Searcharxiv⌕ Search

arXiv subjects

Mikhail V. Ignatyev

Publications and source records attributed to Mikhail V. Ignatyev.

16 recordsLinked to original sources
arXiv2020

Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$, $\mathfrak{n}$ be the nilradical of $\mathfrak{b}$, and $U(\mathfrak{n})$ be the universal enveloping algebra of $\mathfrak{n}$. We study primitive ideals of $U(\mathfrak{n})$. Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center $Z(\mathfrak{n})$ of $U(\mathfrak{n})$. We present an explicit characterization of the centrally generated primitive ideals of $U(\mathfrak{n})$ in terms of the Dixmier map and the Kostant cascade in the case when $\mathfrak{g}$ is a simple algebra of exceptional type. (For classical simple Lie algebras, a similar characterization was obtained by Ivan Penkov and the first author.) As a corollary, we establish a classification of centrally generated primitive ideals of $U(\mathfrak{n})$ for an arbitrary semisimple algebra $\mathfrak{g}$.

math.RT↗
arXiv2020

On tangent cones to Schubert varieties in type $E$

We consider tangent cones to Schubert subvarieties of the flag variety $G/B$, where $B$ is a Borel subgroup of a reductive complex algebraic group $G$ of type $E_6$, $E_7$ or $E_8$. We prove that if $w_1$ and $w_2$ form a good pair of involutions in the Weyl group $W$ of $G$ then the tangent cones $C_{w_1}$ and $C_{w_2}$ to the corresponding Schubert subvarieties of $G/B$ do not coincide as subschemes of the tangent space to $G/B$ at the neutral point.

math.AG↗
arXiv2020

The orbit method for locally nilpotent infinite-dimensional Lie algebras

Let $\mathfrak{n}$ be a locally nilpotent infinite-dimensional Lie algebra over $\mathbb{C}$. Let $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$ be its universal enveloping algebra and its symmetric algebra respectively. Consider the Jacobson topology on the primitive spectrum of $\mathrm{U}(\mathfrak{n})$ and the Poisson topology on the primitive Poisson spectrum of $\mathrm{S}(\mathfrak{n})$. We provide a homeomorphism between the corresponding topological spaces (on the level of points, it gives a bijection between the primitive ideals of $\mathrm{U}(\mathfrak{n})$ and $\mathrm{S}(\mathfrak{n})$). We also show that all primitive ideals of $\mathrm{S}(\mathfrak{n})$ from an open set in a properly chosen topology are generated by their intersections with the Poisson center. Under the assumption that $\mathfrak{n}$ is a nil-Dynkin Lie algebra, we give two criteria for primitive ideals $I(λ)\subset\mathrm{S}(\mathfrak{n})$ and $J(λ)\subset\mathrm{U}(\mathfrak{n})$, $λ\in\mathfrak{n}^*$, to be nonzero. Most of these results generalize the known facts about primitive and Poisson spectrum for finite-dimensional nilpotent Lie algebras (but note that for a finite-dimensional nilpotent Lie algebra all primitive ideals $I(λ)$, $J(λ)$ are nonzero).

math.RT↗
arXiv2018

Centrally generated primitive ideals of $U(\mathfrak{n})$ in types $B$ and $D$

We study the centrally generated primitive ideals of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the (locally) nilpotent radical of a (splitting) Borel subalgebra of a simple complex Lie algebra $\mathfrak{g}=\mathfrak{o}_{2n+1}(\mathbb{C})$, $\mathfrak{o}_{2n}(\mathbb{C})$, $\mathfrak{o}_{\infty}(\mathbb{C})$. In the infinite-dimensional setting, there are infinitely many isomorphism classes of Lie algebras $\mathfrak{n}$, and we fix $\mathfrak{n}$ with "largest possible" center of $U(\mathfrak{n})$. We characterize the centrally generated primitive ideals of $U(\mathfrak{n})$ in terms of the Dixmier map and the Kostant cascade.

math.RT↗
arXiv2018

Gradedness of the set of rook placements in $A_{n-1}$

A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in $A_{n-1}$ with respect to a slightly different order and prove that this poset is graded.

math.RT↗
arXiv2018

On involutions in the Weyl group and $B$-orbit closures in the orthogonal case

We study coadjoint $B$-orbits on $\mathfrak{n}^*$, where $B$ is a Borel subgroup of a complex orthogonal group $G$, and $\mathfrak{n}$ is the Lie algebra of the unipotent radical of $B$. To each basis involution $w$ in the Weyl group $W$ of $G$ one can assign the associated $B$-orbit $Ω_w$. We prove that, given basis involutions $σ$, $τ$ in $W$, if the orbit $Ω_σ$ is contained in the closure of the orbit $Ω_τ$ then $σ$ is less than or equal to $τ$ with respect to the Bruhat order on $W$. For a basis involution $w$, we also compute the dimension of $Ω_w$ and present a conjectural description of the closure of $Ω_w$.

