Searcharxiv⌕ Search

arXiv subjects

Dmitri I. Panyushev

Publications and source records attributed to Dmitri I. Panyushev.

At least 19 recordsLinked to original sources
arXiv2026

The index of subalgebras and strange coadjoint orbits

For an algebraic group $Q$ with $\mathsf{Lie\,} Q=\mathfrak q$, we develop a method for estimating the index of a subalgebra $\mathfrak h$ in $\mathfrak q$ via the use of coadjoint $Q$-orbits in $\mathfrak q^*$. Let $\mathfrak q^ξ$ denote the stabiliser of $ξ\in\mathfrak q^*$. In the special case when $\mathfrak q^ξ\oplus\mathfrak h=\mathfrak q$, our estimate implies that $\mathsf{ind\,}\mathfrak h=0$. Using our theory, we also answer a question of Duflo. An orbit $Q{\cdot}η\subset\mathfrak q^*$ is said to be strange, if $\mathfrak q^η\oplus\mathfrak h=\mathfrak q$ for some $\mathfrak h$. In the second part of the paper, we study strange orbits for a semisimple algebra $\mathfrak g$. It is shown that an orbit ${\mathcal O}\subset\mathfrak g\simeq\mathfrak g^*$ is strange whenever the complexity of ${\mathcal O}$ is at most 1. Furthermore, if ${\mathcal S}\subset\mathfrak g$ is a sheet containing a strange nilpotent orbit, then all orbits in ${\mathcal S}$ are strange. We also show that strange orbits in $\mathfrak{sl}_n$ are not as sparse, as one might expect, and discuss some conjectures on strange orbits.

math.RT↗
arXiv2024

Nilpotent orbits and their secant varieties

Let $G$ be a simple algebraic group and $\mathcal O$ a nilpotent orbit in $\mathfrak g$. Let ${\mathbf{CS}}(\mathcal O)$ denote the affine cone over the secant variety of $\overline{\mathbb P\mathcal O}\subset \mathbb P\mathfrak g$. Using the theory of doubled actions of $G$, we describe ${\mathbf{CS}}(\mathcal O)$ for all $\mathcal O$. We compute $\dim{\mathbf{CS}}(\mathcal O)$ using the complexity and rank of the $G$-variety $\mathcal O$ and show that there is an abelian subalgebra $\mathfrak t_{\mathcal O}\subset\mathfrak g$ such that ${\mathbf{CS}}(\mathcal O)$ is the closure of $G{\cdot}\mathfrak t_\mathcal O$. Another observation is that ${\mathbf{CS}}(\mathcal O)$ coincide with the closure of the image of the moment map associated with the cotangent bundle of $\mathcal O$. We also compute the complexity and rank for all nilpotent orbits.

math.AG↗
arXiv2024

Projections of nilpotent orbits in a simple Lie algebra and shared orbits

Let $G$ be a simple algebraic group with $\mathfrak g=Lie(G)$ and $\mathcal O\subset\mathfrak g$ a nilpotent orbit. If $H$ is a reductive subgroup of $G$ with $Lie(H)=\mathfrak h$, then $\mathfrak g=\mathfrak h\oplus\mathfrak m$, where $\mathfrak m=\mathfrak h^\perp$. We consider the natural projections $ϕ: \bar{\mathcal O}\to\mathfrak h$ and $ψ:\bar{\mathcal O}\to\mathfrak m$, and two related properties of the pair $(H,\mathcal O)$: $(P_1)$: $\bar{\mathcal O}\cap\mathfrak m={0}$ and $(P_2)$: $H$ has a dense orbit in $\mathcal O$. We show that $(P_1)$ implies $(P_2)$ for all $\mathcal O$ and these properties are equivalent for $\mathcal O=\mathcal O_{min}$, the minimal nilpotent orbit. If $(P_1)$ holds, then $ϕ$ is finite, and $ϕ(\bar{\mathcal O})$ is the closure of a nilpotent H-orbit $\mathcal O'$. We prove that $\mathcal O$ is contained in the closure of the G-orbit $G{\cdot}\mathcal O'$ and obtain the classification of pairs $(H,\mathcal O)$ with property $(P_1)$. The orbit $\mathcal O'$ is "shared" in the sense of Brylinski and Kostant. Using our classification, we detect an omission in the list of pairs $(H,G)$ having a shared orbit that is given in "Nilpotent orbits, normality, and hamiltonian group actions", J.A.M.S., 7 (1994), 269--298. It is also proved that if $(P_1)$ holds for $(H, \mathcal O_{min})$, then both varieties $ϕ(\mathcal O_{min})$ and $ψ(\mathcal O_{min})$ generate the same closed subvariety of $\mathfrak g$.

