Spherical Nilpotent Orbits Research Articles

Fully commutative elements and spherical nilpotent orbits

Let g be a complex simple Lie algebra, with fixed Borel subalgebra b⊂g and with Weyl group W. Expanding on previous work of Fan and Stembridge in the simply laced case, this note aims to study the fully commutative elements of W, and their connections with the spherical nilpotent orbits in g. If g is not of type G2, it is shown that an element w∈W is fully commutative if and only if the subalgebra of b determined by the inversions of w lies in the closure of a spherical nilpotent orbit. A similar characterization is also given for the ad-nilpotent ideals of b, which are parametrized by suitable elements in the affine Weyl group of g thanks to the work of Cellini and Papi.

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical Hermitian cases

Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. We show that all such orbit closures are normal and describe the K-module structure of their ring of regular functions.

Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $\sigma$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy representation of $G_0$. Given an abelian subalgebra $\mathfrak a$ of $\mathfrak g$ contained in $\mathfrak g_1$ and stable under the action of some Borel subgroup $B_0 \subset G_0$, we classify the $B_0$-orbits in $\mathfrak a$ and we characterize the sphericity of $G_0 \mathfrak a$. Our main tool is the combinatorics of $\sigma$-minuscule elements in the affine Weyl group of $\mathfrak g$ and that of strongly orthogonal roots in Hermitian symmetric spaces.

Projective normality of model varieties and related results

We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over M M or over a closed G G -stable subvariety of M M is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spaces and of the closures of the spherical nilpotent orbits. Then we focus on a particular case proving two specific conjectures of Adams, Huang and Vogan on an analogue of the model orbit of the group of type E 8 \mathsf E_8 .

Regular functions of symplectic spherical nilpotent orbits and their quantizations

We study the ring of regular functions of classical spherical orbits R ( O ) R(\mathcal {O}) for G = S p ( 2 n , C ) G = Sp(2n,\mathbb {C}) . In particular, treating G G as a real Lie group with maximal compact subgroup K K , we focus on a quantization model of O \mathcal {O} when O \mathcal {O} is the nilpotent orbit ( 2 2 p 1 2 q ) (2^{2p}1^{2q}) . With this model, we verify a conjecture by McGovern and another conjecture by Achar and Sommers related to the character formula of such orbits. Assuming the results in a preprint of Barbasch, we will also verify the Achar-Sommers conjecture for a larger class of nilpotent orbits.

On the quantization of spherical nilpotent orbits

On the quantization of spherical nilpotent orbits

Classification of strict wonderful varieties

In the setting of strict wonderful varieties we answer positively to Luna's conjecture, saying that wonderful varieties are classified by combinatorial objects, the so-called spherical systems. In particular, we prove that strict wonderful varieties are mostly obtained from symmetric spaces, spherical nilpotent orbits or model spaces. To make the paper self-contained as much as possible, we shall gather some known results on these families and more generally on wonderful varieties.

Spherical nilpotent orbits in positive characteristic

Spherical nilpotent orbits in positive characteristic

On the evaluation of some Selberg-like integrals

Several methods of evaluation are presented for a family { I n , d , p } of Selberg-like integrals that arise in the computation of the algebraic–geometric degrees of a family of spherical nilpotent orbits associated to the symmetric space of a simple real Lie group. Adapting the technique of Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of certain iterated sums over permutation groups. The resulting formula, however, is only valid when the integrand involves an even power of the Vandermonde determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral I n , d , p as a product of polynomial of fixed degree times a particular product of gamma factors; thereby identifying the asymptotics of the integrals with respect to their parameters. Lastly, we derive a recursive formula for evaluation of another general class of Selberg-like integrals, by applying some of the technology of generalized hypergeometric functions.

On a class of multiplicity-free nilpotent $K_{\mathbb{C}}$-orbits

Let $G$ be a real, connected, noncompact, semisimple Lie group, let $K_{\mathbb{C}}$ be the complexification of a maximal compact subgroup $K$ of $G$, and let $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ be the corresponding Cartan decomposition of the complexified Lie algebra of $G$. Sequences of strongly orthogonal noncompact weights are constructed and classified for each real noncompact simple Lie group of classical type. We show that for each partial subsequence {$\gamma _{1},\ldots ,\gamma _{i}$} there is a corresponding family of nilpotent $K_{\mathbb{C}}$-orbits in $\mathfrak{p}$, ordered by inclusion and such that the representation of $K$ on the ring of regular functions on each orbit is multiplicity-free. The $K$-types of regular functions on the orbits and the regular functions on their closures are both explicitly identified and demonstrated to coincide, with one exception in the Hermitian symmetric case. The classification presented also includes the specification of a base point for each orbit and exhibits a corresponding system of restricted roots with multiplicities. A formula for the leading term of the Hilbert polynomials corresponding to these orbits is given. This formula, together with the restricted root data, allows the determination of the dimensions of these orbits and the algebraic-geometric degree of their closures. In an appendix, the location of these orbits within D. King’s classification of spherical nilpotent orbits in complex symmetric spaces is depicted via signed partitions and Hasse diagrams.