math.RT↗
arXiv2017

Ind-varieties of generalized flags: a survey of results

This is a review of results on the structure of the homogeneous ind-varieties $G/P$ of the ind-groups $G=\mathrm{GL}_{\infty}(\mathbb{C})$, $\mathrm{SL}_{\infty}(\mathbb{C})$, $\mathrm{SO}_{\infty}(\mathbb{C})$, $\mathrm{Sp}_{\infty}(\mathbb{C})$, subject to the condition that $G/P$ is a inductive limit of compact homogeneous spaces $G_n/P_n$. In this case the subgroup $P\subset G$ is a splitting parabolic subgroup of $G$, and the ind-variety $G/P$ admits a "flag realization". Instead of ordinary flags, one considers generalized flags which are, generally infinite, chains $\mathcal{C}$ of subspaces in the natural representation $V$ of $G$ which satisfy a certain condition: roughly speaking, for each nonzero vector $v$ of $V$ there must be a largest space in $\mathcal{C}$ which does not contain $v$, and a smallest space in $\mathcal{C}$ which contains $v$. We start with a review of the construction of the ind-varieties of generalized flags, and then show that these ind-varieties are homogeneous ind-spaces of the form $G/P$ for splitting parabolic ind-subgroups $P\subset G$. We also briefly review the characterization of more general, i.e. non-splitting, parabolic ind-subgroups in terms of generalized flags. In the special case of an ind-grassmannian $X$, we give a purely algebraic-geometric construction of $X$. Further topics discussed are the Bott--Borel--Weil Theorem for ind-varieties of generalized flags, finite-rank vector bundles on ind-varieties of generalized flags, the theory of Schubert decomposition of $G/P$ for arbitrary splitting parabolic ind-subgroups $P\subset G$, as well as the orbits of real forms on $G/P$ for $G=\mathrm{SL}_{\infty}(\mathbb{C})$.

math.AG↗
arXiv2017

Real group orbits on flag ind-varieties of $\mathrm{SL}(\infty,\mathbb{C})$

We consider the complex ind-group $G=\mathrm{SL}(\infty,\mathbb{C})$ and its real forms $G^0=\mathrm{SU}(\infty,\infty)$, $\mathrm{SU}(p,\infty)$, $\mathrm{SL}(\infty,\mathbb{R})$, $\mathrm{SL}(\infty,\mathbb{H})$. Our main objects of study are the $G^0$-orbits on an ind-variety $G/P$ for an arbitrary splitting parabolic ind-subgroup $P\subset G$. We prove that the intersection of any $G^0$-orbit on $G/P$ with a finite-dimensional flag variety $G_n/P_n$ from a given exhaustion of $G/P$ via $G_n/P_n$ for $n\to\infty$, is a single $(G^0\cap G_n)$-orbit. We also characterize all ind-varieties $G/P$ on which there are finitely many $G^0$-orbits, and provide criteria for the existence of open and closed $G^0$-orbits on $G/P$ in the case of infinitely many $G^0$-orbits.

math.AG↗
arXiv2015

Tangent cones to Schubert varieties in types $A_n$, $B_n$ and $C_n$

Let $G$ be a complex reductive group, $T$ be a maximal torus of $G$, $B$ be a Borel subgroup of $G$ containing $T$, $W$ be the Weyl group of $G$ with respect to $T$. To each element $w$ of $W$ one can associate the Schubert subvariety $X_w$ of the flag variety $G/B$, the tangent cone to $X_w$ at the identity point $p$ considered as a subcheme of the tangent space $T_p(G/B)$, and the reduced tangent cone to $X_w$ at $p$ considered as a subvariety of $T_p(G/B)$. Let $w_1$, $w_2$ be distinct involutions in $W$. We prove that if $G$ is of type $B_n$ or $C_n$, then the tangent cones corresponding to $w_1$ and $w_2$ are distinct. We also prove that if $G$ is of type $A_n$ or $C_n$, then the reduced tangent cones corresponding to $w_1$ and $w_2$ are distinct.