math.RT↗
arXiv2024

Orbits and invariants for coisotropy representations

For a subgroup $H$ of a reductive group $G$, let $\mathfrak m\subset \mathfrak g^*$ be the cotangent space of $eH\in G/H$. The linear action $(H:\mathfrak m)$ is the coisotropy representation. It is known that the complexity and rank of $G/H$ (denoted $c$ and $r$, respectively) are encoded in properties of $(H:\mathfrak m)$. We complement existing results on $c$, $r$, and $(H:\mathfrak m)$, especially for quasiaffine varieties $G/H$. If the algebra of invariants $k[\mathfrak m]^H$ is finitely generated, then we establish a connection between the nullcones in $\mathfrak m$ and $\mathfrak g^*$. Two other topics considered are (i) a relationship between varieties $G/H$ of complexity at most 1 and the homological dimension of the algebra of invariants $k[\mathfrak m]^H$ and (ii) the Poisson structure of $k[\mathfrak m]^H$ and Poisson-commutative subalgebras in $k[\mathfrak m]^H$ with maximal transcendence degree.

math.RT↗
arXiv2023

The Frobenius semiradical, generic stabilisers, and Poisson centre for nilradicals

Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak n$ the nilradical of a parabolic subalgebra of $\mathfrak g$. We consider some properties of the coadjoint representation of $\mathfrak n$ and related algebras of invariants. This includes (i) the problem of existence of generic stabilisers, (ii) a description of the Frobenius semiradical of $\mathfrak n$ and the Poisson centre $Z(\mathfrak n)$ of the symmetric algebra $S(\mathfrak n)$, (iii) the structure of $S(\mathfrak n)$ as $Z(\mathfrak n)$-module, and (iv) the description of square integrable (= quasi-reductive) nilradicals. Our main technical tools are the Kostant cascade in the set of positive roots of $\mathfrak g$ and the notion of optimisation of $\mathfrak n$.

math.RT↗
arXiv2022

Combinatorial and geometric constructions associated with the Kostant cascade

Let $\mathfrak g$ be a complex simple Lie algebra and $\mathfrak b=\mathfrak t\oplus\mathfrak u^+$ a fixed Borel subalgebra. Let $Δ^+$ be the set of positive roots associated with $\mathfrak u^+$ and $\mathcal K\subsetΔ^+$ the Kostant cascade. We elaborate on some constructions related to $\mathcal K$ and applications of $\mathcal K$. This includes the cascade element $x_{\mathcal K}$ in the Cartan subalgebra $\mathfrak t$ and properties of certain objects naturally associated with $\mathcal K$: an abelian ideal of $\mathfrak b$, a nilpotent $G$-orbit in $\mathfrak g$, and an involution of $\mathfrak g$.

math.RT↗
arXiv2021

Commutative polarisations and the Kostant cascade

Let $\mathfrak g$ be a complex simple Lie algebra. We classify the parabolic subalgebras $\mathfrak p$ of $\mathfrak g$ such that the nilradical of $\mathfrak p$ has a commutative polarisation. The answer is given in terms of the Kostant cascade. It requires also the notion of an optimal nilradical and some properties of abelian ideals in a Borel subalgebra of $\mathfrak g$. Some invariant-theoretic consequences of the existence of a commutative polarisation are also discussed.

math.RT↗
arXiv2021

Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties

Let $G$ be a simple algebraic group with $\mathfrak g=\mathsf{Lie} G$ and $\mathcal O_{\sf min}\subset\mathfrak g$ the minimal nilpotent orbit. For a $\mathbb Z_2$-grading $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$, let $G_0$ be a connected subgroup of $G$ with $\mathsf{Lie} G_0=\mathfrak g_0$. We study the $G_0$-equivariant projections $φ:\overline{\mathcal O_{\sf min}}\to \mathfrak g_0$ and $ψ:\overline{\mathcal O_{\sf min}}\to\mathfrak g_1$. It is shown that the properties of $\overline{φ(\mathcal O_{\sf min})}$ and $\overline{ψ(\mathcal O_{\sf min})}$ essentially depend on whether the intersection $\mathcal O_{\sf min}\cap\mathfrak g_1$ is empty or not. If $\mathcal O_{\sf min}\cap\mathfrak g_1\ne\varnothing$, then both $\overline{φ(\mathcal O_{\sf min})}$ and $\overline{ψ(\mathcal O_{\sf min})}$ contain a 1-parameter family of closed $G_0$-orbits, while if $\mathcal O_{\sf min}\cap\mathfrak g_1=\varnothing$, then both are $G_0$-prehomogeneous. We prove that $\overline{G{\cdot}φ(\mathcal O_{\sf min})}=\overline{G{\cdot}ψ(\mathcal O_{\sf min})}$. Moreover, if $\mathcal O_{\sf min}\cap\mathfrak g_1\ne\varnothing$, then this common variety is the affine cone over the secant variety of $\mathbb P(\mathcal O_{\sf min})\subset\mathbb P(\mathfrak g)$. As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple $G$-module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of $\mathfrak g$ in place of $\mathfrak g_0$ or spherical nilpotent $G$-orbits in place of $\mathcal O_{\sf min}$.

