Related papers: A short proof of Timashev's theorem on the real co…

For a connected linear algebraic group $G$ defined over $\mathbb{R}$, we compute the component group $\pi_0G(\mathbb{R})$ of the real Lie group $G(\mathbb{R})$ in terms of a maximal split torus $T_{\text{s}}\subseteq G$. In particular, we…

We show that a connected split reductive group G over a field of characteristic 0 is uniquely determined up to isomorphism by specifying a maximal torus T of G, the set of isomorphism classes of irreducible representations of G, and the…

The efficacy of using complexifications to understand the structure of real algebraic groups is demonstrated. In particular the following results are proved: a) If L is an algebraic subgroup of a connected real algebraic group G such that…

We compute explicitly the group of connected components $\pi_0G(\mathbb{R})$ of the real Lie group $G(\mathbb{R})$ for an arbitrary (not necessarily linear) connected algebraic group $G$ defined over the field $\mathbb{R}$ of real numbers.…

We extend Raimi's classical partition theorem to the continuous setting of the circle and $n$-dimensional torus. Building on recent work of Hegyv\'ari, Pach, and Pham in finite groups, we prove that there exist measurable partitions of the…

We give a self-contained introduction to linear algebraic and semialgebraic groups over real closed fields, and we generalize several key results about semisimple Lie groups to algebraic and semialgebraic groups over real closed fields. We…

Let $p$ be a prime integer, $k$ be a $p$-closed field of characteristic $\neq p$, $T$ be a torus defined over $k$, $F$ be a finite $p$-group, and $1\to T \to G \to F \to 1$ be an exact sequence of algebraic groups. Extending earlier work of…

Following up on our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential p-dimension of linear algebraic groups G whose connected component G^0 is a torus.

Let G be a connected reductive group over the field of real numbers R. Using results of our previous joint paper, we compute combinatorially the first Galois cohomology set H^1(R,G) in terms of reductive Kac labelings. Moreover, we compute…

We describe functorially the first Galois cohomology set $H^1({\mathbb R},G)$ of a connected reductive algebraic group $G$ over the field $\mathbb R$ of real numbers in terms of a certain action of the Weyl group on the real points of order…

We give a presentation of various results on zero-groups in o-minimal structures together with some new observations. In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real…

For a real reductive group $G$, we investigate the structure of the Casselman algebra $\mathcal{S}(G)$ and its similarities to the structure of the reduced group $C^*$-algebra $C_r^*(G)$. We demonstrate that the two algebras are assembled…

We show that for a reductive group $G$ over a field $k$ the $\mathbb{A}^1$-Euler characteristic of the variety of maximal tori in $G$ is an invertible element of the Grothendieck-Witt ring $\mathrm{GW}(k)$, settling the weak form of a…

This is a companion paper to our previous work, where we proved the finiteness of the Tate-Shafarevich group for an arbitrary torus $T$ over a finitely generated field $K$ with respect to any divisorial set $V$ of places of $K$. Here, we…

Let $G$ be a real or $p$-adic reductive group. We consider the tempered dual of $G$, and its connected components. For real groups, Wassermann proved in 1987, by noncommutative-geometric methods, that each connected component has a simple…

Let $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\mathcal{F}$ associates to $\mathbf{G}$ a linear algebraic group…

Let $G$ be a connected compact Lie group, and let $M$ be a connected Hamiltonian $G$-manifold with equivariant moment map $\phi$. We prove that if there is a simply connected orbit $G\cdot x$, then $\pi_1(M)\cong\pi_1(M/G)$; if additionally…

We derive the canonical forms of super Riemannian metrics and the local isometry groups of such metrics. For certain super metrics we also compute the simply connected covering groups of the local isometry groups and interpret these as…

To a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad construct a cohomological invariant called the splitting invariant, which is an important component of…

Let $r$ and $q$ be positive integers and $n=qr+1.$ Let $G = SL(n, \mathbb{C})$ and $T$ be a maximal torus of $G.$ Let $P^{\alpha_r}$ be the maximal parabolic subgroup of $G$ corresponding to the simple root $\alpha_r.$ Let $\omega_r$ be the…