Hamilton’s maximum principle for systems states that given a reaction-diffusion equation (semi-linear heat-type equation) for sections of a vector bundle over a manifold, if the solution is initially in a subset invariant under parallel translation and convex in the fibers and if the ODE associated to the PDE preserves the subset, then the solution remains in the subset for positive time. We ge...

Let G be a connected reductive algebraic group defined over Fq, where q is a power of a prime p that is good for G. Let F be the Frobenius morphism associated with the Fq-structure on G and set G = G , the fixed point subgroup of F . Let P be an F -stable parabolic subgroup of G and let U be the unipotent radical of P; set P = P and U = U . Let Guni be the set of unipotent elements in G. In thi...

Let G be a (real or complex) semisimple Lie group, whose Lie algebra g is endowed with a so called |k|–grading, i.e. a grading of the form g = g−k ⊕ · · · ⊕ gk, such that no simple factor of G is of type A1. Let P be the subgroup corresponding to the subalgebra p = g0 ⊕ · · · ⊕ gk. The aim of this paper is to clarify the geometrical meaning of Cartan connections corresponding to the pair (G,P )...

This article is motivated by the theory of distinguished curves in parabolic geometries, as developed in [2]. A parabolic geometry is, by definition, modelled on a homogeneous space of the form G/P where G is a real semisimple Lie group and P is a parabolic subgroup. (There is also a complex theory which corresponds to the choices of complex G’s and P ’s with specific curvature restrictions for...

This note aims to demonstrate that every parabolic geometry has a naturally defined per-Courant algebröıd structure. If the geometry is regular, this structure is a Courant algebröıd if and only if the the curvature κ of the Cartan connection vanishes. This note assumes familiarity with both parabolic geometry, and Courant algebröıds. See [ČS00] and [ČG02] for a good introduction to the first c...

Abstract We study cells with respect to the p -canonical basis of Hecke algebra a crystallographic Coxeter system (see Jensen and Williamson (2017) (2020)) their compatibility standard parabolic subgroups. show that after induction surrounding bigger group cell module right -cell in subgroup decomposes as direct sum modules. Along way, we state some new positivity properties basis.

Let G be the group scheme SL d + 1 over Z and let Q parabolic subgroup corresponding to simple roots ? 2 , ? ? . Then / is -scheme of partial flags { D ? H V } We will calculate cohomology modules line bundles this flag scheme. prove that only non-trivial ones are isomorphic kernel or cokernel certain matrices with multinomial coefficients.

A Schubert variety in the complete flag manifold $$GL_n/B$$ is Levi-spherical if action of a Borel subgroup Levi standard parabolic has an open dense orbit. We give combinatorial classification these varieties. This establishes conjecture latter two authors, and new formulation terms Coxeter elements. Our proof uses contributes to theory key polynomials (type Demazure module characters).

Suppose $p \geq 5$ is a prime number, and let $G = \textrm{SL}_2(\mathbb{Q}_p)$. We calculate the derived functors $\textrm{R}^n\mathcal{R}_B^G(\pi)$, where $B$ Borel subgroup of $G$, $\mathcal{R}_B^G$ right adjoint smooth parabolic induction constructed by Vign\'eras, $\pi$ any smooth, absolutely irreducible, mod $p$ representation $G$.