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Oksana Yakimova

Publications and source records attributed to Oksana Yakimova.

35 records · Page 2Linked to original sources
arXiv2017

Nilpotent gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

It has been shown that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L^1(N)^K commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N)^K of K-invariant Schwartz functions on N and the space S(Σ) of functions on the Gelfand spectrum Σ of L^1(N)^K which extend to Schwartz functions on Rd, once Σ is suitably embedded in Rd. We call this property (S). We present here a general bootstrapping method which allows to establish property (S) to new nilpotent pairs (N,K), once the same property is known for a class of quotient pairs of (N,K) and a K-invariant form of Hadamard formula holds on N. We finally show how our method can be recursively applied to prove property (S) for a significant class of nilpotent Gelfand pairs.

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arXiv2017

Semi-direct products of Lie algebras and covariants

The coadjoint representation of a connected algebraic group $Q$ with Lie algebra $\mathfrak q$ is a thrilling and fascinating object. Symmetric invariants of $\mathfrak q$ (= $\mathfrak q$-invariants in the symmetric algebra $S(\mathfrak q)$) can be considered as a first approximation to the understanding of the coadjoint action $(Q:\mathfrak q^*)$ and coadjoint orbits. In this article, we study a class of non-reductive Lie algebras, where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant-theoretic properties. If $G$ is a semisimple group with Lie algebra $\mathfrak g$ and $V$ is $G$-module, then we define $\mathfrak q$ to be the semi-direct product of $\mathfrak g$ and $V$. Then we are interested in the case, where the generic isotropy group for the $G$-action on $V$ is reductive and commutative. It turns out that in this case symmetric invariants of $\mathfrak q$ can be constructed via certain $G$-equivariant maps from $\mathfrak g$ to $V$ ("covariants").

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arXiv2017

On seaweed subalgebras and meander graphs in type D

In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}_n$ and computed their index using certain graphs, which we call type-${\sf A}$ meander graphs. Then the subalgebras of seaweed type, or just "seaweeds", have been defined by Panyushev (2001) for arbitrary reductive Lie algebras. Recently, a meander graph approach to computing the index in types ${\sf B}$ and ${\sf C}$ has been developed by the authors. In this article, we consider the most difficult and interesting case of type ${\sf D}$. Some new phenomena occurring here are related to the fact that the Dynkin diagram has a branching node.

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arXiv2016

On seaweed subalgebras and meander graphs in type C

In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}(n)$ and computed their index using certain graphs. In this article, those graphs are called type-A meander graphs. Then the subalgebras of seaweed type, or just "seaweeds", have been defined by Panyushev (2001) for arbitrary simple Lie algebras. Namely, if $\mathfrak p_1,\mathfrak p_2\subset\mathfrak g$ are parabolic subalgebras such that $\mathfrak p_1+\mathfrak p_2=\mathfrak g$, then $\mathfrak q=\mathfrak p_1\cap\mathfrak p_2$ is a seaweed in $\mathfrak g$. A general algebraic formula for the index of seaweeds has been proposed by Tauvel and Yu (2004) and then proved by Joseph (2006). If $\mathfrak p_1$ and $\mathfrak p_2$ are "adapted" to a fixed triangular decomposition of $\mathfrak g$, then $\mathfrak q$ is said to be standard. The number of standard seaweeds is finite. In this paper, elaborating on the "graphical" approach of Dergachev and Kirillov, we introduce the type-C meander graphs, i.e., the graphs associated with the standard seaweeds of $\mathfrak{sp}(2n)$, and give a formula for the index in terms of these graphs. We also note that the very same graphs can be used in case of the odd orthogonal Lie algebras. We also provide several applications of our formula to the Frobenius seaweeds in $\mathfrak{sp}(2n)$. In particular, using a natural partition of the set $\mathcal F_n$ of standard Frobenius seaweeds, we prove that $\# \mathcal F_n$ strictly increases for the passage from $n$ to $n+1$. The similar monotonicity question is open for the standard Frobenius seaweeds in $\mathfrak{sl}(n)$, even for the passage from $n$ to $n+2$.

