g be a connected real reductive group with maximal compact subgroup k of the same rank as g in the recent paper of huang pand ic and vogan it was shown that the admissible stable parabolic subalgebras q of g are in one to one correspondence with the faces of w intersecting the k dominant weyl chamber and that a q 0 modules can be classified by their dirac cohomology in geometric terms they described in detail the cases when g 0 is of type a b f and c except for g 0 sp 2 n r we will describe faces corresponding to a q 0 modules for g 0 sp 2 n r and for g 0 of type d and g 2

dirac cohomology for graded affine hecke algebras we define an analogue of the casimir element for a graded affine hecke algebra mathbb h and then introduce an approximate square root called the dirac element using it we define the dirac cohomology hd x of an mathbb h module x and show that hd x carries a representation of a canonical double cover of the weyl group widetilde w our main result shows that the widetilde w structure on the dirac cohomology of an irreducible mathbb h module x determines the central character of x in a precise way this can be interpreted as p adic analogue of a conjecture of vogan for harish chandra modules we also apply our results to the study of unitary representations of mathbb h

the mizohata structure on the sphere we prove that a compact surface that admits a mizohata structure is homeomorphic to a sphere and that there exists exactly one such structure l up to conjugation by diffeomorphisms we also characterize the range and the kernel of the operator 0 induced by l i e obtained from the exterior derivative on functions by passing to the quotient modulo l

dirac series for some real exceptional lie groups abstract up to equivalence this paper classifies all the irreducible unitary representations with non zero dirac cohomology for the following simple real exceptional lie groups ei e 6 6 eiv e 6 26 fi f 4 4 fii f 4 20 along the way we find an irreducible unitary representation of f 4 4 whose dirac index vanishes while its dirac cohomology is non zero this disproves a conjecture raised in 2015 asserting that there should be no cancellation between the even part and the odd part of the dirac cohomology

symmetric subalgebras noncohomologous to zero in a complex semisimple lie algebra and restrictions of symmetric invariants abstract let g be a semisimple involutory lie algebra and g k the corresponding symmetric pair let h be a fundamental cartan subalgebra of g containing a cartan subalgebra t of k the semisimple involutory lie algebras g with the symmetric subalgebra k noncohomologous to zero in g are completely classified by showing that k is noncohomologous to zero in g if and only if the spin representation of g is primary based on this result we then determine the image of the restriction map s h w g h s t w k t where w g h and w k t are the respective weyl groups

dirac cohomology and geometric quantization let g be a connected real semisimple lie group having a finite center and a compact cartan subgroup t with lie algebra t0 let be a g t invariant symplectic form on xdg t0 we incorporate dirac cohomology into the geometric quantization of x and study the resulting multiplicity free unitary g t representation on a hilbert space h x we also perform symplectic reduction of x and show that our quantiza tion method satisfies the principle quantization commutes with reduction as an application we construct various models of discrete series

signatures of invariant hermitian forms on irreducible highest weight modules perhaps the most important problem in representation theory in the 1970s and early 1980s was the determination of the multiplicity of com position factors in a verma module this problem was settled by the proof of the kazhdan lusztig conjecture which states that the multiplicities may be computed via kazhdan lusztig polynomials in this paper we introduce signed kazhdan lusztig polynomials a variation of kazhdan lusztig polyno mials which encodes signature information in addition to composition factor multiplicities and jantzen filtration level careful consideration of gabber and joseph s proof of kazhdan and lusztig s recursive formula for computing kazhdan lusztig polynomials and an application of jantzen s determinant for mula lead to a recursive formula for the signed kazhdan lusztig polynomials we use these polynomials to compute the signature of an invariant hermitian form on an irreducible highest weight module such a formula has applications to unitarity testing 1 1 the unitary dual problem in the 1930s i m gelfand introduced a broad programme in abstract harmonic analysis which would permit the transfer of dif ficult problems in areas as distinct from analysis as topology to more tractable problems in algebra fourier analysis is just one incarnation of this programme an unresolved component in gelfand s programme is the classification of the irre ducible unitary representations of a group known as the unitary dual problem in the case of a real reductive lie group the problem is equivalent to identifying all irreducible harish chandra modules which admit a positive definite invariant hermitian form as harish chandra modules may be constructed via an algebraic method introduced by zuckerman in 1978 known as cohomological induction it is of interest to study signatures of invariant hermitian forms on cohomologically induced modules and to understand how positivity can fail cohomological induction is a two step process in which we compose an induction functor with a zuckerman functor i the intermediate module in cohomological induction is a generalized verma module which admits an invariant hermitian form if the module to which induction was applied admits an invariant hermitian form formulas for the signatures of invariant hermitian forms on these intermediate modules may be used to compute signatures of forms on corresponding cohomo logically induced modules eg wal84 this motivates the study of invariant hermitian forms on verma modules see wal84 yee05 and irreducible highest weight modules

