We start by briefly explaining key results on the representation theory of semisimple Lie algebras in large enough positive characteristic. After that, we start a new topic: the complex representation theory of finite groups of Lie type and Hecke algebras. Today we consider the most basic group: G := GLn(Fq). We introduce the Hecke algebra as the endomorphism algebra of the G-module C[B \ G]. W...

Given a free action of compact Lie group G on unital C?-algebra A and spectral triple the corresponding fixed point algebra AG, we present systematic in-depth construction that is build upon geometry AG G. We compare our with selection established examples.

The main purpose of the paper is to study hyperkähler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkähler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [10] [11], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra a...

In the representation theory, the classification and the construction of the irreducible representations are essential themes. In the first half of the twentieth century, I. Schur[11] introduced a prominent method to obtain the finite dimensional irreducible representations of the general linear group GL(n,C), or equivalently of its Lie algebra gl(n,C), which we call Schur-Weyl reciprocity at p...

‎highest weight modules of the double affine lie algebra $widehat{widehat{mathfrak{sl}}}_{2}$ are studied under a‎ ‎new triangular decomposition‎. ‎singular vectors of verma modules are‎ ‎determined using a similar condition with horizontal affine lie‎ ‎subalgebras‎, ‎and highest weight modules are described under the‎ ‎condition $c_1>0$ and $c_2=0$.

Let g be a simple, finite-dimensional Lie (super)algebra, equipped with an embedding of sl2 inducing the minimal gradation on g. The corresponding minimal Walgebra W(g, e−θ) introduced by Kac and Wakimoto has strong generators in weights 1, 2, 3 2 , and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra A(g) where g ⊂ g denote...

Theorem 1. Let G ⊂ GL(n,R) be a real reductive group with Cartan decomposition g = k⊕ p, g being the Lie algebra of G. Let GC be the subgroup of GL(n,C) with Lie algebra g ⊕ ig and K̃ the subgroup of GC whose Lie algebra is k⊕ ip. Let ΩC be a complex homogeneous space for GC and φ a K̃-invariant strictly plurisubharmonic function on ΩC. If Ω is a G-orbit in ΩC and f = φ|Ω has a critical point, th...