coxeter elements and kazhdan lusztig cells abstract by the correspondence between coxeter elements of a coxeter system w s and the acyclic orientations of the coxeter graph we study some properties of elements in the set c0 w we show that when w is of finite affine or hyperbolic type any w c0 w satisfies w lr w j with l wj j m w for some j s now assume that w is of finite or affine type we give an explicit description for all the distinguished involutions d of w with d l w for some w e w which verifies a conjecture proposed in jian yi shi adv sci china math 3 1990 79 98 conjecture 8 10 in our case we show that any left cell of w containing some element of c0 w is left connected which verifies a conjecture of lusztig ryoshi hotta ed in problems from the conference on algebraic groups and representations held at katata august 29 september 3 1983 in our case

the distinguished involutions with a value n2 3n 3 in the weyl group of type dn let w s be a weyl group and h its associated hecke algebra let a z u u 1 be the laurent polynomial ring kazhdan and lusztig representation of coxeter groups and hecke algebras invent math 53 1979 165 184 introduced two a bases tw w w and cw w w for the hecke algebra h associated to w let yw y wul w l y ty then yw w w is also an a base for the hecke algebra in this paper we give an explicit expression for certain kazhdan lusztig basis elements cw as a linear combination of yx s in the hecke algebra of type dn in fact this gives also an explicit expression for certain kazhdan lusztig basis elements cw as a linear combination of tx s in the hecke algebra of type dn thus we describe also explicitly the kazhdan lusztig polynomials for certain elements of the weyl group we study also the joint relation among some elements in w and some distinguished involutions with a value n2 3n 3 in the weyl group of type dn

an oka principle for equivariant isomorphisms let g be a reductive complex lie group acting holomorphically on normal stein spaces x and y which are locally g biholomorphic over a common categorical quotient q when is there a global g biholomorphism x y if the actions of g on x and y are what we with justification call generic we prove that the obstruction to solving this local to global problem is topological and provide sufficient conditions for it to vanish our main tool is the equivariant version of grauert s oka principle due to heinzner and kutzschebauch we prove that x and y are g biholomorphic if x is k contractible where k is a maximal compact subgroup of g or if x and y are smooth and there is a g diffeomorphism x y over q which is holomorphic when restricted to each fibre of the quotient map x q we prove a similar theorem when is only a g homeomorphism but with an assumption about its action on g finite functions when g is abelian we obtain stronger theorems our results can be interpreted as instances of the oka principle for sections of the sheaf of g biholomorphisms from x to y over q this sheaf can be badly singular even for a low dimensional representation of sl2 our work is in part motivated by the linearisation problem for actions on n it follows from one of our main results that a holomorphic g action on n which is locally g biholomorphic over a common quotient to a generic linear action is linearisable

the joint relations and the set d1 in certain crystallographic groups abstract for any coxeter group w d kazhdan and g lusztig 1979 invent math 53 165 184 introduced some joint relations among the elements of w by which certain representations of w and its associated hecke algebra h are constructed recently in a letter to lusztig t a springer obtained some interesting results concerning the connection between some non zero structural coefficients of h and the joint relations in w when w is a certain crystallographic group he investigated some properties of a subset d 1 of w springer denoted d 1 by f in his letter which plays an important role for the joint relations in w in the present paper we shall develop some results of springer we shall deduce further joint relations in w from some non zero structural coefficients of h in particular we shall make some detailed discussion for the set d 1 this includes the approach to the proof of a conjecture of lusztig concerning the characterization of the set d 1 the connection between the finiteness of d 1 and some joint relations in w the explicit description for some elements of d 1 and the connectedness of the set d 0 d 2 1 see 3 2 and 5 2 for the notations

