Geometric quantization of real minimal nilpotent orbits

According to a theorem of Brieskorn and Slodowy, the intersection of the nilpotent cone of a simple Lie algebra with a transverse slice to the subregular nilpotent orbit is a simple surface singularity. At the opposite extremity of the poset of nilpotent orbits, the closure of the minimal nilpotent orbit is also an isolated symplectic singularity, called a minimal singularity. For classical Lie algebras, Kraft and Procesi showed that these two types of singularities suffice to describe all generic singularities of nilpotent orbit closures: specifically, any such singularity is either a simple surface singularity, a minimal singularity, or a union of two simple surface singularities of type A2k−1. In the present paper, we complete the picture by determining the generic singularities of all nilpotent orbit closures in exceptional Lie algebras (up to normalization in a few cases). We summarize the results in some graphs at the end of the paper.In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities that do not occur in the classical types. Three of these are unibranch non-normal singularities: an SL2(C)-variety whose normalization is A2, an Sp4(C)-variety whose normalization is A4, and a two-dimensional variety whose normalization is the simple surface singularity A3. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the singularity of the nilpotent cone at a point in the subregular orbit.

Construction process for simple Lie algebras

We continue investigations that are concerned with the complexity of nilpotent orbits in semisimple Lie algebras. We give a characterization of the spherical nilpotent orbits in terms of minimal Levi subalgebras intersecting them. This provides a kind of canonical form for such orbits. A description minimal non-spherical orbits in all simple Lie algebras is obtained. The theory developed for the adjoint representation is then extended to Vinberg’s θ -groups. This yields a description of spherical nilpotent orbits for the isotropy representation of a symmetric variety.

This paper studies how many orthogonal bi-invariant complex structures exist on a metric Lie algebra over the real numbers. Recently, it was shown that irreducible Lie algebras which are additionally 2-step nilpotent admit at most one orthogonal bi-invariant complex structure up to sign. The main result generalizes this statement to metric Lie algebras with any number of irreducible factors and which are not necessarily 2-step nilpotent. It states that there are either 0 or $$2^k$$ such complex structures, with k the number of irreducible factors of the metric Lie algebra. The motivation for this problem comes from differential geometry, for instance to construct non-parallel Killing-Yano 2-forms on nilmanifolds or to describe the compact Chern-flat quasi-Kahler manifolds. The main tool we develop is the unique orthogonal decomposition into irreducible factors for metric Lie algebras with no non-trivial abelian factor. This is a generalization of a recent result which only deals with nilpotent Lie algebras over the real numbers. Not only do we apply this fact to describe the orthogonal bi-invariant complex structures on a given metric Lie algebra, but it also gives us a method to study different inner products on a given Lie algebra, computing the number of irreducible factors and orthogonal bi-invariant complex structures for varying inner products.

We consider a complex simple Lie algebra $${\mathfrak{g}}$$ , with the action of its adjoint group. Among the three canonical nilpotent orbits under this action, the minimal orbit is the non zero orbit of smallest dimension. We are interested in equivariant deformation quantization: we construct $${\mathfrak{g}}$$ -invariant star-products on the minimal orbit and on its closure, a singular algebraic variety. We shall make use of Hochschild homology and cohomology, of some results about the invariants of the classical groups, and of some interesting representations of simple Lie algebras. To the minimal orbit is associated a unique, completely prime two-sided ideal of the universal enveloping algebra $${{\rm U}(\mathfrak{g})}$$ . This ideal is primitive and is called the Joseph ideal. We give explicit expressions for the generators of the Joseph ideal and compute the infinitesimal characters.

There is a natural duality between orbits γ of a real form G of a complex semisimple group Gℂ on a homogeneous rational manifold Z=Gℂ/P and those κ of the complexification Kℂ of any of its maximal compact subgroups K: (γ,κ) is a dual pair if γ∩κ is a K-orbit. The cycle space C(γ) is defined to be the connected component containing the identity of the interior of {g:g(κ)∩γ is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C(γ) is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C(γ) is a certain explicitly computable universal domain.

Given a Lie algebra, there uniquely exists a Poisson algebra which is called a Lie-Poisson algebra over the Lie algebra. We will prove that given a Loday/Leibniz algebra there exists uniquely a noncommutative Poisson algebra over the Loday algebra. The noncommutative Poisson algebras are called the Loday-Poisson algebras. In the super/graded cases, the Loday-Poisson bracket is regarded as a noncommutative version of classical (linear) Schouten-Nijenhuis bracket. It will be shown that the Loday-Poisson algebras form a special subclass of Aguiar's dual-prePoisson algebras. We also study a problem of deformation quantization over the Loday-Poisson algebra. It will be shown that the polynomial Loday-Poisson algebra is deformation quantizable and that the associated quantum algebra is Loday's associative dialgebra.

