We consider the parabolic obstacle type problem  Hu = fχΩ in Q − 1 , u = |∇u| = 0 on Q1 \ Ω, where Ω is an unknown open subset of Q1 . This problem has its origin in parabolic potential theory. When f is Hölder continuous we can, under a combination of energetic and geometric assumptions, prove the optimal C x ∩ C t regularity of the solution.

In this paper we describe the relationship between finite free resolutions of perfect ideals in split format (for Dynkin formats) and certain intersections opposite Schubert varieties with big cell for homogeneous spaces $G/P$ where $P$ is a maximal parabolic subgroup.

Let G be a rank 2 complete affine or hyperbolic Kac–Moody group over a finite field k. Then G is locally compact and totally disconnected. Let B− = HU− be the negative minimal parabolic subgroup of G, where H is the analog of a diagonal subgroup and U− is generated by all negative real root groups. Let w1 and w2 be the generators of the Weyl group. Let P − 1 = B − t Bw1B be the negative standar...

The orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential to the study of symmetric k-varieties and their representations. This paper gives an algorithm to compute these orbits and most of the combinatorial structure of the orbit decomposition. There are several ways to describe these orbits, see for example [22, 28, 35]. Fundamental in all these descriptions ar...

where the product is over all reflection hyperplanes of G. Let A : V/G -+ @ be the map induced by 6, thus A is the discriminant of G. A subgroup of G is called parabolic if it is generated by all reflections of G fixing elementwise a given subspace of V. The degrees of G are denoted by dl , d2, . . . , d,,. We call a degree of G primitive if it is bigger than the degrees of all proper parabolic...

The action of a reductive algebraic group G on G/P−, where P− is a parabolic subgroup, differentiates to a representation of the Lie algebra g of G by vector fields on U+, the unipotent radical of a parabolic opposite to P−. We show that both the regular functions on U+ and the polynomial vector fields on U+ form g-modules that are dual to parabolically induced modules, construct an explicit co...

In this paper we arrive at explicit formulae for the innnitesi-mal generators of the action of a complex simple Lie group G on the mani-fold M = G=P where P is a maximal parabolic subgroup. These formulae are obtained by assuming that local coordinates on M are furnished by the nilpotent subalgebra n complementary to the maximal parabolic subalgebra p corresponding to P. For the classical isoge...

Let G be a classical algebraic group, X a maximal rank reductive subgroup and P a parabolic subgroup. This paper classifies when X\G/P is finite. Finiteness is proven using geometric arguments about the action of X on subspaces of the natural module for G. Infiniteness is proven using a dimension criterion which involves root systems. 1 Statement of results Let G be a classical algebraic group ...

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a “matrix problem”. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a ...

We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representat...