Rutwig Campoamor-Stursberg
We describe the invariants for the coadjoint representation of all real Lie algebras with nontrivial Levi decomposition up to dimension eight.
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We determine all the contractions within the class of finite-dimensional real Lie algebras whose coadjoint orbits have dimensions $\le2$.
Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were the subject of much study. However, they remain hard for calculation and application. We propose simple solutions for the following problems: an explicit parameterization of the orbit by means of a generalized stereographic projection, obtaining a Kählerian structure on the orbit, introducing basis two-forms for the cohomology group of the orbit.
Invariant star products are constructed on minimal coadjoint orbits of all the simple Lie algebras. Explicit expressions are given for the generators of the Joseph ideals and the associated infinitesimal characters.
We present a one-to-one correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits for a class of pro-Lie groups including all connected locally compact nilpotent groups and arbitrary infinite direct products of nilpotent Lie groups.
In this article, we give a simple explicit construction of an affine slice for the coadjoint action of a certain class of biparabolic (also called seaweed) subalgebras of a semisimple Lie algebra over an algebraically closed field of characteristic zero. In particular, this class includes all Borel subalgebras.
We describe the admissible coadjoint orbits of a compact connected Lie group and their spin-c quantization.
Let $L$ be a restricted Cartan type Lie algebra over an algebraically closed field $k$ of characteristic $p>3$, and let $G$ denote the automorphism group of $L$. We prove that there are no nontrivial invariants of $L^*$ under the coadjoint action of $G$, i.e., $k[L^*]^G=k$. This property characterises the Cartan type algebras among the restricted simple Lie algebras.
Let {\Lnk} be the class of all $n$-dimensional real solvable Lie algebras having $k$-dimensional derived ideals. In 2020 the authors et al. gave a classification of all non 2-step nilpotent Lie algebras of {\Li}. We propose in this paper to study representations of these Lie algebras as well as their corresponding connected and simply connected Lie groups. That is, for each algebra, we give an upper bound of the minimal degree of a faithful representation. Then, we give a geometrical description of coadjoint orbits of corresponding groups. Moreover, we show that the characteristic property of …
We study the coadjoint representation of contractions of reductive Lie algebras associated with symmetric decompositions. Let $\frak g=\frak g_0\oplus \frak g_1$ be a symmetric decomposition of a reductive Lie algebra $\frak g$. Then the semi-direct product of $\frak g_0$ and the $\frak g_0$-module $\frak g_1$ is a contraction of $\frak g$. We conjecture that these contractions have many properties in common with reductive Lie algebras. In particular, it is proved that in many cases the algebra of invariants is polynomial. We also discuss the so-called "codim--2 property" for coadjoint represe…
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 among nilpotent coadjoint orbits in the dual of Lie algebras of type $B,C,F_4$ in characteristic $2$ and in the dual of Lie algebra of type $G_2$ in characteristic $3$. In each case we give an explicit description of the nilpotent pieces in the dual defined in \cite{CP}, which are in general unions of nilpotent coadjoint orbits, coincide with the earlier case…
A complete description of the coadjoint orbits for A_{n-1}^{+}, the nilpotent Lie algebra of n-by-n strictly upper triangular matrices, has not yet been obtained, though there has been steady progress on it ever since the orbit method was devised. We apply methods developed by Andre' to find defining equations for the elementary coadjoint orbits for the maximal nilpotent Lie subalgebras of the orthogonal Lie algebras, and we also determine all the possible dimensions of coadjoint orbits in the case of A_{n-1}^{+}.
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 alg…
Square-integrable representations and the coadjoint action of solvable Lie groups
本文利用余伴随作用的广义轨道刻画可解李群的平方可积表示，建立 I 型平方可积表示拟等价类与单连通开轨道的双射，并分析本原理想空间性质。
We determine the space of primary ideals in the group algebra $L^1(G)$ of a connected nilpotent Lie group by identifying for every $π\in\hat G $ the family ${\mathcal I}^π$ of primary ideals with hull $\{π\}$ with the family of invariant polynomials of a certain finite dimensional subspace ${\mathcal P}_Q^π$ of the space of polynomials ${\mathcal P}(G) $ on $G $.