We define new quantizations of the Heisenberg group by introducing new quantizations in the universal enveloping algebra of its Lie algebra. Matrix coefficients of the Stone--von Neumann representation are preserved by these new multiplications on the algebra of functions on the Heisenberg group. Some of the new quantizations provide also a new multiplication in the algebra of theta functions; we obtain in this way Sklyanin algebras.

Let (g, [•, •]) be a Lie algebra with an integrable complex structure J. The ±i eigenspaces of J are complex subalgebras of gC isomorphic to the algebra (g, [*]J) with bracket [X * Y]J = 2 ([X, Y] - [JX, JY]). We consider here the case where these subalgebras are nilpotent and prove that the original (g, [•, •]) Lie algebra must be solvable. We consider also the 6-dimensional case and determine explicitly the possible nilpotent Lie algebras (g, [*]J). Finally we produce several examples illustrating different situations, in particular we show that for each given s there exists g with complex structure J such that (g, [*]J) is s-step nilpotent. Similar examples of hypercomplex structures are also built.

Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are equivariant with respect to the action of a finite group. In the first part of this paper, we define global Weyl modules for equivariant map algebras satisfying a mild assumption. We then identify a commutative algebra A that acts naturally on the global Weyl modules, which leads to a Weyl functor from the category of A-modules to the category of modules for the equivariant map algebra in question. These definitions extend the ones previously given for generalized current algebras (i.e. untwisted map algebras) and twisted loop algebras. In the second part of the paper, we restrict our attention to equivariant map algebras where the group involved is abelian, acts on the target Lie algebra by diagram automorphisms, and freely on (the set of rational points of) the scheme. Under these additional assumptions, we prove that A is finitely generated and the global Weyl module is a finitely generated A-module. We also define local Weyl modules via the Weyl functor and prove that these coincide with the local Weyl modules defined directly in a previous paper. Finally, we show that A is the algebra of coinvariants of the analogous algebra in the untwisted case.

This dissertation is devoted to the study of quantizations of nilpotent orbits and their covers, and their applications to the theory of unipotent representations. It combines the results from the papers written by the author during his PhD. In the first part, we construct a bijection between the sets of quantizations of a nilpotent orbit in a classical simple Lie algebra, and quantizations of its function algebra. In the second part, we study the geometry of nilpotent orbit covers, and the sets of quantizations of their function algebras. After that, we use the canonical quantizations of these function algebras to give new definitions of unipotent ideals and unipotent representations of a semisimple complex Lie algebra. Our definitions are vindicated by the many favorable properties which these ideals and bimodules enjoy. First of all, we show that both unipotent ideals and bimodules have nice geometric classifications. Unipotent ideals are classified by certain geometrically-defined equivalence classes of covers of nilpotent orbits. Unipotent bimodules are classified by irreducible representations of certain finite groups. We show that all unipotent ideals are maximal, and for classical groups all unipotent bimodules are unitary. In addition, we show that all unipotent bimodules are, as $G$-representations, of a very special form, proving a conjecture of Vogan. Finally, we show that all special unipotent bimodules are unipotent. The final assertion is proved using a certain refinement of Barbasch-Vogan-Lusztig-Spaltenstein duality, inspired by the symplectic duality. Along the way, we establish combinatorial algorithms (in classical and exceptional types) for computing the infinitesimal characters of unipotent ideals--Author's abstract

Let S S be a real semisimple Lie group with Lie algebra s = k + p \mathfrak {s}=\mathfrak {k} +\mathfrak {p} . The Kostant–Sekiguchi correspondence is a bijection between nilpotent S S orbits on s \mathfrak {s} and nilpotent K C K_{\mathbb {C}} orbits on p C \mathfrak {p}_{\mathbb {C}} . In this note we prove that the closure relations among nilpotent orbits are preserved under the Kostant–Sekiguchi correspondence. The techniques rely on work of M. Vergne and P. Kronheimer.

Multiplicity One Theorems for Generalized Gelfand-Graev Representations of Semisimple Lie Groups and Whittaker Models for the Discrete Series

The classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.

These are notes for four lectures on higher structures in M-theory as presented at workshops at the Erwin Schroedinger Institute and Tohoku University. The first lecture gives an overview of systems of multiple M5-branes and introduces the relevant mathematical structures underlying a local description of higher gauge theory. In the second lecture, we develop the corresponding global picture. A construction of non-abelian superconformal gauge theories in six dimensions using twistor spaces is discussed in the third lecture. The last lecture deals with the problem of higher quantization and its relation to loop space. An appendix summarizes the relation between 3-Lie algebras and Lie 2-algebras.