unitary equivalences for reductive p adic groups we establish a transfer of unitarity for a bernstein component of the category of smooth representations of a reductive p adic group to the associated hecke algebra in the framework of the theory of types whenever the hecke algebra is an affine hecke algebra with geometric parameters in the sense of lusztig possibly extended by a group of automorphisms of the root datum it is known that there is a large class of such examples detailed in the paper as a consequence we establish relations between the unitary duals of different groups in the spirit of endoscopy

the unitary dual of p adic g_2 in this paper is completly solved the unitarizability problem for the p adic reductive group of type g 2 the unitary dual of p adic g 2 is described in terms of supercuspidal representations of its levi subgroups

the homomorphisms between scalar generalized verma modules associated to maximal parabolic subalgebras let g be a finite dimensional simple lie algebra over c we classify the homomorphisms between g modules induced from one dimensional modules of maximal parabolic subalgebras 1

the langlands parameters of subquotients of certain derived functor modules letgbe a noncompact simple lie group with finite center letkbe a maximal compact subgroup and let g0 k0 p0be the corresponding decomposition of the lie algebra suppose rank g rank k and let t0be a compact cartan subalgebra and b be a borel subalgebra letab be a derived functor module with infinitesimal character which is nondominant with respect to a noncompact simple root suppose that 2 p iskdominant so that thektype with highest weight occurs with multiplicity one inab we develop conditions on the roots of g under which a functor of cohomological induction maps a certain module of parabolic induction to another module of parabolic induction extending a result due to vogan this allows us in many cases to determine the langlands parameters of the subquotient ofab containing via a conjectured method of knapp

explicit induction principle and symplectic orthogonal theta lifts abstract during the last two decades great efforts have been devoted to the calculation of the local theta correspondence for reductive dual pairs however uniform formulas remain elusive for real dual pairs of type i the purpose of this paper is twofold to formulate an explicit version of induction principle for dual pairs o p q s p 2 n r with p q even and to apply it to obtain a complete and explicit description of the local theta correspondence when p q 4 our approach is very elementary by analysis on the infinitesimal characters and k types under the theta correspondence

unitary dual of the non split inner form of sp 8 f we classify the non cuspidal part of the unitary dual of the non quasi split inner form of sp 8 f where f is a non archimedean field of characteristic zero we obtain a conjectural description of the discrete l packets which contain representations of sp 4 f and its non split inner form

induction functor in noncommutative equivariant cohomology and dirac cohomology abstract the aim of this paper is to put some recent results of huang and pandzic conjectured by vogan and kostant on dirac cohomology in a broader perspective this is achieved by introducing an induction functor in the noncommutative equivariant cohomology in this context the results of huang pandzic and kostant are interpreted as special cases corresponding to the manifold being a point of more general results on noncommutative equivariant cohomology introduced by alekseev and meinrenken