on certain distinguished involutions in the weyl group of type dn let w s be a weyl group let a z u u 1 be the laurent polynomial ring in an indeterminate u kazhdan and lusztig invent math 33 1979 165 184 introduced two a bases tw w w and cw w w for the hecke algebra h associated to w let yw y wul w l y ty then yw w w is also an a basis for the hecke algebra in this paper we assume w of type dn and we express certain kazhdan lusztig basis elements cw as a linear combination of yx s this in turn gives an explicit expression for certain kazhdan lusztig basis elements cw as a linear combination of tx s thus we describe explicitly the kazhdan lusztig polynomials for certain pairs of elements of w we also study the joint relation among some elements in w in particular we find certain distinguished involutions in the two sided cell t of w with a value 12 n2 n 4t2 2t for 1 2t n and n even 12 n2 n 4t2 2t for n odd where the two sided cell t does not contain the longest element w0 j in subgroup wj of w for any j s

left cells in the affine weyl group of typec4 we find a representative set of left cells of the affine weyl groupwaof typec4as well as its left cell graphs by applying an algorithm this algorithm was designed in my previous paper tohoku j math 46 1994 105 124 it is reformulated and improved in a more efficient form here these representatives of left cells are presented as the vertices of so called essential graphs so that the generalized invariants of left cells ofwaare actually described explicitly the latter almost characterize the left cells ofwa some comments and conjectures are proposed on cells of affine weyl groups mostly for the case of typecl l 2

relative monodromy weight filtrations in this paper we establish the existence of relative weight filtrations of cones of monodromy logarithms and identities between relative weight filtrations in a context more general than m kashiwara s infinitesimal mixed hodge modules this generalizes a theorem of m kashiwara and adds conceptual clarity to the problem our approach to working with relative weight filtrations w n w is to split the weight filtrations w with canonical splittings introduced by p deligne these splittings agree with the splittings that occur in the mathfrak sl 2 theory of variations of pure hodge structure of w schmid and of e cattani a kaplan and w schmid but are more general as they don t require filtrations w to be monodromy weight filtrations

polarized mixed hodge structures and the local monodromy of a variation of hodge structure a variation of polarized hodge structures on the complement of a divisor with normal crossings gives rise locally on the divisor to a commuting set ni of nilpotent endomorphisms of the vector space underlying the variation the logarithms of the unipotent parts of the various picard lefschetz transformations these reflect properties of the singularities of the period map associated to the variation cf 5 7 9 and sect 3 for example a variation depending on a single parameter defines asymptotically a mixed hodge structure whose weight filtration is the monodromy weight filtration of the corresponding endomorphism n 9 11 this paper is mainly concerned with the properties of the set n arising in the several parameters case in particular theorem 3 3 asserts that all the elements in the open polyhedral cone c spanned over r by the n s determine the same monodromy weight filtration it also describes the relationship between this common filtration and those associated to the faces of the cone this statement was conjectured by deligne based on his analogous result 4 1 9 2 for the q cone associated to those variations which arise from families of polarized non singular algebraic varieties in w 1 we discuss some general properties of commuting sets of nilpotent endomorphisms and their associated filtrations section2 focuses on the case when each element in such set defines in the sense of 2 4 a polarized mixed hodge structure with a fixed hodge filtration f as is the case for the monodromy cone c of a variation these conditions imply cf 2 16 a generic uniqueness for the monodromy weight filtration we also consider those pola

variation of mixed hodge structure i around 1970 griffiths introduced the notion of a variation of hodge structure on a complex manifold s see 17 w 2 it constitutes the axiomatization of the features possessed by the local systems of cohomology associated to a family of compact k ihler manifolds esp smooth projective varieties parametrized by s these are for a variation of hodge structure u of weight m 0 a locally constant sheaf u of q vector spaces or u of real vector spaces 1 hodge filtration horizontality a decreasing filtration o r of gs by holomorphic subbundles such that the evident connection v on takes o p into q 2 hodge decompositions if gs denotes the sheaf of functions on s and if

a homotopy theorem for oka theory we prove a homotopy theorem for sheaves its application shortens and simplifies the proof of many oka principles such as gromov s oka principle for elliptic submersions

