Using principal series Harish-Chandra modules, local cohomology with support in Schubert cells and twisting functors we construct certain modules parametrized by the Weyl group and a highest weight in the subcategory O of the category of representations of a complex semisimple Lie algebra. These are in a sense modules between a Verma module and its dual. We prove that the three different approa...

For a connected, semisimple, simply connected algebraic group G defined and isotropic over a field k, the corresponding Tits building is used to study central extensions of the abstract group G(k). When k is a nonArchimedean local field and A is a finite, abelian p-group where p is the characteristic of the residue field of k, then with G of k-rank at least 2, we show that the group H2(G(k), A)...

In this paper, we show that every surjective $n$-homomorphism ($n$-anti-homomorphism) from a Banach algebra $A$ into a semisimple Banach algebra $B$ is continuous.

Since the invention of canonical bases of quantized enveloping algebras, one of the main problems has been to establish what they look like. Explicit formulas are only known in a few cases corresponding to root systems of low rank, namely A1 (trivial), A2 ([Lusztig 90]), A3 ([Xi 99a]), and B2 ([Xi 99b]). Furthermore, there is evidence suggesting that for higher ranks the formulas become so comp...

We make a first step towards a classification of simple generalized HarishChandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras (g, k), we construct, via cohomological induction, the fundamental series F ·(p, E) of generalized Harish-Chandra modules. We then use F ·(p, E) to characterize any simpl...

The goal of Section 2 is to provide a proof of Theorem 2.0.1. Section 3 introduces the necessary facts about Lie algebras and representation theory, with the goal being the proof of Proposition 3.5.7 (ultimately as an application of Theorem 2.0.1), and Proposition 3.3.1. In Section 4 we prove the main theorem, using Propositions 3.3.1 and 3.5.7. In Section 5, we apply the theorem to the special...

For a connected semisimple algebraic group G, we consider some special infinite series of tensor products of simple G-modules whose G-fixed point spaces are at most one-dimensional. We prove that their existence is closely related to the existence of open G-orbits in multiple flag varieties and address the problem of classifying such series. Let G be a connected simply connected semisimple alge...

For $\mathbb{G}$ an algebraic (or more generally, a bornological) quantum group and $\mathbb{B}$ closed subgroup of $\mathbb{G}$, we build in this paper induction module by explicitly defining, on the convolution algebra inner product which takes its value $\mathbb{B}$, as original approach Rieffel. In context, study link with functor defined Vaes. last part, illustrate our result parabolic com...