Related papers: Semi-simple groups that are quasi-split over a tam…

Let G be a reductive group over a non-archimedean local field k. We provide necessary conditions and sufficient conditions for all tori of G to split over a tamely ramified extension of k. We then show the existence of good semisimple…

Let $K$ be a $p$-adically closed field and $G$ a group interpretable in $K$. We show that if $G$ is definably semisimple (i.e. $G$ has no definable infinite normal abelian subgroups) then there exists a finite normal subgroup $H$ such that…

Let $F$ be a nonarchimedean local field, let $E$ be a Galois quadratic extension of $F$ and let $G$ be a quasisplit group defined over $F$; a conjecture by Dipendra Prasad states that the Steinberg representation of $G(E)$ is then…

A semisimple element $s$ of a connected reductive group $G$ is said {\it quasi-isolated} (respectively {\it isolated}) if $C_G(s)$ (respectively $C_G^0(s)$) is not contained in a Levi subgroup of a proper parabolic subgroup of $G$. We study…

A semisimple algebraic tensor category over an algebraically closed field k of characteristic zero is the representation category of all finite dimensional twisted super representations of an affine reductive supergroup G over k. Such a…

We study infinite groups interpretable in power bounded $T$-convex, $V$-minimal or $p$-adically closed fields. We show that if $G$ is an interpretable definably semisimple group (i.e., has no definable infinite normal abelian subgroups)…

Let $L/K$ be a finite Galois extension of fields with Galois group $G$. It is known that $L/K$ admits exactly two Hopf-Galois structures when $G$ is non-abelian simple. In this paper, we extend this result to the case when $G$ is…

Let $R$ be a discrete valuation ring of field of fractions $K$ and of residue field $k$ of characteristic $p > 0$. In an earlier work, we studied the question of extending torsors on $K$-curves into torsors over $R$-regular models of the…

We introduce the notion of a quasi-connected reductive group over an arbitrary field to be an almost direct product of a connected semisimple group and a quasi-torus (a smooth group of multiplicative type). We show that a linear algebraic…

Let $K$ be a complete discrete valuation field. Let $\mathcal{O}_K$ be its ring of integers. Let $k$ be its residue field which we assume to be algebraically closed of characteristic exponent $p\geq1$. Let $G/K$ be a semi-abelian variety.…

Let $C$ be a smooth, projective and geometrically connected curve defined over a finite field $\mathbb{F}_q(C)$. Given a semisimple $C-S$-group scheme $\underline{G}$ where $S$ is a finite set of closed points of $C$, we describe the set of…

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is…

Let $K$ be a complete discretely valued field. An extension $L/K$ is "weakly totally ramified" if the residue extension is purely inseparable. We sharpen a result of Ax by showing that any Galois-invariant disk in the algebraic closure of…

Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the connected part of the…

We show that if G is an anisotropic, semisimple, absolutely almost simple, simply connected group over a field k, then two elements of G over any field extension of k are R-equivalent if and only if they are A^1-equivalent. As a…

The main result of this paper (Theorem B) asserts that under natural conditions, any weakly-split Tits system in G(k), G a reductive or quasi-reductive group over an arbitrary field k, is the standard one.

Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field…

A topological group is (openly) almost-elliptic if it contains a(n open) dense subset of elements generating relatively-compact cyclic subgroups. We classify the (openly) almost-elliptic connected locally compact groups as precisely those…

We study quasi-semisimple elements of disconnected reductive algebraic groups over an algebraically closed field. We describe their centralizers, define isolated and quasi-isolated quasi-semisimple elements and classify their conjugacy…

Let $K$ be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let $X$ be a connected, proper scheme over $\mathcal{O}_K$, and let $U$ be the complement in $X$ of a divisor with simple normal…