homotopy principles for equivariant isomorphisms let g be a reductive complex lie group acting holomorphically on stein manifolds x and y let px x qx and py y qy be the quotient mappings when is there an equivariant biholomorphism of x and y a necessary condition is that the categorical quotients qx and qy are biholomorphic and that the biholomorphism sends the luna strata of qx isomorphically onto the corresponding luna strata of qy fix we demonstrate two homotopy principles in this situation the first result says that if there is a g diffeomorphism x y inducing which is g biholomorphic on the reduced fibres of the quotient mappings then is homotopic through g diffeomorphisms satisfying the same conditions to a g equivariant biholomorphism from x to y the second result roughly says that if we have a g homeomorphism x y which induces a continuous family of gequivariant biholomorphisms of the fibres px 1 q and py 1 q for q qx and if x satisfies an auxiliary property which holds for most x then is homotopic through g homeomorphisms satisfying the same conditions to a g equivariant biholomorphism from x to y our results improve upon those of our earlier paper j reine angew math 706 2015 193 214 and use new ideas and techniques

the second lowest two sided cell in an affine weyl group abstract let w a be an irreducible affine weyl group with w 0 the associated weyl group the present paper is to study the second lowest two sided cell qr of w a let n qr be the number of left cells of w a in qr we conjecture that the equality n qr 1 2 w 0 should always hold when w a is either a n 1 n 2 or of rank 4 this equality can be verified by the existing data see 0 3 then the main result of the paper is to prove the inequality n qr 1 2 w 0 in all cases

asymptotics of degenerations of mixed hodge structures abstract we construct a hermitian metric on the classifying spaces of graded polarized mixed hodge structures and prove analogs of the strong distance estimate 6 between an admissible period map and the approximating nilpotent orbit we also consider the asymptotic behavior of the biextension metric introduced by hain 12 analogs of the norm estimates of 19 and the asymptotics of the naive limit hodge filtration considered in 21

expression of certain kazhdan lusztig basis elements cw over the hecke algebra of type an abstract in this paper we determine expression of kazhdan lusztig basis elements c w over the hecke algebra of type a n where w a j i is an element in weyl group of type a n we also find joined relation of some elements in w

oka s principle for holomorphic sections of elliptic bundles 2 the h principle for locally trivial fibrations 2 1 the h principle over small subsets 2 2 the h principle over totally real submanifolds 2 3 local extension of the h principle 2 4 localizable extensions 2 5 the h principle for cm bundles 2 6 manifolds with totally real souls 2 7 totally real extensions 2 8 nicely localizable extensions 2 9 the basic h principle for z x 2 10 homomorphisms and holomorphic maps of rank r

fully commutative elements in the weyl and affine weyl groups abstract let w be a weyl or an affine weyl group and let w c be the set of fully commutative elements in w we associate each w w c to a digraph g w by using g w we give a graph theoretic description for lusztig s a function on w c and describe explicitly all the distinguished involutions of w c the results verify two conjectures in our case one was proposed by myself in adv sci china math 3 1990 79 98 conjecture 8 10 and the other by lusztig in t asai et al open problems in algebraic groups in r hotta ed problems from the conference on algebraic groups and representations katata august 29 september 3 1983

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

joint relations on elements of the symmetric group abstract we introduce two methods which reduce the checking of the joint relations on element pairs x y i e checking if x y 0 see 1 2 inside a two sided cell of the symmetric group s n to some special cases up to star operations one involves the involutions in s n and the other involves the set for n see 3 1 and 5 1 we give a detailed investigation for the properties of the set in terms of standard tableaux also we study the sets u s and u s and give an affirmative answer to a question of n h xi see proposition 2 9

kazhdan lusztig coefficients for an affine weyl group of type b 2 in this paper we compute the leading coefficients u w of the kazhdan lusztig polynomials pu w for an affine weyl group of type b 2 by using the a function of a coxeter group defined by lusztig we compute most u w explicitly with part of these values u w we show that a conjecture of lusztig on distinguished involutions is true for an affine weyl group of type b 2 we also show that the conjectural formula concerning some semilinear equations needs a modification