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arXiv2013

On maximal commutative subalgebras of Poisson algebras associated with involutions of semisimple Lie algebras

For any involution $σ$ of a semisimple Lie algebra $\mathfrak g$, one constructs a non-reductive Lie algebra $\mathfrak k$, which is called a $\mathbb Z_2$-contraction of $\mathfrak g$. In this paper, we attack the problem of describing maximal commutative subalgebras of the Poisson algebra $S(\mathfrak k)$. This is closely related to the study of the coadjoint representation of $\mathfrak k$ and the set, $\mathfrak k^*_{reg}$, of the regular elements of $\mathfrak k^*$. By our previous results, in the context of $\mathbb Z_2$-contractions, the argument shift method provides maximal commutative subalgebras of $S(\mathfrak k)$ whenever $codim(\mathfrak k^* \setminus \mathfrak k^*_{reg})\ge 3$. Our main result here is that $codim(\mathfrak k^* \setminus \mathfrak k^*_{reg})\ge 3$ if and only if the Satake diagram of $σ$ has no trivial nodes. (A node is trivial, if it is white, has no arrows attached, and all adjacent nodes are also white.) The list of suitable involutions is provided. We also describe certain maximal commutative subalgebras of $S(\mathfrak k)$ if the (-1)-eigenspace of $σ$ in $\mathfrak g$ contains regular elements.

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arXiv2013

Parabolic contractions of semisimple Lie algebras and their invariants

Let $G$ be a connected semisimple algebraic group with Lie algebra $g$ and $P$ a parabolic subgroup of $G$ with $Lie(P)=p$. The parabolic contraction of $g$ is the semi-direct product of $p$ and a $p$-module $g/p$ regarded as an abelian ideal. We are interested in the polynomial invariants of the adjoint and coadjoint representations of $q$. In the adjoint case the algebra of invariants is easy to describe and turns out to be a graded polynomial algebra. The coadjoint case is more complicated. Here we found a connection between symmetric invariants of $q$ and symmetric invariants of centralisers $g_e\subset g$, where $e$ is a Richardson element with polarisation $p$. Using this connection and results of Panyushev, Premet, and Yakimova (see arxiv:0610049), we prove that the algebra of symmetric invariants of $q$ is free for all parabolics in types $A$ and $C$ and some parabolics in type $B$. The technique also applies to minimal parabolics in all types. For a Borel subalgebra, one gets a contraction of $g$ recently introduced by E.Feigin (arXiv:1007.0646 and arXiv:1101.1898) and studied from invariant-theoretic point of view in our previous paper (arxiv:1107.0702).

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arXiv2012

Nilpotent Gelfand pairs and spherical transforms of Schwartz functions III. Isomorphisms between Schwartz spaces under Vinberg's condition

Let (N,K) be a nilpotent Gelfand pair, i.e., N is a nilpotent Lie group, K a compact group of automorphisms of N, and the algebra D(N)^K of left-invariant and K-invariant differential operators on N is commutative. In these hypotheses, N is necessarily of step at most two. We say that (N,K) satisfies Vinberg's condition if K acts irreducibly on $n/[n,n]$, where n= Lie(N). Fixing a system D of d formally self-adjoint generators of D(N)^K, the Gelfand spectrum of the commutative convolution algebra L^1(N)^K can be canonically identified with a closed subset S_D of R^d. We prove that, on a nilpotent Gelfand pair satisfying Vinberg's condition, the spherical transform establishes an isomorphism from the space of $K$-invariant Schwartz functions on N and the space of restrictions to S_D of Schwartz functions in R^d.

math.FA↗
arXiv2012

One-parameter contractions of Lie-Poisson brackets

We consider contractions of Lie and Poisson algebras and the behaviour of their centres under contractions. A polynomial Poisson algebra A=K[W] is said to be of Kostant type, if its centre Z(A) is freely generated by homogeneous polynomials F_1,...,F_r such that they give Kostant's regularity criterion on W (d_xF_i are linear independent if and only if the Poisson tensor has the maximal rank at x). If the initial Poisson algebra is of Kostant type and F_i satisfy a certain degree-equality, then the contraction is also of Kostant type. The general result is illustrated by two examples. Both are contractions of a simple Lie algebra g corresponding to a decomposition g=h \oplus V, where h is a subalgebra. Here A=S(g)=K[g^*], Z(A)=S(g)^g, and the contracted Lie algebra is a semidirect product of h and an Abelian ideal isomorphic to g/h as an h-module. In the first example, h is a symmetric subalgebra and in the second, it is a Borel subalgebra and V is the nilpotent radical of an opposite Borel.

