Nilpotent Orbit Closures Research Articles

Minimal set of generators of ideals defining nilpotent orbit closures

Over a field of characteristic 0 0 , we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of n × n n \times n matrices. This modifies a conjecture of Weyman and provides a complete answer to it.

Birational geometry for the covering of a nilpotent orbit closure

Let \({\mathfrak {g}}\) be a complex classical simple Lie algebra and let O be a nilpotent orbit of \({\mathfrak {g}}\). The fundamental group \(\pi _1(O)\) is finite. Take the universal covering \(\pi ^0: X^0 \rightarrow O\). Then \(\pi ^0\) extends to a finite cover \(\pi : X \rightarrow {\bar{O}}\). By using the Kirillov–Kostant form \(\omega _{KK}\) on O, the normal affine variety X becomes a conical symplectic variety. In this article we give an explicit construction of a Q-factorial terminalization of X.

Partial implosions and quivers

We propose magnetic quivers for partial implosion spaces. Such partial implosions involve a choice of parabolic subgroup, with the Borel subgroup corresponding to the standard implosion. In the subregular case we test the conjecture by verifying that reduction by the Levi group gives the appropriate nilpotent orbit closure. In the case of a parabolic corresponding to a hook diagram we are also able to carry out this verification provided we work at nonzero Fayet-Iliopoulos parameters.

Birational geometry for the covering of a nilpotent orbit closure II

Let O be a nilpotent orbit of a complex semisimple Lie algebra g and let π:X→O¯ be the finite covering associated with the universal covering of O. In the previous article [14] we have explicitly constructed a Q-factorial terminalization X˜ of X when g is classical. In this article we count how many non-isomorphic Q-factorial terminalizations X has. We construct the universal Poisson deformation of X˜ over H2(X˜,C) and look at the action of the Weyl group W(X) on H2(X˜,C). The main result is an explicit geometric description of W(X).

Magnetic lattices for orthosymplectic quivers

For any gauge theory, there may be a subgroup of the gauge group which acts trivially on the matter content. While many physical observables are not sensitive to this fact, the choice of the precise gauge group becomes crucial when the magnetic lattice of the theory is considered. This question is addressed in the context of Coulomb branches for 3d mathcal{N} = 4 quiver gauge theories, which are moduli spaces of dressed monopole operators. We compute the Coulomb branch Hilbert series of many unitary-orthosymplectic quivers for different choices of gauge groups, including diagonal quotients of the product gauge group of individual factors, where the quotient is by a trivially acting subgroup. Choosing different such diagonal groups results in distinct Coulomb branches, related as orbifolds. Examples include nilpotent orbit closures of the exceptional E-type algebras and magnetic quivers that arise from brane physics. This includes Higgs branches of theories with 8 supercharges in dimensions 4, 5, and 6. A crucial ingredient in the calculation of exact refined Hilbert series is the alternative construction of unframed magnetic quivers from resolved Slodowy slices, whose Hilbert series can be derived from Hall-Littlewood polynomials.

A quantum Mirković-Vybornov isomorphism

We present a quantization of an isomorphism of Mirković and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in g l N \mathfrak {gl}_N to a slice between spherical Schubert varieties in the affine Grassmannian of P G L n PGL_n (with weights encoded by the Jordan types of the nilpotent orbits). A quantization of the former variety is provided by a parabolic W-algebra and of the latter by a truncated shifted Yangian. Building on earlier work of Brundan and Kleshchev, we define an explicit isomorphism between these non-commutative algebras and show that its classical limit is a variation of the original isomorphism of Mirković and Vybornov. As a corollary, we deduce that the W-algebra is free as a left (or right) module over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and Ovsienko.

Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional mathcal{N}=4 theories

The Coulomb branches of certain 3-dimensional mathcal{N}=4 quiver gauge theories are closures of nilpotent orbits of classical or exceptional Lie algebras. The monopole formula, as Hilbert series of the associated Coulomb branch chiral ring, has been successful in describing the singular hyper-Kähler structure. By means of the monopole formula with background charges for flavour symmetries, which realises real mass deformations, we study the resolution properties of all (characteristic) height two nilpotent orbits. As a result, the monopole formula correctly reproduces (i) the existence of a symplectic resolution, (ii) the form of the symplectic resolution, and (iii) the Mukai flops in the case of multiple resolutions. Moreover, the (characteristic) height two nilpotent orbit closures are resolved by cotangent bundles of Hermitian symmetric spaces and the unitary Coulomb branch quiver realisations exhaust all the possibilities.

NILPOTENT SUBSPACES AND NILPOTENT ORBITS

Let $G$ be a semisimple complex algebraic group with Lie algebra $\mathfrak{g}$. For a nilpotent $G$-orbit ${\mathcal{O}}\subset \mathfrak{g}$, let $d_{{\mathcal{O}}}$ denote the maximal dimension of a subspace $V\subset \mathfrak{g}$ that is contained in the closure of ${\mathcal{O}}$. In this note, we prove that $d_{{\mathcal{O}}}\leq {\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ and this upper bound is attained if and only if ${\mathcal{O}}$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V={\textstyle \frac{1}{2}}\dim {\mathcal{O}}$, then $V$ is the nilradical of a polarisation of ${\mathcal{O}}$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_{2}$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits ${\mathcal{O}}$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$, or (2) is the only $B$-stable subspace $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$.

Branes and the Kraft-Procesi transition: classical case

Moduli spaces of a large set of 3d mathcal{N}=4 effective gauge theories are known to be closures of nilpotent orbits. This set of theories has recently acquired a special status, due to Namikawa’s theorem. As a consequence of this theorem, closures of nilpotent orbits are the simplest non-trivial moduli spaces that can be found in three dimensional theories with eight supercharges. In the early 80’s mathematicians Hanspeter Kraft and Claudio Procesi characterized an inclusion relation between nilpotent orbit closures of the same classical Lie algebra. We recently [1] showed a physical realization of their work in terms of the motion of D3-branes on the Type IIB superstring embedding of the effective gauge theories. This analysis is restricted to A-type Lie algebras. The present note expands our previous discussion to the remaining classical cases: orthogonal and symplectic algebras. In order to do so we introduce O3-planes in the superstring description. We also find a brane realization for the mathematical map between two partitions of the same integer number known as collapse. Another result is that basic Kraft-Procesi transitions turn out to be described by the moduli space of orthosymplectic quivers with varying boundary conditions.

Filtrations on Springer fiber cohomology and Kostka polynomials

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.

Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops

In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure B(1)‾ of type A, and study relations between an NCCR and crepant resolutions Y and Y+ of B(1)‾. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions Y and Y+ of B(1)‾ as moduli spaces of representations of the quiver. We also study the Kawamata–Namikawa's derived equivalence between crepant resolutions Y and Y+ of B(1)‾ in terms of an NCCR. We also show that the P-twist on the derived category of Y corresponds to a certain operation of the NCCR, which we call multi-mutation, and that a multi-mutation is a composition of Iyama–Wemyss's mutations.

A characterization of nilpotent orbit closures among symplectic singularities

We prove that a conical symplectic variety with maximal weight 1 is isomorphic to one of the following: (i) an affine space with the standard symplectic form (ii) a nilpotent orbit closure of a complex semisimple Lie algebra with the Kirillov-Kostant form.

Generic singularities of nilpotent orbit closures

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.

On quantization of a nilpotent orbit closure in $G_2$

On quantization of a nilpotent orbit closure in $G_2$

JOSEPH IDEALS AND LISSE MINIMAL -ALGEBRAS

We consider a lifting of Joseph ideals for the minimal nilpotent orbit closure to the setting of affine Kac–Moody algebras and find new examples of affine vertex algebras whose associated varieties are minimal nilpotent orbit closures. As an application we obtain a new family of lisse ($C_{2}$-cofinite)$W$-algebras that are not coming from admissible representations of affine Kac–Moody algebras.

