integral crystalline cohomology over very ramified valuation rings in this paper we intend to show how fontaine s comparison theory fol fo2 between crystalline and p adic etale cohomology can be extented to very ramified base rings so far it had been developed either for qp coefficients or zp coefficients if the base is unramified over the witt vectors see fa2 for example it states that for any p adic discrete valuation ring v there exists a ring b v with the following property if x denotes a smooth and proper v scheme the p adic etale cohomology of the generic fiber x xv k is related to the crystalline cohomology of x v0 vo defined below by an isomorphism

harmonically induced representations of solvable lie groups soit un groupe de lie g d algebre de lie g et h une polarisation pour l orbite on etudie la representation obtenue quand h est totalement complexe non necessairement positive et g resoluble

non elliptic laplace equations on nilpotent lie groups soit n un groupe de lie connexe nilpotent simplement connexe qui admet une structure complexe invariante a gauche on considere une 2 forme non degeneree j invariante alternee sur 7b n qui definit une structure pseudo kahler sur la variete complexe n on etudie le laplacien complexe definie a partir de la forme non positive agissant sur soit l 2 n l 2 n ou certains fibres de ligne sur ces espaces

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

a characterization of banach spaces containing it is proved that a banach space contains a subspace isomorphic to 11 if and only if it has a bounded sequence with no weak cauchy subsequence the proof yields that a sequence of subsets of a given set has a subsequence that is either convergent or boolean independent a bounded sequence of elements fn in a banach space b is said to be equivalent to the usual 11 basis provided there is a a 0 so that for all n and choices of scalars c cn

on the classification of homogeneous affine tube domains with large automorphism groups in arbitrary dimensions abstract we classify tube domains in c n 1 n 1 with affinely homogeneous base of their boundary and a with positive definite levi form and b with lorentzian type levi form and affine isotropy of dimension at least n 2 n 3 2

the almost product structure of newton strata in the deformation space of a barsotti tate group with crystalline tate tensors in this paper we construct the almost product structure of the minimal newton stratum in deformation spaces of barsotti tate groups with crystalline tate tensors similar to oort s and mantovan s construction for shimura varieties of pel type it allows us to describe the geometry of the newton stratum in terms of the geometry of two simpler objects the central leaf and the isogeny leaf this yields the dimension and the closure relations of the newton strata in the deformation space in particular their nonemptiness shows that a generalisation of grothendieck s conjecture of deformations of barsotti tate groups with given newton polygon holds as an application we determine analogous geometric properties of the newton stratification of shimura varieties of hodge type and prove the equidimensionality of rapoport zink spaces of hodge type

towards a classification of homogeneous tube domains in c 4 we classify the tube domains in c with affinely homogeneous base whose boundary contains a non degenerate affinely homogeneous hypersurface it follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains that do not have bounded realisations these domains lie to either side of a pair of levi indefinite hypersurface using the geometry of these two hypersurfaces we find the automorphism groups of the domains

gauges of baire class one functions abstract let k be a compact metric space and f k r be a bounded baire class one function we proved that for any e 0 there exists an upper semicontinuous positive function of f with finite oscillation index and f x f y e whenever d x y min x y

on functions of finite baire index it is proved that every function of finite baire index on a separable metric space k is a d function i e a difference of bounded semi continuous functions on k in fact it is a strong d function meaning it can be approximated arbitrarily closely in d norm by simple d functions it is shown that if the n th derived set of k is non empty for all finite n there exist d functions on k which are not strong d functions further structural results for the classes of finite index functions and strong d functions are also given

singular unitary representations and indefinite harmonic theory abstract square integrable harmonic spaces are defined and studied in a homogeneous indefinite metric setting in the process dolbeault cohomologies are unitarized and singlar unitary representations are obtained and studied

dieudonne crystals and wach modules for p divisible groups let k be a perfect field of characteristic p 2 and k an extension of f fracw k contained in some f pr using crystalline dieudonne theory we provide a classification of p divisible groups over r ok t1 td in terms of finite height modules over s w k u t1 td when d 0 such a classification is a consequence of a special case of the theory of kisin ren in this setting our construction gives an independent proof of this result and moreover allows us to recover the dieudonne crystal of a p divisible group from the wach module associated to its tate module by berger breuil or by kisin ren

