Related papers: Minimal special degenerations and duality

The nilpotent cone of a simple Lie algebra is partitioned into locally closed subvarieties called special pieces, each containing exactly one special orbit. Lusztig conjectured that each special piece is the quotient of some smooth variety…

According to a well-known 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…

We classify the nilpotent orbits in a simple Lie algebra for which the restriction of the adjoint quotient map to a Slodowy slice is the universal Poisson deformation of its central fibre. This generalises work of Brieskorn and Slodowy on…

The minimal degeneration singularities in the affine Grassmannians of simple simply-laced algebraic groups are determined to be either Kleinian singularities of type A, or closures of minimal orbits in nilpotent cones. The singularities for…

This is an expository article on the singularities of nilpotent orbit closures in simple Lie algebras over the complex numbers. It is slanted towards aspects that are relevant for representation theory, including Maffei's theorem relating…

Motivated by geometric Langlands, we initiate a program to study the mirror symmetry between nilpotent orbit closures of a semisimple Lie algebra and those of its Langlands dual. The most interesting case is $B_n$ via $C_n$. Classically,…

The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible…

We introduce minimal Richardson orbits and pseudo-polarizations for nilpotent orbits in classical Lie algebras of types B, C, and D. For any nilpotent orbit, we classify all minimal Richardson orbits containing it and thereby determine the…

We study general properties of quantisations of Slodowy slices and discuss in detail the case of the minimal nilpotent orbit. Associated varieties of related primitive ideals of U(g) are determined, in the general case, and the de Vos-van…

This short note is a supplement to the previous article with the same title. Here we treat a conical symplectic variety obtained as a finite covering of a (not necessarily normal) nilpotent orbit closure of a complex semisimple Lie algebra.

We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group G with Lie algebra g. These slices…

We first show the closure of the minimal nilpotent adjoint orbit Omin^{D_n} in so_{2n} is isomorphic to the affinization of T^*(SL_{n-1}/[P,P]) where P is the parabolic subgroup P_{(1,1,n-3)} of SL_{n-1}(C). Then we prove that the closure…

Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of…

The Slodowy slice is a flat Poisson deformation of its nilpotent part, and it was demonstrated by Lehn-Namikawa-Sorger that there is an interesting infinite family of nilpotent orbits in symplectic Lie algebras for which the slice is not…

We establish a novel connection between the minimal nilpotent orbit $\mathbb{O}_n$ in $\mathfrak{sl}_n$ and the minimal nilpotent orbit closure $\overline{\mathbf{O}}_n$ in $\mathfrak{so}_{2n+2}$, which differs from the shared-orbit…

Let G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a…

We show that the numbers of nilpotent coadjoint orbits in the dual of exceptional Lie algebra $G_2$ in characteristic $3$ and in the dual of exceptional Lie algebra $F_4$ in characteristic $2$ are finite. We determine the closure relation…

Let us fix a complex simple Lie algebra and its non-compact real form. This paper focuses on non-zero adjoint nilpotent orbits in the complex simple Lie algebra meeting the real form. We show that the poset consisting of such nilpotent…

This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts…

For an irreducible smooth representation of a connected reductive $p$-adic group, two important associated invariants are the wavefront set and the (partly conjectural) Langlands parameter. While a wavefront set consists of $p$-adic…