We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble the Frobenius maps, and allow us to classify definably simple stable groups in the specific case when they admit such automorphisms.

In this paper we establish Springer correspondence for the symmetric pair $(\text{SL}(N),\text{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$. Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry.

We collect evidence in support of a conjecture of Griffiths, Green, and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi–Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions $1\,\le \,d\,\le \,6$ arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is ${{G}_{2}}$ ) of the family of 6-folds, and the theory of boundary components of Mumford–Tate domains.

Let $k$ be an infinite field. Let $R$ be the semi-local ring of a finite family of closed points on a $k$-smooth affine irreducible variety, let $K$ be the fraction field of $R$, and let $G$ be a reductive simple simply connected $R$-group scheme isotropic over $R$. Our Theorem 1.1 states that for any Noetherian $k$-algebra $A$ the kernel of the map

$$\begin{eqnarray}H_{\acute{\text{e}}\text{t}}^{1}(R\otimes _{k}A,G)\rightarrow H_{\acute{\text{e}}\text{t}}^{1}(K\otimes _{k}A,G)\end{eqnarray}$$

$R$

$K$

$A=k$

$R$

This research was partially supported by the National Security Agency. It is well known from work of Bruhat and Tits that an affine building has a spherical building at infinity. The paper studies the structure at infinity for an affine {\em twin} building. It is shown that the twinning restricts the structure at infinity in a natural way, producing two smaller spherical buildings that are canonically isomorphic to one another. In the process it is shown that to each wall and panel of these buildings at infinity is attached a twin tree.

In this paper we describe a canonical basis for the equivariant $K$-theory (with respect to a ${{\mathbf{C}}^{*}}$-action) of the variety of Borel subalgebras containing a subregular nilpotent element of a simple complex Lie algebra of type $D$ or $E$.

We continue in this paper our study of conjugacy classes of a reductive monoid $M$. The main theorems establish a strong connection with the Bruhat-Renner decomposition of $M$. We use our results to decompose the variety ${{M}_{\text{nil}}}$ of nilpotent elements of $M$ into irreducible components. We also identify a class of nilpotent elements that we call standard and prove that the number of conjugacy classes of standard nilpotent elements is always finite.

Let M be a connected linear algebraic monoid with zero and a reductive unit group. We show that there exist reductive groups G1,..., Gt, each with an automorphism, such that the conjugacy classes of M are in a natural bijective correspondence with the twisted conjugacy classes of Gi, i = 1,..., t.