dirac cohomology unitary representations and a proof of a conjecture of vogan u radu je dokazana slutnja d vogana koja odredjuje infinitesimalni karakter ireducibilnog g k modula koji ima diracovu kohomologiju razlicitu od nula infinitesimalni karakter je dan kao rho translat najvece tezine k tipa u diracovoj kohomologiji diracova kohomologija se definira kao ker d im d cap ker d gdje d oznacava diracov operator na zadanom g k modulu tenzoriranom sa spin modulom

global properties of a class of vector fields in the plane on etudie l existence des solutions globales au probleme lu o du o sur r 2 ou l est un champ vectoriel sans singularite on se restreint au cas d operateurs localement integrables et satisfaisant certaines proprietes

invariant syzygies and semisimple groups the purpose of this paper is to explore some intriguing connections between lie theory relative homological algebra invariant theory and representations of finite dimensional algebras in the process some results of hilbert 20 cartan and eilenberg 7 hochschild 21 gelfand 171 and others are generalized and a question of zl is partly answered in theorem 1 5 duflo seems to have stronger unpublished results on this problem sections 1 to 3 take place in a fairly general setting whereas section 4 focuses on semisimple lie groups the problem considered here goes back in some sense to hilbert 20 this paper was published 102 years ago to explain this let me start by singling out two out of many fundamental contributions of the great mathematician the first one is invariant theory namely the study of the algebra sv n consisting of those polynomial functions on a finite dimensional vector space v which are invariant under the action of a group h of linear transformations the second one is the so called syzygy theorem which asserts in today s language that the global dimension of the full symmetric algebra sv is dim v this prompts the question is there an algebra d attached to v as naturally as sv is on which h acts by automorphisms such that the global dimension of the algebra b dh of invariants is dim v the most natural candidate is

a characterization of low weight words that span generalized reed muller codes we consider the generalized reed muller code r sub fq spl rho m of order spl rho and length q sup m m 1 over the field f sub q where q p sup t for prime p and t spl ges 1 in particular we are interested in the case that t 1 so that q is not prime and the order spl rho is at least q as shown by ding and key under these conditions unless spl rho is very large i e spl rho m 1 q 1 p sup t 1 2 the code is not spanned by its minimum weight words furthermore there was no known characterization of words with small weight that span the code in this correspondence we characterize a set of words that span the code and show that their weight is upper bounded by q sup spl lceil m q 1 spl rho q q p spl rceil which is at most quadratic in the weight of the minimum weight words

dirac cohomology and translation functors abstract we study the relationship between the dirac cohomology of a g k module x and the dirac cohomology of a jantzen zuckerman translate of x more precisely we show that if x is unitary and if some submodule x of a translate of x has nonzero dirac cohomology then x has nonzero dirac cohomology we also show that the space of harmonic spinors i e the kernel of the dirac operator related to x embeds into a certain product of harmonic spinors for x and harmonic spinors for the finite dimensional module used to define the translation this generalizes with a simpler proof results of mehdi and parthasarathy 2008 mp1 and mehdi and parthasarathy 2010 mp2

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

the unitary dual for complex classical lie groups in this paper we classify the unitary spectrum of the complex classical groups let g be any complex simply connected semisimple lie group there are two well known ways of constructing unitary representations from lower groups unitary induction from a parabolic subgroup and complementary series a reasonable way to describe the unitary dual is to find a distinguished set of representations for each group such that every other unitary representation is obtained from this set by the above constructions from strictly smaller parabolic subgroups this is entirely similar to the classification of admissible modules where the distinguished representations are the discrete series for integral infinitesimal character such a set is provided by the special unipotent representations introduced in bv1 their existence and unitarity was conjectured earlier by a they were shown to have many interesting properties such as the fact that their annihilator is a maximal primitive ideal from this it was possible to obtain character formulas for them however it is clear that this set is not large enough to describe the entire unitary spectrum for example in type c the metaplectic representation is not special unipotent or even one of the larger class of representations considered in a the definition of special unipotent representations suggests that one should consider representations with maximal primitive ideal and infinitesimal character as small as possible it is not clear which of them should be unitary imposing the condition that the annihilator be a completely prime ideal which is necessary for unitarity does not seem to be very helpful either thus one is left with the task of explicitly computing the g invariant form and ruling out the cases when it is indefinite this is an almost impossible calculation a rough estimate says that one needs to compute signatures on the k types in the principal series which have length ii 2p ql

