We generalize the construction of dual BGG complexes in Faltings–Chai and Mokrane–Tilouine–Polo (from the case of Siegel modular varieties) to all smooth integral models of PEL-type Shimura varieties.

Construction of affine hypersurfaces with prescribed inte rsection cohomology) Let �(q) = a1q + ··· + adqd be a polynomial of degree d, with non-negative integral coefficients and without constant term. Leta = �(1) and N = 2d + a. We exhibit a quasi-homogeneous hypersurface V� � CN+1 such that the m-th intersection

Let G be a chordal graph, X(G) the complement of the associated complex arrangement and Gamma(G) the fundamental group of X(G). We show that Gamma(G) is a limit of colored braid groups over the poset of simplices of G. When G = G_T is the comparability graph associated with a rooted tree T, a case recently investigated by the first author, the result takes the following very simple form: Gamma(G_T) is a limit over T of colored braid groups.

For a semisimple adjoint algebraic group $G$ and a Borel subgroup $B$, consider the double classes $BwB$ in $G$ and their closures in the canonical compactification of $G$: we call these closures large Schubert varieties. We show that these varieties are normal and Cohen-Macaulay; we describe their Picard group and the spaces of sections of their line bundles. As an application, we construct geometrically van der Kallen's filtration of the algebra of regular functions on $B$. We also construct a degeneration of the flag variety $G/B$ embedded diagonally in $G/B\times G/B$, into a union of Schubert varieties. This leads to formulae for the class of the diagonal in $T$-equivariant $K$-theory of $G/B\times G/B$, where $T$ is a maximal torus of $B$.

À tout polynôme P à coefficients entiers positifs et terme constant égal à 1, de degré d, nous associons une paire d’éléments (y,w) dans le groupe symétrique Sn, où n = d + 1 + P(1), pour laquelle nous montrons que le polynôme de Kazhdan-Lusztig Py,u. est égal à P. Cette paire vérifie ℓ{w) - ℓ(y) = 2d + P(l) - 1, où ℓ(w) désigne le nombre d’inversions de w.

To each polynomial P P with integral nonnegative coefficients and constant term equal to 1 1 , of degree d d , we associate a certain pair of elements ( y , w ) (y,w) in the symmetric group S n S_n , where n = 1 + d + P ( 1 ) n = 1 + d + P(1) , such that the Kazhdan-Lusztig polynomial P y , w P_{y,w} equals P P . This pair satisfies ℓ ( w ) − ℓ ( y ) = 2 d + P ( 1 ) − 1 \ell (w) - \ell (y) = 2d + P(1) - 1 , where ℓ ( w ) \ell (w) denotes the number of inversions of w w .

Let G be a connected semisimple algebraic group, B a Borel subgroup, T a maximal torus in B with Weyl group W, and Q a subgroup containing B. For \(w \in W\), let \(X_{wQ}\) denote the Schubert variety \(\overline{BwQ}/Q\). For \(y\in W\) such that \(X_{yQ}\subseteq X_{wQ}\), one knows that ByQ / Q admits a T-stable transversal in \(X_{wQ}\), which we denote by \({\cal N}_{yQ,wQ}\). We prove that, under certain hypotheses, \({\cal N}_{yQ, wQ}\) is isomorphic to the orbit closure of a highest weight vector in a certain Weyl module. We also obtain a generalisation of this result under slightly weaker hypotheses. Further, we prove that our hypotheses are satisfied when Q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight, and \(X_{yQ}\) is an irreducible component of the boundary of \(X_{wQ}\) (that is, the complement of the open orbit of the stabiliser in G of \(X_{wQ}\)). As a consequence, we describe the singularity of \(X_{wQ}\) along ByQ / Q and obtain that the boundary of \(X_{wQ}\) equals its singular locus.

Let g be a semi-simple complex Lie algebra, U = U(g) its enveloping algebra, and A a minimal primitive factor of U, with central character chi. Under the assumption that chi is regular and integral, we prove that the Dynkin diagram of g is a Morita invariant of A. Further, a slight refinement implies that the flag variety of g is determined, within all generalized flag varieties, by its ring of differential operators.Then we derive the following consequences. First, if U(g) congruent to U(g'), for some Lie algebra g', then g' congruent to g. Second, any automorphism of U acts on the centre, and on some dense open subset of the primitive spectrum, as a diagram automorphism. We conjecture that this result holds true on the whole primitive spectrum, and give K-theoretic versions of the result and the conjecture. We also improve a key result of [1]. Finally, when chi is only assumed to be regular, we prove, using a result of Soergel, that the Weyl group of g is a Morita invariant of A. (C) Elsevier, Paris.

Let ? be a semisimple Lie algebra over k, an algebraically closed field of characteristic zero, and let ?⊂? be a Cartan subalgebra inside a Borel subalgebra of ?. Let ? be the enveloping algebra of ?. For μ∈? * let M(μ) denote the corresponding Verma module and let ? μ = ?/ Ann  M(μ). Let W be the Weyl group and let W 0 μ be the stabiliser of μ in W. We prove the following theorem, which affirms a conjecture of T.J. Hodges.

Let g denote a semisimple Lie algebra over an algebraically closed field k of characteristic zero and G, a finite group of k-automorphisms of the enveloping algebra U of g. In this paper, it is proved that, if the subalgebra UG is k-isomorphic to an enveloping algebra, then G is trivial. A similar result for Weyl algebras over k is also obtained.