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arXiv2011

Coadjoint orbits of reductive type of seaweed Lie algebras

A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of its dual of reductive type, that is such that the quotient of its stabiliser by the centre of Q is a reductive subgroup of GL(q), where q=Lie(Q). Due to results of M. Duflo, coadjoint representation of a quasi-reductive Q possesses a so called maximal reductive stabiliser and knowing this subgroup, defined up to a conjugation in Q, one can describe all coadjoint orbits of reductive type. In this paper, we consider quasi-reductive parabolic subalgebras of simple complex Lie algebras as well as all seaweed subalgebras of gl(n) and describe the classes of their maximal reductive stabilisers.

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arXiv2011

A remarkable contraction of semisimple Lie algebras

Recently, E.Feigin introduced a very interesting contraction $\mathfrak q$ of a semisimple Lie algebra $\mathfrak g$ (see arXiv:1007.0646 and arXiv:1101.1898). We prove that these non-reductive Lie algebras retain good invariant-theoretic properties of $\mathfrak g$. For instance, the algebras of invariants of both adjoint and coadjoint representations of $\mathfrak q$ are free, and also the enveloping algebra of $\mathfrak q$ is a free module over its centre.

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arXiv2011

Nilpotent Gelfand pairs and spherical transforms of Schwartz functions II. Taylor expansion on singular sets

This paper is a continuation of [8], in the direction of proving the conjecture that the spherical transform on a nilpotent Gelfand pair (N,K) establishes an isomorphism between the space of K-invariant Schwartz functions on N and the space of Schwartz functions restricted to the Gelfand spectrum properly embedded in a Euclidean space. We prove a result, of independent interest for the representation theoretical problems that are involved, which can be viewed as a generalised Hadamard lemma for K-invariant functions on N. The context is that of nilpotent Gelfand pairs satisfying Vinberg's condition. This means that the Lie algebra n of N (which is step 2) decomposes as a direct sum of [n,n] and a K-invariant irreducible subspace.

math.AC↗
arXiv2010

On the derived algebra of a centraliser

Let $\g$ be a classical Lie algebra, $e$ a nilpotent of $\g$ element and $\gt g_e$ the centraliser of $e$ in $\g$. We prove that $\g_e=[\g_e,\g_e]$ if and only if $e$ is rigid. It is also shown that if $e$ is contained in $[\g_e,\g_e]$, then the nilpotent radical of $\g_e$ coincides with $[\g(1)_e,\g_e]$, where $\g(1)_e$ is an eigenspace of a characteristic of $e$ with the eigenvalue 1.

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arXiv2010

Nilpotent Gelfand Pairs and Spherical Transforms of Schwartz Functions I. Rank-One Actions on the Centre

The spectrum of a Gelfand pair of the form (K lx N, K), where N is a nilpotent group, can be embedded in a Euclidean space Rd . The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on Rd has been proved already when N is a Heisenberg group and in the case where N = N3,2 is the free two-step nilpotent Lie group with three generators, with K = SO3 [2, 3, 11]. We prove that the same identification holds for all pairs in which the K-orbits in the centre of N are spheres. In the appendix, we produce bases of K-invariant polynomials on the Lie algebra n of N for all Gelfand pairs (K lx N, K) in Vinberg's list [27, 30]. (The references numbers refers to the bibliography at the end of the article)

math.FA↗
arXiv2008

Surprising properties of centralisers in classical Lie algebras

Let $g$ be a classical Lie algebra, i.e., either $gl_n$, $sp_n$, or $so_n$ and let $e\in g$ be a nilpotent element. We study various properties of centralisers $g_e$. The first four sections deal with rather elementary questions, like the centre of $g_e$, commuting varieties associated with $g_e$, or centralisers of commuting pairs. The second half of the paper addresses problems related to different Poisson structures on $g_e^*$ and symmetric invariants of $g_e$.

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arXiv2006

Symmetric pairs and associated commuting varieties

We obtain a series of new results on the problem of irreducibility of commuting varieties associated with symmetric pairs or, in other words, $Z_2$-graded simple Lie algebras. In particular, we present many examples of reducible commuting varieties and show that the number of irreducible components can be arbitrarily large.

math.AG↗
arXiv2004

Principal Gelfand pairs

Let X=G/K be a connected Riemannian homogeneous space of a real Lie group G. The homogeneous space X is called commutative if the algebra of G-invariant differential operators on X is commutative. We prove an effective commutativity criterion and classify commutative spaces under two mild technical constraints. In particular, we obtain several new examples of commutative homogeneous spaces that are not of Heisenberg type.

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