We give a branching law for subgroups fixed by an involution. As an application we give a generalization of the Cartan-Helgason theorem and a noncompact analogue of the Borel-Weil theorem. 0. Introduction 0.1. Let G C be a simply-connected complex semisimple Lie group and let g = Lie G C. Let g o be a real form of g and let G be the real semisimple Lie subgroup of G C corresponding to g o. Usin...

After dimensional reduction to three dimensions, the lowest order effective actions for pure gravity, M-theory and the Bosonic string admit an enhanced symmetry group. In this paper we initiate study of how this enhancement is affected by the inclusion of higher derivative terms. In particular we show that the coefficients of the scalar fields associated to the Cartan subalgebra are given by we...

We show that the vertex algebra W1+∞ with central charge −1 is isomorphic to a tensor product of the simple W3 algebra with central charge −2 and a Heisenberg vertex algebra generated by a free bosonic field. We construct a family of irreducible modules of the W3 algebra with central charge −2 in terms of free fields and calculate the full character formulas of these modules with respect to the...

This paper shows that the highest weights of the K-types of any irreducible admissible representation of SU(1, n) are determined by certain restriction maps from u to u ∩ k cohomology. In particular, the image of these maps determines a set of points in a Cartan subalgebra. It is proved that the highest weights of the K-types are given by intersecting a translate of the root lattice with the cl...

Let G be a (real or complex) semisimple Lie group, whose Lie algebra g is endowed with a so called |k|–grading, i.e. a grading of the form g = g−k ⊕ · · · ⊕ gk, such that no simple factor of G is of type A1. Let P be the subgroup corresponding to the subalgebra p = g0 ⊕ · · · ⊕ gk. The aim of this paper is to clarify the geometrical meaning of Cartan connections corresponding to the pair (G,P )...

Given a reductive Lie algebra g and choice of Cartan subalgebra h, the HarishChandra map gives an isomorphism between the center z of the universal enveloping algebra U(g) of g and the algebra of Weyl group invariant symmetric tensors of h, denoted S(h) . We define these objects and give a sketch of the proof, using sl(2,C) as a motivating example. The proofs and examples mirror those found in ...

Let L be a finite-dimensional, semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let H be a fixed Cartan subalgebra of L, and Φ be the root system. Fix a base ∆ = {α1, · · · , αl} of Φ. Let Λ denote the set of dominant, integral linear functions on H. Theorem 0.1. There is a one-to-one correspondence Λ ∼ −→ {isomorphism classes of finite-dimensional irreducible L-...

Multiplicity-free Hamiltonian group actions are the symplectic analogs of multiplicity-free representations, that is, representations in which each irreducible appears at most once. The most well-known examples are toric varieties. The purpose of this paper is to show that under certain assumptions multiplicity-free actions whose moment maps are transversal to a Cartan subalgebra are in one-to-...

Let Am,n be the tensor product of Laurent polynomial algebra in m even variables and exterior n odd over complex field C, Witt superalgebra Wm,n Lie superderivations Am,n. In this paper, we classify simple weight modules with finite-dimensional spaces respect to standard Cartan subalgebra Wm,0. Every such module is either a quotient or highest type.

In a previous work [AS2] we showed how to attach to a pointed Hopf algebra A with coradical kΓ, a braided strictly graded Hopf algebra R in the category Γ Γ YD of Yetter-Drinfeld modules over Γ. In this paper, we consider a further invariant of A, namely the subalgebra R of R generated by the space V of primitive elements. Algebras of this kind are known since the pioneering work of Nichols. It...