相关论文: Normalizers of maximal tori and real forms of Lie …

The normalizer $N_G(H_G)$ of a maximal torus $H_G$ in a semisimple complex Lie group $G$ does not in general allow a presentation as a semidirect product of $H_G$ and the corresponding Weyl group $W_G$. Meanwhile, splitting holds for…

Let $G$ be a finite group of Lie type $E_6$ over $F_q$ (adjoint or simply connected) and $W$ be the Weyl group of $G$. We describe maximal tori $T$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. It is well known that…

Let $G$ be a finite group of Lie type $E_l$ with $l\in\{6,7,8\}$ over $F_q$ and $W$ be the Weyl group of $G$. We describe all maximal tori $T$ of $G$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. Let $T$ correspond to…

Let $G$ be a finite group of Lie type and $T$ a maximal torus of $G$. In this paper we complete the study of the question of the existence of a complement for the torus $T$ in its algebraic normalizer $N(G,T)$. It is proved that every…

We determine the groups which can appear as the normalizer of a maximal torus in a connected 2-compact group. The technique depends on using ideas of Tits to give a novel description of the normalizer of the torus in a connected compact Lie…

Let N be the normalizer of a maximal torus T in a split reductive group over F_q, and let w be an involution in the Weyl group N/T. We construct a section of W satisfying the braid relations, such that the image of the lift n of w under the…

Let N be the normalizer of a maximal torus T in a split reductive group over F_q and let w be an involution in the Weyl group N/T. We construct explicitly a lifting n of w in N such that the image of n under the Frobenius map is equal to…

Let $G$ be a finite simple group of Lie type and let $T$ be a maximal torus of $G$. It is well known that if the defining field of $G$ is large enough, then the normaliser of $T$ in $G$ is equal to the algebraic normaliser $N(G,T)$. We…

In this note, we present some results that partially answer the following question. Let $G$ be a simple real Lie group; what is the set of representations $V$ of $G$ in which the longest element $w_0$ of the restricted Weyl group $W$ acts…

Given a simple connected compact Lie group $K$ and a maximal torus $T$ of $K$, the Weyl group $W=N_K(T)/T$ naturally acts on $T$. First, we use the combinatorics of the (extended) affine Weyl group to provide an explicit $W$-equivariant…

Let $G$ be a connected reductive group over a non-archimedean local field $F$ and $I$ be an Iwahori subgroup of $G(F)$. Let $I_n$ is the $n$-th Moy-Prasad filtration subgroup of $I$. The purpose of this paper is two-fold: to give some nice…

This note constitutes a brief survey of our recent work on the problem of determining, for a given real Lie group~$G$, the set of representations~$V$ in which the longest element~$w_0$ of the restricted Weyl group~$W$ acts nontrivially on…

For a cyclic group $A$ and a connected Lie group $G$ with an $A$-module structure (with the additional conditions that $G$ is compact and the $A$-module structure on $G$ is 1-semisimple if $A\cong\ZZ$), we define the twisted Weyl group…

Let $G$ be a finite group of Lie type $F_4$ with the Weyl group $W$. For every maximal torus $T$ of $G$, we find the minimal order of a supplement to $T$ in its algebraic normalizer $N(G,T)$. In particular, we obtain all maximal tori having…

We prove that the set of elements of a given finite order in the connected component $N_w$ of the normalizer $N_G(T)$ of a maximal torus $T$ of a semisimple group $G$ is either empty or a disjoint union of finitely many irreducible…

Associated to any compact quantum group $G\subset U_N^+$ is a canonical family of group dual subgroups $\widehat{\Gamma}_Q\subset G$, parametrized by unitaries $Q\in U_N$, playing the role of "maximal tori" for $G$. We present here a series…

Let $\mathbf{S}_k$ denote a maximal torus in the complex Lie group $\mathbf{G} = \mathrm{SL}_n(\mathbb{C})/C_k$ and let $T_k$ denote a maximal torus in its compact real form $\mathrm{SU}_n(\mathbb{C})/C_k$, where $k$ divides $n$. Let $W$…

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…

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…

Let $G$ be a noncompact semisimple algebraic group with trivial center, $S < G$ a maximal split torus, $H < G$ the centralizer of $S$ in $G$ and $\Gamma < G$ an irreducible lattice. Consider the group measure space von Neumann algebra…