The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems for quadratic Hamiltonians of $\mathbb R^{2n}$ on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra $\mathfrak g$ that admits an ad-invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that one on $\mathbb R^{2n}$. This system is a Lax pair equation whose solution can be computed with help of the Adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits …
Nilpotent Lie groups with stepwise square integrable representations were recently investigated by J.A.~Wolf. We give an alternative approach to these representations by relating them to the stratifications of the duals of nilpotent Lie algebras, thus proving that they correspond to a subset with relative Hausdorff topology, dense interior, and total Plancherel measure in the unitary dual of the Lie group under consideration.
Let $G$ be a semisimple Lie group and let $\g =\n_- +\hh +\n$ be a triangular decomposition of $\g= \hbox{Lie}\,G$. Let $\b =\hh +\n$ and let $H,N,B$ be Lie subgroups of $G$ corresponding respectively to $\hh,\n$ and $\b$. We may identify $\n_-$ with the dual space to $\n$. The coadjoint action of $N$ on $\n_-$ extends to an action of $B$ on $\n_-$. There exists a unique nonempty Zariski open orbit $X$ of $B$ on $\n_-$. Any $N$-orbit in $X$ is a maximal coadjoint orbit of $N$ in $\n_-$. The cascade of orthogonal roots defines a cross-section $\r_-^{\times}$ of the set of such orbits leading to…
On the regular representation of solvable Lie groups with open coadjoint quasi-orbits
本研究对正则表示为因子表示的连通单连通可解李群给出李理论内蕴刻画，得到超有限II$_\text{∞}$因子标准表示的几何模型，并证明具有开余伴随轨道的这类群其正则表示为I型。
Characteristically nilpotent Lie groups with flat coadjoint orbits
针对模中心平方可积但中心商不允许伸缩族的情形，本文构造了维数11、具有平坦余伴随轨道的特征幂零李群$G(α,β)$双参数族。
Let $\mathfrak{g}$ be a finite-dimensional real or complex Lie algebra, and let $μ\in \mathfrak{g}^{*}$. In the first part of the paper, the relation is discussed between the derived algebra of the stabilizer of $μ$ and the set of coadjoint orbits which have the same dimension as the orbit of $μ$. In the second part, semisimple Lie algebras are considered, and the relation is discussed between the derived algebra of a centralizer and sheets.
The purpose of this paper is to apply deformation quantization to the study of the coadjoint orbit method in the case of real reductive groups. We first prove some general results on the existence of equivariant deformation quantization of vector bundles on closed Lagrangian subvarieties, which lie in smooth symplectic varieties with Hamiltonian group actions. Then we apply them to orbit method and construct nontrivial irreducible Harish-Chandra modules for certain coadjoint orbits. Our examples include new geometric construction of representations associated to certain orbits of real exceptio…
Given a quasi-Hermitian semisimple Lie algebra, we describe possible spectra of the sum of two admissible elements from its dual vector space.
In this paper, we develop the fundamentals of Lie-Poisson theory for direct limits $G=\dirlim G_{n}$ of complex algebraic groups $G_{n}$ and their Lie algebras $\fg=\dirlim \fg_{n}$. We show that $\fg^{*}=\invlim\fg_{n}^{*}$ has the structure of a Poisson provariety and that each coadjoint orbit of $G$ on $\fg^{*}$ has the structure of an ind-variety. We construct a weak symplectic form on every coadjoint orbit and prove that the coadjoint orbits form a weak symplectic foliation of the Poisson provariety $\fg^{*}$. We apply our results to the specific setting of $G=GL(\infty)=\dirlim GL(n,\C)$…
A regular factor is a factor algebra of the unitriangular Lie algebra with respect to some regular ideal. In the paper we construct system of generators of the field of invariants for the coadjoint representation of an arbitrary regular factor.
We correspond to any factor algebra of the unitriangular Lie algebra with respect to a regular ideal some permutation. In terms of this permutation one can construct a diagram, that allows to calculate index and maximal dimension of coadjoint representation.