math.RT↗
arXiv2021

Periodic automorphisms, compatible Poisson brackets, and Gaudin subalgebras

Let $\mathfrak g$ be a finite-dimensional Lie algebra. The symmetric algebra $\mathcal S(\mathfrak g)$ is equipped with the standard Lie-Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on $\mathcal S(\mathfrak g)$ to any finite order automorphism $θ$ of $\mathfrak g$. We study related Poisson-commutative subalgebras $\mathcal C$ of $\mathcal S(\mathfrak g)$ and associated Lie algebra contractions of $\mathfrak g$. To obtain substantial results, we have to assume that $\mathfrak g$ is semisimple. Then we can use Vinberg's theory of $θ$-groups and the machinery of Invariant Theory. If $\mathfrak g=\mathfrak h\oplus\dots \oplus \mathfrak h$ (sum of $k$ copies), where $\mathfrak h$ is simple, and $θ$ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra $\mathcal C$ is polynomial and maximal. Furthermore, we quantise this $\mathcal C$ using a Gaudin subalgebra in the enveloping algebra $\mathcal U(\mathfrak g)$.

math.RT↗
arXiv2021

Nilpotent orbits and mixed gradings of semisimple Lie algebras

Let $σ$ be an involution of a complex semisimple Lie algebra $\mathfrak g$ and $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$ the related $\mathbb Z_2$-grading. We study relations between nilpotent $G_0$-orbits in $\mathfrak g_0$ and the respective $G$-orbits in $\mathfrak g$. If $e\in\mathfrak g_0$ is nilpotent and $\{e,h,f\}\subset\mathfrak g_0$ is an $\mathfrak{sl}_2$-triple, then the semisimple element $h$ yields a $\mathbb Z$-grading of $\mathfrak g$. Our main tool is the combined $\mathbb Z\times\mathbb Z_2$-grading of $\mathfrak g$, which is called a mixed grading. We prove, in particular, that if $e_σ$ is a regular nilpotent element of $\mathfrak g_0$, then the weighted Dynkin diagram of $e_σ$, $\mathcal D(e_σ)$, has only isolated zeros. It is also shown that if $G{\cdot}e_σ\cap\mathfrak g_1\ne\varnothing$, then the Satake diagram of $σ$ has only isolated black nodes and these black nodes occur among the zeros of $\mathcal D(e_σ)$. Using mixed gradings related to $e_σ$, we define an inner involution $\checkσ$ such that $σ$ and $\checkσ$ commute. Here we prove that the Satake diagrams for both $\checkσ$ and $σ\checkσ$ have isolated black nodes.

math.RT↗
arXiv2020

Reductive subalgebras of semisimple Lie algebras and Poisson commutativity

Let $\mathfrak g$ be a semisimple Lie algebra, $\mathfrak h\subset\mathfrak g$ a reductive subalgebra such that $\mathfrak h^\perp$ is a complementary $\mathfrak h$-submodule of $\mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra ${\mathcal S}(\mathfrak g)$ by taking the subalgebra ${\mathcal Z}$ generated by the bi-homogeneous components of all $H\in{\mathcal S}(\mathfrak g)^{\mathfrak g}$. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras ${\mathcal Z}$. As a by-product, we prove that ${\mathcal Z}$ is Poisson commutative if $\mathfrak h$ is abelian and describe ${\mathcal Z}$ in the special case when $\mathfrak h$ is a Cartan subalgebra. In this case, ${\mathcal Z}$ appears to be polynomial and has the maximal transcendence degree $(\mathrm{dim}\,\mathfrak g+\mathrm{rk}\,\mathfrak g)/2$.

math.RT↗
arXiv2019

Casimir elements associated with Levi subalgebras of simple Lie algebras and their applications

Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak h$ a Levi subalgebra, and $C_{\mathfrak h}\in U(\mathfrak h)$ the Casimir element defined via the restriction of the Killing form on $\mathfrak g$ to $\mathfrak h$. We study $C_{\mathfrak h}$-eigenvalues in $\mathfrak g/\mathfrak h$ and related $\mathfrak h$-modules. Without loss of generality, one may assume that $\mathfrak h$ is a maximal Levi. Then $\mathfrak g$ is equipped with the natural $\mathbb Z$-grading $\mathfrak g=\bigoplus_{i\in\mathbb Z}\mathfrak g(i)$ such that $\mathfrak g(0)=\mathfrak h$ and $\mathfrak g(i)$ is a simple $\mathfrak h$-module for $i\ne 0$. We give explicit formulae for the $C_\mathfrak h$-eigenvalues in each $\mathfrak g(i)$, $i\ne 0$, and relate eigenvalues of $C_\mathfrak h$ in $\bigwedge^\bullet\mathfrak g(1)$ to the dimensions of abelian subspaces of $\mathfrak g(1)$. We also prove that if $\mathfrak a\subset\mathfrak g(1)$ is abelian, whereas $\mathfrak g(1)$ is not, then $\dim\mathfrak a\le \dim\mathfrak g(1)/2$. Moreover, if $\dim\mathfrak a=(\dim\mathfrak g(1))/2$, then $\mathfrak a$ has an abelian complement. The $\mathbb Z$-gradings of height $\le 2$ are closely related to involutions of $\mathfrak g$, and we provide a connection of our theory to (an extension of) the "strange formula" of Freudenthal-de Vries.

math.RT↗
arXiv2018

Glorious pairs of roots and Abelian ideals of a Borel subalgebra

Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$. Let $Δ^+$ be the corresponding (po)set of positive roots and $θ$ the highest root. A pair $\{η,η'\}\subset Δ^+$ is said to be glorious, if $η,η'$ are incomparable and $η+η'=θ$. Using the theory of abelian ideals of $\mathfrak b$, we (1) establish a relationship of $η,η'$ to certain abelian ideals associated with long simple roots, (2) provide a natural bijection between the glorious pairs and the pairs of adjacent long simple roots (i.e., some edges of the Dynkin diagram), and (3) point out a simple transform connecting two glorious pairs corresponding to the incident edges in the Dynkin digram. In types ${\bf DE}$, we prove that if $\{η,η'\}$ corresponds to the edge through the branching node of the Dynkin diagram, then the meet $η\wedgeη'$ is the unique maximal non-commutative root. There is also an analogue of this property for all other types except type ${\bf A}$. As an application, we describe the minimal non-abelian ideals of $\mathfrak b$.

math.CO↗
arXiv2018

Abelian ideals of a Borel subalgebra and root systems, II

Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$ and $\mathfrak{Ab}$ the set of abelian ideals of $\mathfrak b$. Let $Δ^+$ be the corresponding set of positive roots. We continue our study of combinatorial properties of the partition of $\mathfrak{Ab}$ parameterised by the long positive roots. In particular, the union of an arbitrary set of maximal abelian ideals is described, if $\mathfrak g\ne\mathfrak{sl}_n$. We also characterise the greatest lower bound of two positive roots, when it exists, and point out interesting subposets of $Δ^+$ that are modular lattices.

math.RT↗
arXiv2018

Semi-direct products involving $Sp_{2n}$ or $Spin_n$ with free algebras of symmetric invariants

This is a part of an ongoing project, the goal of which is to classify all semi-direct products $\mathfrak s=\mathfrak g{\ltimes} V$ such that $\mathfrak g$ is a simple Lie algebra, $V$ is a $\mathfrak g$-module, and $\mathfrak s$ has a free algebra of symmetric invariants. In this paper, we obtain such a classification for the representations of the orthogonal and symplectic algebras.

math.RT↗
arXiv2017

Abelian ideals and amazing roots

Let $\mathfrak g$ be a simple Lie algebra with a Borel subalgebra $\mathfrak b$. To any long positive root $γ$, one associates two ideals of $\mathfrak b$: the abelian ideal $I(γ)_{max}$ and not necessarily abelian ideal $I\langle{\succcurlyeq}γ\rangle$. It is known that $I(γ)_{max} \subset I\langle{\succcurlyeq}γ\rangle$, and $γ$ is said to be amazing if the equality holds. The set of amazing roots, $\mathcal A$, is closed under the operation `$\vee$' in $Δ^+$, and $γ\in\mathcal A$ is said to be primitive, if it cannot be written as $γ_1\veeγ_2$ with incomparable amazing roots $γ_1,γ_2$. We classify the amazing roots and notice that the number of primitive roots equals $\mathsf{rk}(\mathfrak g)$. Moreover, if $Π$ (resp. $\mathcal A_{\sf pr}$) is the set of simple (resp. primitive) roots, then there is a natural bijection $Π\longleftrightarrow \mathcal A_{\sf pr}$. We also describe the set $\mathcal A\cap{\mathcal H}$, where ${\mathcal H}$ is the Heisenberg subset of $Δ^+$.

math.RT↗