math.AG↗
arXiv2013

Rook placements in $A_n$ and combinatorics of $B$-orbit closures

Let $G$ be a complex reductive group, $B$ be a Borel subgroup of G, $\nt$ be the Lie algebra of the unipotent radical of $B$, and $\nt^*$ be its dual space. Let $Φ$ be the root system of $G$, and $Φ^+$ be the set of positive roots with respect to $B$. A subset of $Φ^+$ is called a rook placement if it consists of roots with pairwise non-positive inner products. To each rook placement $D$ one can associate the coadjoint orbit $Ω_D$ of $B$ in $\nt^*$. By definition, $Ω_D$ is the orbit of $f_D$, where $f_D$ is the sum of root covectors corresponging to the roots from $D$. We find the dimension of $Ω_D$ and construct a polarization of $\nt$ at $f_D$. We also study the partial order on the set of rook placements induced by the incidences among the orbits associated with rook placements.

math.RT↗
arXiv2012

Kostant--Kumar polynomials and tangent cones to Schubert varieties for involutions in $A_n$, $F_4$ and $G_2$

Let $G$ be a reductive complex algebraic group, $T$ a maximal torus of $G$, $B$ a Borel subgroup of $G$ containing $T$, $Φ$ the root system of $G$ w.r.t. $T$, $W$ the Weyl group of $Φ$. Denote by $\Fo = G/B$ the flag variety, by $X_w$ the Schubert subvariety of $\Fo$ associated with an element $w\in W$, and by $C_w$ the tangent cone to $X_w$ at the point $p = eB$. Then $C_w$ is a subscheme of the tangent space $T_pX_w\subseteq T_p\Fo$. Suppose $w$, $w'$ are distinct involutions in $W$. Using the so-called Kostant--Kumar polynomials, we show that if every irreducible component of $Φ$ is of type $A_n$, $F_4$ or $G_2$, then $C_w$ and $C_{w'}$ do not coincide.

math.RT↗
arXiv2011

Combinatorics of $B$-orbits and Bruhat--Chevalley order on involutions

Let $B$ be the group of invertible upper-triangular complex $n\times n$ matrices, $\mathfrak{u}$ the space of upper-triangular complex matrices with zeroes on the diagonal and $\mathfrak{u}^*$ its dual space. The group $B$ acts on $\mathfrak{u}^*$ by $(g.f)(x)=f(gxg^{-1})$, $g\in B$, $f\in\mathfrak{u}^*$, $x\in\mathfrak{u}$. To each involution $σ$ in $S_n$, the symmetric group on $n$ letters, one can assign the $B$-orbit $Ω_σ\in\mathfrak{u}^*$. We present a combinatorial description of the partial order on the set of involutions induced by the orbit closures. The answer is given in terms of rook placements and is dual to A. Melnikov's results on $B$-orbits on $\mathfrak{u}$. Using results of F. Incitti, we also prove that this partial order coincides with the restriction of the Bruhat--Chevalley order to the set of involutions.

math.RT↗
arXiv2011

The Bruhat--Chevalley order on involutions of the hyperoctahedral group and combinatorics of $B$-orbit closures

Let $G$ be the symplectic group, $Φ=C_n$ its root system, $B\subset G$ its standard Borel subgroup, $W$ the Weyl group of $Φ$. To each involution $σ\in W$ one can assign the $B$-orbit $Ω_σ$ contained in the dual space of the Lie algebra of the unipotent radical of $B$. We prove that $Ω_σ$ is contained in the Zariski closure of $Ω_τ$ if and only of $σ\leqτ$ with respect to the Bruhat--Chevalley order. We also prove that $\dimΩ_σ$ is equal to $l(σ)$, the length of $σ$ in $W$.

math.RT↗
arXiv2011

Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups

Let $Φ$ be a classical root system and $k$ be a field of sufficiently large characteristic. Let $G$ be the classical group over $k$ with the root system $Φ$, $U$ be its maximal unipotent subgroup and $\mathfrak{u}$ be the Lie algebra of $U$. Let $D$ be an orthogonal subset of $Φ$ and $Ω$ be a coadjoint orbit of $U$ associated with $D$. We construct a polarization of $\mathfrak{u}$ at the canonical form on $Ω$. We also find the dimension of $Ω$ in terms of the Weyl group of $Φ$. As a corollary, we determine all possible dimensions of irreducible complex represenations of the group $U$ for the case of finite field $k$.

math.RT↗
arXiv2011

Orthogonal subsets of root systems and the orbit method

Let $k$ be the algebraic closure of a finite field, $G$ a Chevalley group over $k$, $U$ the maximal unipotent subgroup of $G$. To each orthogonal subset $D$ of the root system of the group $G$ and each set $ξ$ of $|D|$ non-zero scalars from $k$ one can assign the coadjoint orbit of the group $U$. We prove that the dimension of such an orbit does not depend on $ξ$. We also give an upper bound of the dimension in terms of the Weyl group.

math.RT↗