Let $G$ be a simply connected, nilpotent Lie group with Lie algebra $\gee$. The group $G$ acts on the dual space $\gee^*$ by the coadjoint action. %% which partitions $\gee^*$ into coadjoint orbits. By the orbit method of Kirillov, the simple unitary representations of $G$ are in bijective correspondence with the coadjoint orbits in $\gee^*$, which in turn are in bijective correspondence with the primitive ideals of the universal enveloping algebra of $\gee$. The number of simple $\gee$-modules which have a common eigenvector for a particular subalgebra of $\gee$ and are annihilated by a parti…
Coadjoint orbits and multiplicity free spaces of compact Lie groups are important examples of symplectic manifolds with Hamiltonian groups actions. Constructing action-angle variables on these spaces is a challenging task. A fundamental result in the field is the Guillemin-Sternberg construction of Gelfand-Zeitlin integrable systems for the groups $K=U(n), SO(n)$. Extending these results to groups of other types is one of the goals of this paper. Partial tropicalizations are Poisson spaces with constant Poisson bracket built using techniques of Poisson-Lie theory and the geometric crystals of …
Consider the restriction of an irreducible unitary representation $π$ of a Lie group $G$ to its subgroup $H$. Kirillov's revolutionary idea on the orbit method suggests that the multiplicity of an irreducible $H$-module $ν$ occurring in the restriction $π|_H$ could be read from the coadjoint action of $H$ on $O^G \cap pr^{-1}(O^H)$ provided $π$ and $ν$ are "geometric quantizations" of a $G$-coadjoint orbit $O^G$ and an $H$-coadjoint orbit $O^H$,respectively, where $pr: \sqrt{-1} g^{\ast} \to \sqrt{-1} h^{\ast}$ is the projection dual to the inclusion $h \subset g$ of Lie algebras. Such results…
Let $G$ be a complex simply-connected semisimple Lie group and let $\frak{g}= Lie G$. Let $\frak{g} = \frak{n}_- +\frak{h} + \frak{n}$ be a triangular decomposition of $\frak{g}$. One readily has that $Cent\,U({\frak n})$ is isomorphic to the ring $S({\frak n})^{\frak\n}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogonal roots, some time ago we proved that $S({\frak n})^{{\frak n}$ is a polynomial ring $\Bbb C [ξ_1,...,ξ_m]$ where $m$ is the cardinality of ${\cal B}$. Using this result we establish that the maximal coadjoint of $N = exp \frak {n}$ has codimension $m$…
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 se…
Let $\g$ be a simple Lie algebra of type A or C. We show that the coadjoint representation of any seaweed subalgebra of $\g$ has some properties similar to that of the adjoint representation of a reductive Lie algebra. Namely, a) the field of invariants is rational and b) there exists a generic stabiliser whose identity component is a torus. Our main tool for this is a result about coadjoint representations of some N-graded Lie algebras, which can be regarded as an extension of Rais' theorem for the index of semi-direct products. For all other simple types, we give a uniform description of a p…
Let $G$ be a complex simply-connected semisimple Lie group and let $\frak{g}= Lie G$. Let $\frak{g} = \frak{n}_- +\frak{h} + \frak{n}$ be a triangular decomposition of $\frak{g}$. One readily has that $Cent\,U({\frak n})$ is isomorphic to the ring $S({\frak n})^{\frak\n}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogonal roots, some time ago we proved that $S({\frak n})^{{\frak n}$ is a polynomial ring $\Bbb C [ξ_1,...,ξ_m]$ where $m$ is the cardinality of ${\cal B}$. Using this result we establish that the maximal coadjoint of $N = exp \frak {n}$ has codimension $m$…
Let G_R be a Lie group acting on an oriented manifold M, and let $ω$ be an equivariantly closed form on M. If both G_R and M are compact, then the integral $\int_M ω$ is given by the fixed point integral localization formula (Theorem 7.11 in [BGV]). Unfortunately, this formula fails when the acting Lie group G_R is not compact: there simply may not be enough fixed points present. A proposed remedy is to modify the action of G_R in such a way that all fixed points are accounted for. Let G_R be a real semisimple Lie group, possibly noncompact. One of the most important examples of equivariantly …
A unitary representation of a, possibly infinite dimensional, Lie group G is called semi-bounded if the corresponding operators idπ(x) from the derived representations are uniformly bounded from above on some non-empty open subset of the Lie algebra g. Not every Lie group has non-trivial semibounded unitary representations, so that it becomes an important issue to decide when this is the case. In the present paper we describe a complete solution of this problem for the class of generalized oscillator groups, which are semidirect products of Heisenberg groups with a one-parameter group γ. For t…
Cotangent Bundles as Coadjoint Orbits and Asymptotic Character Formulas
该研究将余切丛几何量子化纳入Kirillov轨道方法框架，通过微分同胚群构造的无限维李群实现余切丛为余伴随轨道，并借助切群胚得到两个渐近特征公式。
Consider a compact Lie group and a closed subgroup. Generalizing a result of Klyachko, we give a necessary and sufficient criterion for a coadjoint orbit of the subgroup to be contained in the projection of a given coadjoint orbit of the ambient group. The criterion is couched in terms of the ``relative'' Schubert calculus of the flag varieties of the two groups.
