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The irreducibility of a subspace U ?= V under the κα implies that, for generic values of the parameter, the braid group Bg acts irreducibly on U.
      
We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin's braid group Bn on the zero-weight spaces of all simple Usln-modules for n≥4.
      
About Knop's Action of the Weyl Group on the Set of Orbits of a Spherical Subgroup in the Flag Manifold
      
Let G be a connected reductive algebraic group defined on an algebraically
      
Modules over the small quantum group and semi-infinite flag manifold
      
Let G be a reductive connected algebraic group over an algebraically
      
We show that the structure of a block outside the critical hyperplanes of category O over a symmetrizable Kac-Moody algebra depends only on the corresponding integral Weyl group and its action on the parameters of the Verma modules.
      
Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0.
      
Let q be a power of p and let G(q) be the finite group of Fq-rational points of G.
      
If the Lie group SU(n + 1) acts nontrivially and in a locally smooth
      
Let k be a perfect field and G an algebraic group defined over k.
      
The group of direct isometries of the hyperbolic space ${\Bbb H}^n \mbox{is} G=\mbox{SO}_0(n,1).$ This isometric action admits many differentiable compactifications into an action on the closed n-dimensional ball.
      
In the algebraic setting we prove that if a complex linear group G acts on a Kahler manifold in a symplectically asystatic fashion, then the G-orbits are spherical.
      
Let ${\bf B}_n({\mathbb C})$ be the group of invertible upper-triangular matrices acting on ${\mathfrak n}_n$ by conjugation.
      
It is proved that for any prime $p\geqslant 5$ the group $G_2(p)$ is a quotient of $(2,3,7;2p) = \langle X,Y: X^2=Y^3=(XY)^7 =[X,Y]^{2p}=1 \rangle.$
      
We work on an arbitrary complete algebraic curve, the structure group is an arbitrary semisimple group.
      
The automorphism group ${\rm Aut}\,(X)$ is determined in the last section.
      
Let X = G/K be a connected Riemannian homogeneous space of a real Lie group G.
      
By a case-free approach we give a precise description of the closure of a Steinberg fiber within a twisted wonderful compactification of a simple linear algebraic group.
      
For a split reductive algebraic group, this paper observes a homological interpretation for Weyl module multiplicities in Jantzen's sum formula.
      
 

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