principal series representations and harmonic spinors abstract let g be a real reductive lie group and g h a reductive homogeneous space we consider kostant s cubic dirac operator d on g h twisted with a finite dimensional representation of h under the assumption that g and h have the same complex rank we construct a nonzero intertwining operator from principal series representations of g into the kernel of d the langlands parameters of these principal series are described explicitly in particular we obtain an explicit integral formula for certain solutions of the cubic dirac equation d 0 on g h

howe correspondence for real unitary groups roger howe proved that for any reductive dual pair g g in the symplectic groupsp 2n r there is a one to one correspondence of irreducible admissible representations of some two fold covers ofgandg we determine this correspondence explicitly for dual pairs of the form u p q u r s withr s p q and describe it in terms of langlands parameters in this case the correspondence may be understood in a natural way as a correspondence of representations of the linear groups rather than the appropriate covers we show that every irreducible admissible representation ofu p q occurs in the correspondence with precisely one unitary group of equal rank this result verifies a conjecture of harris kudla and sweet who investigated the correspondence for unitary groups of equal size overp adic fields the correspondence of discrete series representations was determined by j s li for induced representations the correspondence is obtained in a natural way from the corresponding discrete series on the respective levi factors of the parabolic subgroups ofu p q andu r s generalizing a result of li we show that under the correspondence representations with nonzero cohomology are matched in an interesting way with unitarity not necessarily preserved the proof uses the induction principle which is due to kudla and an argument involvingk types and the space of joint harmonics howe

dirac cohomology k characters and branching laws inspired by work of enright and willenbring ew we prove a generalized littlewood s restriction formula in terms of dirac cohomology our approach is to use a character formula of irreducible unitary lowest weight modules instead of the bernstein gelfand gelfand resolution and the proof is much simpler we also show that our branching formula is equivalent to the formula of enright and willenbring in terms of nilpotent lie algebra cohomology this follows from the close relationship between the dirac cohomology and the corresponding nilpotent lie algebra cohomology for unitary representations of semisimple lie groups of hermitian type which was established in hpr

multiplets of representations twisted dirac operators and vogan s conjecture in affine setting abstract we extend classical results of kostant et al on multiplets of representations of finite dimensional lie algebras and on the cubic dirac operator to the setting of affine lie algebras and twisted affine cubic dirac operator we prove in this setting an analogue of vogan s conjecture on infinitesimal characters of harish chandra modules in terms of dirac cohomology for our calculations we use the machinery of lie conformal and vertex algebras

dirac index and associated cycles of harish chandra modules abstract let g r be a simple real linear lie group with maximal compact subgroup k r and assume that rank g r rank k r for any representation x of gelfand kirillov dimension 1 2 dim g r k r we consider the polynomial on the dual of a compact cartan subalgebra given by the dimension of the dirac index of members of the coherent family containing x under a technical condition involving the springer correspondence we establish an explicit relationship between this polynomial and the multiplicities of the irreducible components occurring in the associated cycle of x this relationship was conjectured in 12

minimum weight codewords as generators of generalized reed muller codes we establish the range of values of spl rho where 0 spl les spl rho spl les m q 1 for which the generalized reed muller code r sub fq spl rho m of length q sup m over the field f sub q of order q is spanned by its minimum weight vectors

