Global and local Weyl modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in [CP2] were obtained. In this paper we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that, unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.

Let $$\mathfrak{g}$$ _u be a compact Lie algebra and let $$\mathfrak{g}$$ _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of $$\mathfrak{g}$$ _u of a square root of the discriminant of $$\mathfrak{g}$$ . Generalizing a result of W. Lichtenstein in the case $$\mathfrak{g}$$ _u = $$\mathfrak{s}\mathfrak{u}$$ (n, ℂ) or $$\mathfrak{s}\mathfrak{o}$$ (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( $$\mathfrak{g}$$ ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of $$\mathfrak{g}$$ .

Finite complex reflection groups have the remarkable property that the character field k of their reflection representation is a splitting field, that is, every irreducible complex representation can be realized over k. Here we show that this statement remains true for extensions of finite complex reflection groups by elements in their normalizer. Also, we generalize the corresponding result for cyclotomic Hecke algebras to Hecke algebras attached to extended finite complex reflection groups.

For a connected semisimple algebraic group G over an algebraically closed field k and a fixed pair (B, B^–) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB^– is nonempty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution m_C ∈ 2 W associated to C. We prove that the element m_C is the unique maximal length element in its conjugacy class in W, and we classify all such elements in W. For G = SL(n + 1; k), we describe m_C explicitly for every conjugacy class C, and when w ∈ W ≌ S_n+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty.

We define and study a family of partitions of the wonderful compactification $\overline{G}$ of a semisimple algebraic group G of adjoint type. The partitions are obtained from subgroups of G × G associated to triples (A₁, A₂, a), where A₁ and A₂ are subgraphs of the Dynkin graph Γ of G and a : A₁ → A₂ is an isomorphism. The partitions of $\overline{G}$ of Springer and Lusztig correspond, respectively, to the triples (∅, ∅, id) and (Γ, Γ, id).

We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology.

We show that the well-known fact that the equivariant cohomology (with real coefficients) of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with isotropy rank equal to the rank of the acting group. This is true essentially because the action on this set is always equivariantly formal. In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces.

The first part of this paper describes the construction of pseudo-Riemannian homogeneous spaces with special curvature properties such as Einstein spaces, using corresponding known compact Riemannian ones. This construction is based on the notion of a certain duality between compact and non-compact homogeneous spaces. In the second part we apply this method to obtain pseudo-Riemannian homogeneous manifolds with real Killing spinors. We will prove that under a certain additional condition a dual pseudo-Riemannian space (G′/H′, g′) of a compact Riemannian homogeneous space (G/H, g) with homogeneousSpin-structure admits a homogeneousSpin⁺-structure and theG_invariant Killing spinors on (G/H, g) correspond toG′-invariant Killing spinors on (G′/H′, g′). We can ensure that in most cases the hypothesis onG-invariance is satisfied.