Let B be a reductive Lie subalgebra of a semi-simple Lie algebra F of the same rank both over the complex numbers. To each finite dimensional irreducible representation Vlambda of F we assign a multiplet of irreducible representations of B with m elements in each multiplet, where m is the index of the Weyl group of B in the Weyl group of F. We obtain a generalization of the Weyl character formu...

We consider subrings S of rings R such that R = S⊕V with V either two sided invariant under multiplication by S or invariant under the commutator with S. We show that some important properties of R are inherited by S under such conditions. One is the FCR-property which says that every finite dimensional representation is completely reducible. Another application gives a characterization (in cha...

It is shown that the renormalization group method does not necessarily eliminate all secular terms in perturbation series to partial differential equations and a functional subspace of renormalizable secular solutions corresponds to a choice of scales of independent variables in the reductive perturbation method. Recently a novel method based on the perturbative renormalization group theory has...

The aim of these notes is to develop the theory of reductive group schemes, incorporating some simplifications into the methods of [SGA3]. We assume the reader is familiar with the basic structure theory over an algebraically closed field k (as developed in [Bo91], [Hum87], [Spr]), but a review is given in §1 in order to fix our terminology, set everything in the framework of k-schemes (rather ...

Geometric invariant theory (GIT) was developed in the 1960s by Mumford in order to construct quotients of reductive group actions on algebraic varieties and hence to construct and study a number of moduli spaces, including, for example, moduli spaces of bundles over a nonsingular projective curve [26, 28]. Moduli spaces often arise naturally as quotients of varieties by algebraic group actions,...

In this article, a logahoric Higgs torsor is defined as parahoric with logarithmic field. For connected complex reductive group G, we introduce notion of stability for $$\mathcal {G}_{\varvec{\theta }}$$ -Higgs torsors on smooth algebraic curve X, where scheme X. the case when G general linear $$\textrm{GL}_n$$ , show that condition equivalent to parabolic bundle. A correspondence between semis...