realization of representations of completely solvable lie groups in spaces of l2 harmonic forms abstract let s be a completely solvable ad algebraic connected and simply connected lie group with lie algebra j let j and let p be a totally complex metric polarization for which satisfies the complex pukanszky condition in this work we prove the non triviality and irreducibility of the harmonically induced representation corresponding to the given data for these results to hold it is necessary to replace the invariant metric on the coadjoint orbit by a semi invariant metric it is proven that these results also have applications to the study of homogeneous domains in c n

the l 2 index theorem for homogeneous spaces of lie groups the l2 index theorem for covering spaces of atiyah 3 and singer 20 asserts that given a discrete group f acting smoothly and freely on a manifold m with compact quotient m f m and an elliptic differential operator d on m which is f invariant and thus descends to an elliptic differential operator d on the compact manifold m then the p index indrd dimrkerd dimrkerd of d coincides with the ordinary index ind d dim ker d dim ker d of d here ker d is the space of l2 solutions of du 0 and dim r denotes the dimension function corresponding to the trace on the commutant of f acting on the hilbert space of l2 sections over m naturally associated to p the importance of this theorem lies in the fact that indd 0 implies the existence of nontrivial l2 solutions for the equation du 0 and as such it was used in a crucial way by atiyah and schmid 5 to construct explicit realizations of the discrete series representations for semisimple lie groups indeed if g is a lie group which possesses a discrete torsion free cocompact subgroup f and h is a compact subgroup of g then the l2 index theorem applied to the covering space m g h of m f g h combined with the index formula of atiyah singer 6 yields existence results for l2 solutions of g invariant elliptic equations on the homogeneous space g h it is relevant to note that with d denoting this time a g invariant elliptic differential operator on g h the ratio between the f index of d and the covolume of f gives a number independent of f this real number indg d can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of g on ker d and ker d respectively and hence makes sense for any unimodular lie group g even when it has no discrete cocompact subgroups which outside the semisimple case is the generic

holomorphically induced representations of exponential lie groups abstract we present a slightly modified definition of the concept of holomorphically induced representation we prove that in the exponential solvable case our definition associates a nontrivial irreducible representation to any positive polarization which satisfies the puksanzky condition this is in marked contrast to the results obtained using the usual definition

duflo s conjecture for the branching problem to the iwasawa an subgroup of a semisimple lie group the purpose of this paper is to prove duflo s conjecture for g pi an where g is a real connected simple lie group of hermitian type and pi is a discrete series of g and an is the maximal exponential solvable subgroup for an iwasawa decomposition g kan this is essentially deduced from the following general theorem which we prove in this paper let g be a real connected semisimple lie group with lie algebra mathfrak g then a strongly elliptic g coadjoint orbit mathcal o is holomorphic if and only if text p mathcal o is an open an coadjoint orbit here text p mathfrak g longrightarrow mathfrak a oplus mathfrak n is the natural projection

holomorphically induced representations of some solvable lie groups we study holomorphically induced representations of lie groups g exp g from weak polarizations h at f g when g is a connected and simply connected lie group whose lie algebra is a normal j algebra we obtain a sufficient condition for non vanishing of and the decomposition of into irreducible representations under the assumption that the coadjoint g orbit g f is open and h is a positive weak polarization at f

l adic etale cohomology of shimura varieties of hodge type with non trivial coefficients let mathsf g mathsf x be a shimura datum of hodge type let p be an odd prime such that mathsf g mathbb q p splits after a tamely ramified extension and p not mid pi 1 mathsf g mathrm der under some mild additional assumptions that are satisfied if the associated shimura variety is proper and mathsf g mathbb q p is either unramified or residually split we prove the generalisation of mantovan s formula for the l adic cohomology of the associated shimura variety on the way we derive some new results about the geometry of the newton stratification of the reduction modulo p of the kisin pappas integral model

a motivic conjecture of milne let k be an algebraically closed field of characteristic p 0 let w k be the ring of witt vectors with coefficients ink we prove a motivic conjecture of milne that relates thecohomology with zp coefficients to the crystalline cohomology with integral coefficients in the wider context ofp divisible groups endowed with families of crystalline tensors over a finite discrete valuation ring extension of w k the result extends work of faltings as a main new tool we construct global deformations of p divisible groups endowed with crystalline tensors over certain regular formally smooth schemes over w k