Let g be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For I ⊆ S let g I be the corresponding semi-simple subalgebra of g. Denote by W I the Weyl group of g I and let w • and w I • be the longest elements of W and W I , respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjec...

We study a class of perverse sheaves on the variety of pairs (P, gUP ) where P runs through a conjugacy class of parabolics in a connected reductive group and gUP runs through G/UP . This is a generalization of the theory of character sheaves. 2000 Math. Subj. Class. 20G99.

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a 2-dimensional Artin group the Deligne complex is Gromov hyperbolic preci...

This paper deals with a degenerate parabolic equation vt = ∆v + av1 ∥v∥1 α1 subject to homogeneous Dirichlet condition. The local existence of a nonnegative weak solution is given. The blow-up and global existence conditions of nonnegative solutions are obtained. Moreover, we establish the precise blow-up rate estimates for all the blow-up solutions.

Parabolas and paraboloids are used to introduce curvature, both qualitatively and quantitatively. 1. Introduction. Most of the standard diierential geometry textbooks recognize the oscu-lating paraboloid as a useful interpretation of the second fundamental form of a surface. This interpretation leads to some qualitative conclusions about the shape of the surface in consideration 9, pp.87,91]. T...

In [3] Felder and Veselov considered the standard and twisted actions of a finite Coxeter group W on the cohomology H(MW ) of the complement of the complexified hyperplane arrangement MW of W . The twisted action is obtained by combining the standard action with complex conjugation; we refer the reader to [3] for precise statements. In a case by case argument, Felder and Veselov obtain a formul...

In order to explain what we want to do in this paper, let us begin with an explicit example. Let O be the nilpotent orbit of sl(4,C) with Jordan type [3, 1] (under the adjoint action of G := SL(4,C)). We will denote by Xi,j,k the cotangent bundle T (G/Pi,j,k) of the projective manifold G/Pi,j,k where Pi,j,k is a parabolic subgroup of G with flag type (i, j, k). Then the closure Ō of O admits th...

This paper presents a variation of constants formula for the system of functional parabolic partial differential equations ∂u(t, x) ∂t = D∆u+ Lut + f(t, x), t > 0, u ∈ R ∂u(t, x) ∂η = 0, t > 0, x ∈ ∂Ω u(0, x) = φ(x) u(s, x) = φ(s, x), s ∈ [−τ, 0), x ∈ Ω . Here Ω is a bounded domain in Rn, the n × n matrix D is block diagonal with semi-simple eigenvalues having non negative real part, the operat...

We present some new a priori estimates of the solutions to the second-order elliptic and parabolic interface problems. The novelty of these estimates lies in the explicit appearance of the discontinuous coefficients and the jumps of coefficients across the interface.