Symplectic resolutions for coverings of nilpotent orbits

Let $\text{O}$ be a nilpotent orbit in a semisimple complex Lie algebra $\text{g}$. Denote by G the simply connected Lie group with Lie algebra $\text{g}$. For a G-homogeneous covering $\text{M→}\text{O}$, let X be the normalization of $\text{O}$ in the function field of M. In this Note, we study the existence of symplectic resolutions for such coverings X. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Symplectic resolutions for nilpotent orbits (III)

We prove that two symplectic resolutions of a nilpotent orbit closures in a simple complex Lie algebra of classical type are related by Mukai flops in codimension 2. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Symplectic resolutions for nilpotent orbits (II)

Based on our previous work, Fu (Invent. Math. 151 (2003) 167–186), we prove that, given any two projective symplectic resolutions Z₁ and Z₂ of a nilpotent orbit closure in a complex simple Lie algebra of classical type, Z₁ is deformation equivalent to Z₂. This provides support for a ‘folklore’ conjecture on symplectic resolutions for symplectic singularities. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Symplectic geometry of semisimple orbits

Let G be a complex semisimple group, T G a maximal torus and B a Borel subgroup of G containing T. Let Ω be the Kostant-Kirillov holomorphic symplectic structure on the adjoint orbit O = Ad(G)c G/Z(c), where c Lie(T), and Z(c) is the centralizer of c in G. We prove that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad(c) are real (respectively, purely imaginary). Furthermore, each of these real symplectic manifolds is symplectomorphic to the cotangent bundle of the partial flag manifold G/Z(cc)B, equipped with the Liouville symplectic form. The latter result is generalized to hyperbolic adjoint orbits in a real semisimple Lie algebra.     

Theta lifting of nilpotent orbits for symmetric pairs

We consider a reductive dual pair in the stable range with the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent -orbits, where is a maximal compact subgroup of and we describe the precise -module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair . As an application, we prove sphericality and normality of the closure of certain nilpotent -orbits obtained in this way. We also give integral formulas for their degrees.

     

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold(M,ω) into a smooth symplectic manifold . Then we study how the formality and the Lefschetz property of are compared with that of (M,ω). We also study the formality of the symplectic blow-up of (M,ω) along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.     

On the singularities of nilpotent orbits

Let be a nilpotent orbit of the adjoint action of a complex connected semi-simple Lie group on its Lie algebra. We prove that the normalization of the closure of is Gorenstein and has rational singularities.     

Fano contact manifolds and nilpotent orbits

A contact structure on a complex manifold M is a corank 1 subbundle F of T_M such that the bilinear form on F with values in the quotient line bundle L = T_M/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.?If is a simple Lie algebra, the unique closed orbit in (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.?In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M P(H ₀(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant. Received: July 28, 1997     

Induced nilpotent orbits and birational geometry

In general, a nilpotent orbit closure in a complex simple Lie algebra g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when g is classical. Here, the induced orbits play an important role instead of Richardson orbits.     

Two-row nilpotent orbits of cyclic quivers

We prove that the local intersection cohomology of nilpotent orbit closures of cyclic quivers is trivial when the two orbits involved correspond to partitions with at most two rows. This gives a geometric proof of a result of Graham and Lehrer, which states that standard modules of the affine Hecke algebra of GL_d corresponding to nilpotents with at most two Jordan blocks are multiplicity-free. Received: 7 February 2002 / Published online: 8 November 2002     

Spectral synthesis for coadjoint orbits of nilpotent Lie groups

We determine the space of primary ideals in the group algebra \(L^{1}(G) \) of a connected nilpotent Lie group by identifying for every \(\pi \in \widehat{G} \) the family \(\mathcal I^\pi \) of primary ideals with hull \(\{\pi \} \) with the family of invariant subspaces of a certain finite dimensional sub-space \(\mathcal P_Q^\pi \) of the space of polynomials \(\mathcal P(G) \) on G.     

Symbols and orbits for 3-step nilpotent Lie groups

Sufficient conditions are derived for L²-boundedness of a convolution operator on a 3-step nilpotent Lie group. This is achieved by producing estimates on the Kirillov symbol of the operator, and is closely linked to the co-adjoint orbit structure of the group. A structure theorem for 3-step nilpotent Lie groups with 1-dimensional center is proved.     

Large orbits in actions of nilpotent groups

If a nontrivial nilpotent group acts faithfully and coprimely on a group , it is shown that some element of has a small centralizer in and hence lies in a large orbit. Specifically, there exists such that , where is the smallest prime divisor of .

     

Spherical nilpotent orbits and the Kostant-Sekiguchi correspondence

Let be a connected, linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. The Kostant-Sekiguchi correspondence is a bijection between the nilpotent -orbits in and the nilpotent -orbits in . We show that this correspondence associates each spherical nilpotent -orbit to a nilpotent -orbit that is multiplicity free as a Hamiltonian -space. The converse also holds.

     

Rationality of singularities and the Gorenstein property for nilpotent orbits

Ordzhonikidze Aviation Institute, Moscow. Translated from Funktsional'yi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 76–78, July–September, 1991.     

Complexity of nilpotent orbits and the Kostant-Sekiguchi correspondence

Let G be a connected linear semisimple Lie group with Lie algebra , and let be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in and is the nilpotent -orbit in associated to Ω by the Kostant-Sekiguchi correspondence. We show that the corank of the Hamiltonian K-space Ω is twice the complexity of the variety .