a property of the functions and distributions annihilated by a locally integrable system of complex vector fields page introduction 387 1 basic notation and ingredients 389 2 approximation and constancy on the fibers of the classical solutions of the homogeneous equations 393 3 first consequences and interpretation of the fundamental theorem 398 4 distribution solutions of the homogeneous equations 402 5 regularity of the solutions of the homogeneous equations 409 6 example of a system that is analytic hypo elliptic but not co hypo elliptic 413 references 420

on the dirac series of u p q this paper computes the dirac index of all the weakly fair a mathfrak q lambda modules of u p q we find counter examples to a conjecture of vogan on the unitary dual of u p q which was phrased by trapa in 2001 however we still believe that any irreducible unitary representation of u p q with non zero dirac cohomology must be a weakly fair a mathfrak q lambda module

unitary highest weight modules the unitary highest weight modules of a semisimple lie group g have been classified s 93 unfortunately this classification does not fit well with a more general picture of the unitary dual of g we give such a classification for g a classical group using derived functors and the philosophy of unipotent representations in particular these representations are associated to coadjoint orbits of g let 8 lie g 0 a cartan involution of 6 8 c a derived functor module is defined by the data q i u a o stable parabolic subalgebra of 0 and 17 an irreducible unitary representation of l the stabilizer of l in g considering z7 as a representation of i write z7 a rc where a e c c center of i denotes dual and rt z7l r we define a rc i rc cf definition 6 1 now fix n and vary 1 under certain restrictions on i which amount to a positivity condition on the corresponding bundle over g l a i rr is known to be irreducible and unitary as ii varies within this range the langlands parameters of a a rr vary smoothly however one cannot obtain all unitary highest weight modules under these restrictions the orbit method suggests how to obtain the remaining unitary highest weight modules the modules a rr are associated to coadjoint orbits as follows if rc is trivial we say a rr is associated to the elliptic orbit 0 g ii not all unitary highest weight modules may be obtained with rc trivial an example of which is the oscillator representation of 2n r the metapletic group we define 2n irreducible unitary highest weight modules rc 1 2 n k r is 113 oool 8708 87 7 50

flags in zero dimensional complete intersection algebras and indices of real vector fields we introduce bilinear forms in a flag in a complete intersection local mathbb r algebra of dimension 0 related to the eisenbud levine khimshiashvili bilinear form we give a variational interpretation of these forms in terms of jantzen s filtration and bilinear forms we use the signatures of these forms to compute in the real case the constant relating the gsv index with the signature function of vector fields tangent to an even dimensional hypersurface singularity one being topologically defined and the other computable with finite dimensional commutative algebra methods

conditionally cycle free graphical models for coset codes conditionally cycle free graphical models i e cyclic graphical models which become cycle free after conditioning on a subset of the hidden variables are constructed for coset codes following the description of a general construction procedure examples of a number of families of codes including first order reed muller rm and the delsarte goethals codes are provided for which the proposed procedure yields optimal soft in soft out siso decoding algorithms that are less complex than the best known trellis based algorithms in the case of the first order rm codes which have a recursive coset construction the optimal siso decoding algorithm that results when the proposed construction is applied repeatedly is denoted recursive coset representation rcr decoding connections are made between rcr decoding and existing algorithms that exploit fast hadamard transforms finally the utility of the proposed decoding algorithms are supported by a practically motivated application the construction of serially concatenated codes that have high rates and low error floors extended hamming codes are proposed as outer codes in such constructions with efficient decoding employing the siso algorithms developed herein

corrigendum to dirac cohomology and translation functors j algebra 375 2013 328 336 abstract the statements of theorem 1 3 lemma 3 2 and proposition 5 2 in 4 are incorrect we give counterexamples to these statements and we offer a replacement for theorem 1 3 under stronger assumptions

analysis on the minimal representation of o p q i realization via conformal geometry abstract this is the first in a series of papers devoted to an analogue of the metaplectic representation namely the minimal unitary representation of an indefinite orthogonal group this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of kirillov kostant we begin by applying methods from conformal geometry of pseudo riemannian manifolds to a general construction of an infinite dimensional representation of the conformal group on the solution space of the yamabe equation by functoriality of the constructions we obtain different models of the unitary representation as well as giving new proofs of unitarity and irreducibility the results in this paper play a basic role in the subsequent papers where we give explicit branching formulae and prove unitarization in the various models

