Let m be a Levi factor of a proper parabolic subalgebra q of a complex semisimple Lie algebra g. Let t = centm. A nonzero element ν ∈ t is called a t-root if the corresponding adjoint weight space gν is not zero. If ν is a t-root, some time ago we proved that gν is adm irreducible. Based on this result we develop in the present paper a theory of t-roots which replicates much of the structure of...

We show how to recover a discrete twist over an ample Hausdorff groupoid from pair consisting of algebra and what we call quasi-Cartan subalgebra. identify precisely which twists arise in this way (namely, those that satisfy the local bisection hypothesis), prove assignment twisted Steinberg algebras such our construction are mutually inverse. algebraic pairs correspond effective groupoids prin...

It is known from a work of Feigin and Frenkel that a Wakimoto type, generalized free field realization of the current algebra Ĝk can be associated with each parabolic subalgebra P = (G0+G+) of the Lie algebra G, where in the standard case G0 is the Cartan and P is the Borel subalgebra. In this letter we obtain an explicit formula for the Wakimoto realization in the general case. Using Hamiltoni...

Given a complex semisimple Lie algebra g = k + ik (k is a compact real form of g), let π : g → h be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra h := t + it, where t is a maximal abelian subalgebra of k. Given x ∈ g, we consider π(Ad(K)x), where K is the analytic subgroup G corresponding to k, and show that it is star-shaped. The result extends a resul...

Let G be a (real or complex) semisimple Lie group, whose Lie algebra g is endowed with a so called jkj{grading, i.e. a grading of the form g = g ?k g k , such that no simple factor of G is of type A 1. Let P be the subgroup corresponding to the subalgebra p = g 0 g k. The aim of this paper is to clarify the geometrical meaning of Cartan connections corresponding to the pair (G; P) and to study ...

Let G be a (real or complex) semisimple Lie group, whose Lie algebra g is endowed with a so called jkj{grading, i.e. a grading of the form g = g ?k g k , such that no simple factor of G is of type A 1. Let P be the subgroup corresponding to the subalgebra p = g 0 g k. The aim of this paper is to clarify the geometrical meaning of Cartan connections corresponding to the pair (G; P) and to study ...

Let K r be the group algebra of the symmetric group on r symbols over a eld K of prime characteristic p. This work is motivated by the problem of comparing modules in blocks of K r of a xed core and for varying weight w, where r 2 N is arbitrary. We approach this via polynomial representations of the general linear groups GL n (K) or equivalently via Schur algebras. The main part of the thesis ...

We generalize all known results on rigidity of uniform Roe algebras to the setting arbitrary uniformly locally finite coarse spaces. For instance, we show that isomorphism between spaces whose contain only compact ghost projections implies base are coarsely equivalent. Moreover, if one has property A, then bijectively also provide a characterization for existence an embedding onto hereditary su...

Associated to every subset J of nodes of a Dynkin diagram is a standard parabolic subalgebra pJ of the corresponding Lie algebra g, generated by the positive Borel subalgebra of g together with the negative simple generators fj, j ∈ J . Furthermore one has a decomposition of g as a semi-direct product of pJ and the subalgebra n − J generated by the remaining negative simple generators. In the s...

We study the representation theory of Kazama-Suzuki coset vertex operator superalgebra associated with pair a complex simple Lie algebra and its Cartan subalgebra. In case type $A_{1}$, B.L. Feigin, A.M. Semikhatov, I.Yu. Tipunin introduced another construction, which is "inverse" construction. this paper we generalize latter construction to arbitrary establish categorical equivalence between c...