Some amazing properties of spherical nilpotent orbits

Let G be a simple algebraic group defined over an algebraically closed field k of characteristic zero. Write g for its Lie algebra. Let x ∈ g be a nilpotent element and G·x ⊂ g the corresponding nilpotent orbit. The maximal number m such that (adx) 6= 0 is called the height of x or of G·x, denoted ht(x). Recall that an irreducible G-variety X is called G-spherical if a Borel subgroup of G has an open orbit in X. It was shown in [Pa1] that G·x is G-spherical if and only if (adx) = 0. This means that the spherical nilpotent orbits are precisely the orbits of height 2 and 3. Unfortunately, whereas my proof for the orbits of height 2 and height ≥ 4 was completely general, the argument for the orbits of height 3 explicitly used their classification. In this paper, we give a proof of sphericity that does not rely on the classification of nilpotent orbits, see Theorem 3.3. We begin with some properties of invariants of symplectic representations. For instance, we prove that (1) if H ⊂ Sp(V ) is an irreducible representation without nonconstant invariants, then H = Sp(V ), and (2) if H has non-constant invariants, then it has an invariant of degree 4. Applying these results to nilpotent orbits, we prove that the centraliser zg(x) has a rather specific structure whenever ht(x) is odd. From this description, we deduce a conceptual proof of sphericity in case ht(x) = 3. As another application we compute the index of zg(x). It will be shown that ind zg(x) = rk g, if ht(x) = 3. This confirms Elashvili’s conjecture for such x (see [Pa5, Sect. 3] about this conjecture). In Section 4, we prove that if θ, the highest root of g, is fundamental, then g always has a specific orbit of height 3, which is denoted by O. This orbit satisfies several arithmetical relations. Namely, if g = ⊕ −3≤i≤3 g〈i〉 is the Z-grading associated with an sl2-triple containing e ∈ O, then dim g〈3〉 = 2 and dim g〈1〉 = 2 dim g〈2〉. Furthermore, the weighted Dynkin diagram of O can explicitly be described. Let β be the unique simple root that is not orthogonal to θ and let {αi} be all simple roots adjacent to β on the Dynkin diagram of g. Then one has to put ‘1’ at all αi’s and ‘0’ at all other simple roots (Theorem 4.5). It is curious that these properties of O enables us to give an intrinsic construction of G2-grading in each simple g whose highest root is fundamental.

Spherical Orbits and Abelian Ideals

Let G be a connected reductive complex algebraic group with Lie algebra Lie G=g. Let B be a Borel subgroup of G with unipotent radical Bu . We denote the Lie algebra of B and Bu by b and bu , respectively. The group B acts on any ideal of b by means of the adjoint representation. After a preliminary section we study a relationship between spherical nilpotent orbits and abelian ideals a of b, using the structure theory for these orbits from [10]. The principal result of this section is that, for an abelian ideal a of b, any nilpotent orbit meeting a is a spherical G-variety, see Theorem 2.3. As a consequence of this we obtain a short conceptual proof of a finiteness theorem from [14]. Namely, for a parabolic subgroup P of G and an abelian ideal a of p=Lie P in the nilpotent radical pu=Lie Pu , the group P operates on a with finitely many orbits. The proof of this fact in [14] involved long and tedious case by case considerations. We also prove a partial converse to the result just mentioned. Following [4], we say that an ideal of b is ad nilpotent whenever it consists of nilpotent elements. In case G is simply laced, we show that an ad nilpotent ideal c of b is abelian provided any nilpotent orbit meeting c is spherical, see Proposition 2.7. In Section 3 we consider some properties of ad nilpotent ideals of b. In Theorem 3.2 we give a description of the normaliser of such ideals. This applies in particular to abelian ideals of b. A remarkable theorem of doi:10.1006 aima.2000.1959, available online at http: www.idealibrary.com on

Unipotent representations attached to spherical nilpotent orbits

A coadjoint nilpotent orbit [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] of a complex semisimple Lie group G is said to be spherical if it contains an open orbit of a Borel subgroup. We determine when and how to attach unitary representations to such an orbit for the real orthogonal and symplectic groups. Our results actually extend to a larger class of nilpotent coadjoint orbits.