Tits polygons

We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition. We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We apply this classification to give a “rank 2 2 ” presentation for the group of F F -rational points of an arbitrary exceptional simple group of F F -rank at least 4 4 and to determine defining relations for the group of F F -rational points of an an arbitrary group of F F -rank 1 1 and absolute type D 4 D_4 , E 6 E_6 , E 7 E_7 or E 8 E_8 associated to the unique vertex of the Dynkin diagram that is not orthogonal to the highest root. All of these results are over a field of arbitrary characteristic.

COMPACT ORBITS OF PARABOLIC SUBGROUPS

We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of a compact connected Lie group U with Lie algebra $\mathfrak {u}$ extends holomorphically to an action of the complexified group $U^{\mathbb {C}}$ and that the U-action on Z is Hamiltonian. If $G\subset U^{\mathbb {C}}$ is compatible, there exists a gradient map $\mu _{\mathfrak p}:X \longrightarrow \mathfrak p$ where $\mathfrak g=\mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of $\mathfrak g$ . In this paper, we describe compact orbits of parabolic subgroups of G in terms of the gradient map $\mu _{\mathfrak p}$ .

Overgroups of regular unipotent elements in simple algebraic groups

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.

MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

Let G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

Closures of locally divergent orbits of maximal tori and values of homogeneous forms

Let ${\mathbf {G}}$ be a semisimple algebraic group over a number field K, $\mathcal {S}$ a finite set of places of K, $K_{\mathcal {S}}$ the direct product of the completions $K_{v}, v \in \mathcal {S}$ , and ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map. We describe the closures of the locally divergent orbits ${T\pi (g)}$ where T is a maximal $K_{\mathcal {S}}$ -split torus in G. If $\# S = 2$ then the closure $\overline {T\pi (g)}$ is a finite union of T-orbits stratified in terms of parabolic subgroups of ${\mathbf {G}} \times {\mathbf {G}}$ and, consequently, $\overline {T\pi (g)}$ is homogeneous (i.e. $\overline {T\pi (g)}= H\pi (g)$ for a subgroup H of G) if and only if ${T\pi (g)}$ is closed. On the other hand, if $\# \mathcal {S}> 2$ and K is not a $\mathrm {CM}$ -field then $\overline {T\pi (g)}$ is homogeneous for ${\mathbf {G}} = \mathbf {SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ${\mathbf {G}} \neq \mathbf {SL}_{n}$ . As an application, we prove that $\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$ for the class of non-rational locally K-decomposable homogeneous forms $f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$ .