Jet schemes of the closure of nilpotent orbits

We study in this paper the jet schemes of the closure of nilpotent orbits in a finite-dimensional complex reductive Lie algebra. For the nilpotent cone, which is the closure of the regular nilpotent orbit, all the jet schemes are irreducible. This was first observed by Eisenbud and Frenkel, and follows from a strong result of Musta\u{t}\c{a} (2001). Using induction and restriction of "little" nilpotent orbits in reductive Lie algebras, we show that for a large number of nilpotent orbits, the jet schemes of their closure are reducible. As a consequence, we obtain certain geometrical properties of these nilpotent orbit closures.

Normality of orthogonal and symplectic nilpotent orbit closures in positive characteristic

In this note we investigate the normality of closures of orthogonal and symplectic nilpotent orbits in positive characteristic. We prove that the closure of such a nilpotent orbit is normal provided that neither type d nor type e minimal irreducible degeneration occurs in the closure, and conversely if the closure is normal, then any type e minimal irreducible degeneration does not occur in it. Here, the minimal irreducible degenerations of a nilpotent orbit are introduced by W. Hesselink in [7] (or see [11] from which we take Table 1 for the complete list of all minimal irreducible degenerations). Our result is a weak version in positive characteristic of [11, Theorem 16.2(ii)], one of the main results of [11] over complex numbers.

Crepant Resolutions of a Slodowy Slice in a Nilpotent Orbit Closure in $\mathfrak {sl}_N(\mathbb C)$

One of our results of this article is that every (projective) crepant resolution of a Slodowy slice in a nilpotent orbit closure in \mathfrak {sl}_N(\mathbb C) can be obtained as the restriction of some crepant resolution of the nilpotent orbit closure. We also show that there is a decomposition of the Slodowy slice into other Slodowy slices with good properties. From this decomposition, one can count the number of crepant resolutions.

Existence of Affine Pavings for Varieties of Partial Flags Associated to Nilpotent Elements

The flag variety of a complex reductive linear algebraic group |$G$| is by definition the quotient |$G/B$| by a Borel subgroup. It can be regarded as the set of Borel subalgebras of |${\rm Lie}(G)$|⁠. Given a nilpotent element |$e$| in |${\rm Lie}(G)$|⁠, one calls Springer fiber the subvariety formed by the Borel subalgebras which |$e$| belongs to. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by De Concini, Lusztig, and Procesi asserts that when |$G$| is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In this paper, we study varieties generalizing the Springer fibers in two ways: being contained in a partial flag variety |$G/P$| (the quotient by a parabolic subgroup, instead of a Borel subgroup) and defined by a more general belonging condition (in terms of an ideal of |${\rm Lie}(P))$|⁠. These varieties arise, for instance, as fibers of resolutions of nilpotent orbit closures. The main result of the paper is a generalization of De Concini, Lusztig, and Procesi's theorem to this context.Communicated by Corrado de Concini

HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II

Let G = GL n be the general linear group over an algebraically closed field k and let \( \mathfrak{g}=\mathfrak{g}{\mathfrak{l}}_n \) be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[\( \mathfrak{g} \)] be the algebra of polynomial functions on g and let k[\( \mathfrak{g} \)] G be the algebra of invariants under the conjugation action of G. For all weights χ ∈ ℤ n with χ2 ≤ 0 or χ n − 1 ≥ 0 we give explicit bases for the k[\( \mathfrak{g} \)] G -module k \( {\left[\mathfrak{g}\right]}_{\chi}^U \) of highest weight vectors of weight χ. We also give bases for the vector spaces k[C] χ U where C is a nilpotent orbit closure. This extends earlier results to a much bigger class of weights. To express our semi-invariants in terms of matrix powers we prove certain Cayley-Hamilton type identities.