duals of affine grassmann codes and their relatives affine grassmann codes are a variant of generalized reed muller codes and are closely related to grassmann codes these codes were introduced in a recent work by beelen here we consider more generally affine grassmann codes of a given level we explicitly determine the dual of an affine grassmann code of any level and compute its minimum distance further we ameliorate the results by beelen concerning the automorphism group of affine grassmann codes finally we prove that affine grassmann codes and their duals have the property that they are linear codes generated by their minimum weight codewords this provides a clean analogue of a corresponding result for generalized reed muller codes

duality between gl n r gl n qp and the degenerate affine hecke algebra for gl n we define an exact functor f n k from the category of harish chandra modules for rm gl n bbb r to the category of finite dimensional representations for the degenerate affine hecke algebra for frak g frak l k under certain natural hypotheses we prove that the functor maps standard modules to standard modules or zero and irreducibles to irreducibles or zero

translation principle for dirac index let g be a finite cover of a closed connected transpose stable subgroup of rm gl n bbb r with complexified lie algebra frak g let k be a maximal compact subgroup of g and assume that g and k have equal rank we prove a translation principle for the dirac index of virtual frak g k modules as a byproduct to each coherent family of such modules we attach a polynomial on the dual of the compact cartan subalgebra of frak g this index polynomial generates an irreducible representation of the weyl group contained in the coherent continuation representation we show that the index polynomial is the exact analogue on the compact cartan subgroup of king s character polynomial the character polynomial was defined by king on the maximally split cartan subgroup and it was shown to be equal to the goldie rank polynomial up to a scalar multiple in the case of representations of gelfand kirillov dimension at most half the dimension of g k we also conjecture an explicit relationship between our index polynomial and the multiplicities of the irreducible components occurring in the associated cycle of the corresponding coherent family

the explicit duality correspondence of sp p q o 2n abstract we investigate the type i dual pairs over the quaternion algebra h namely the family of dual pairs sp p q o 2n we give a complete and explicit description of duality correspondence for p q n as well as some of the cases for p q n in terms of the langlands parameters

dirac cohomology of cohomologically induced modules for reductive lie groups we extend the setting and a proof of the vogan s conjecture on dirac cohomology to a possibly disconnected real reductive lie group g in the harish chandra class we show that the dirac cohomology of cohomologically induced module mathcal l s z is completely determined by the dirac cohomology of the inducing module z more precisely we prove that if z is weakly good then the dirac cohomology h d mathcal l s z is equal to mathcal l widetilde k s h d z otimes bbb c rho frak p cap frak u an immediate application is a classification of tempered irreducible unitary representations with nonzero dirac cohomology

jacquet modules and dirac cohomology abstract we show that dirac cohomology of the jacquet module of a harish chandra module is a harish chandra module for the corresponding levi subgroup we obtain an explicit formula of dirac cohomology of the jacquet module for most of the principal series based on our determination of dirac cohomology of irreducible generalized verma modules with regular infinitesimal characters

a vanishing theorem for dirac cohomology of standard modules abstract this paper studies the dirac cohomology of standard modules in the setting of graded hecke algebras with geometric parameters we prove that the dirac cohomology of a standard module vanishes if and only if the module is not twisted elliptic tempered the proof makes use of two deep results one is some structural information from the generalized springer correspondence obtained by s kato and lusztig another one is a computation of the dirac cohomology of tempered modules by barbasch ciubotaru trapa and ciubotaru we apply our result to compute the dirac cohomology of ladder representations for type a n for each of such representations with non zero dirac cohomology we associate to a canonical weyl group representation we use the dirac cohomology to conclude that such representations appear with multiplicity one