In this paper we develop a groupoid approach to some basic topological properties of dual spaces of solvable Lie groups using suitable dynamical systems related to the coadjoint action. One of our main results is that the coadjoint dynamical system of any exponential solvable Lie group is a piecewise pullback of group bundles. Our dynamical system approach to solvable Lie groups also allows us to construct some new examples of connected solvable Lie groups whose $C^*$-algebras admit faithful irreducible representations.
Kirillov's orbit theory provides a powerful tool for the investigation of irreducible unitary representations of many classes of Lie groups. In a previous paper we used a modification hereof, called monomial linearisation, to construct a monomial basis of the regular representation of $p$-Sylow subgroups $U$ of the finite classical groups of untwisted type. In this sequel to this article we determine the stabilizers of special orbit generators and show, that for the groups of Lie type ${\mathfrak{B}}_n$ and ${\mathfrak{D}}_n$ a subclass of the orbit modules decompose the $U$-modules affording …
We study the class of algebraic Lie algebras for which the generic stabilizer of the coadjoint action is reductive modulo the center.
We provide an explicit construction of quasi-invariant measures on polarized coadjoint orbits of a Lie group G. The use of specific (trivial) central extensions of G by the multiplicative group ${R}^+$ allows us to restore the strict invariance of the measures and, accordingly, the unitarity of the quantization of coadjoint orbits. As an example, the representations of $SL(2,\Real)$ are recovered.
We develop a semigroup approach to representation theory for pro-Lie groups satisfying suitable amenability conditions. As an application of our approach, we establish a one-to-one correspondence between equivalence classes of unitary irreducible representations and coadjoint orbits for a class of pro-Lie groups including all connected locally compact nilpotent groups and arbitrary infinite direct products of nilpotent Lie groups. The usual $C^*$-algebraic approach to group representation theory positivey breaks down for infinite direct products of non-compact locally compact groups, hence the…
We describe boundedness and compactness properties for the operators obtained by the Weyl-Pedersen calculus in the case of the irreducible unitary representations of nilpotent Lie groups that are associated with flat coadjoint orbits. We use spaces of smooth symbols satisfying appropriate growth conditions expressed in terms of invariant differential operators on the coadjoint orbit under consideration. Our method also provides conditions for these operators to belong to one of the Schatten ideals of compact operators. In the special case of the Schrödinger representation of the Heisenberg gro…
Partial convex hulls of coadjoint orbits and degrees of invariants
本文通过引入部分凸包概念定义紧李群余伴随轨道的两个数值不变量，证明其与 Littlewood-Richardson 锥相关，并给出不可约酉表示上不变多项式次数的下界。
Nilpotent orbits of classical Lie algebras stable under negation
本文研究经典李代数中 nilpotent 轨道在 negation 下的稳定性，证明在这些轨道上不存在 Gibbs 状态，并对所有经典李代数进行了分类。
The index of a finite-dimensional Lie algebra $g$ is the minimum of dimensions of stabilisers $g_α$ of elements $α\in g^*$. Let $g$ be a reductive Lie algebra and $z(x)$ a centraliser of a nilpotent element $x\in g$. Elashvili has conjectured that the index of the centraliser $z(x)$ equals the index of $g$, i.e., the rank of $g$. Here Elashvili's conjecture is proved for reductive Lie algebras of classical type. It is shown that in cases $g=gl_n$ and $g=sp_{2n}$ the coadjoint action of $z(x)$ has a generic stabiliser. Also, we give an example of a nilpotent element $x\in so_8$ such that the co…
The goal of this diploma thesis is to give a detailed description of Kirillov's Orbit Method for the case of compact connected Lie groups. The theory of Kirillov aims at finding all irreducible unitary representations of a given Lie group $G$. The first chapter is intended to recall some facts about Lie groups. The most important result is that for a functional $λ$ on the Lie algebra $\mathfrak t$ of a maximal torus in a compact connected Lie group the stabilizer is of the form $C(T_1)$ for a smaller torus $T_1$. This allows us to define a complex structure on the coadjoint orbit $\mathcal O_λ…
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 c…
Recently V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from noncommutative symplectic geometry. In this note we generalize this argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg.
CFT characters from localization
该研究类比有限维李群的余伴随轨道方法，提出从2维(S)CFT的余伴随轨道几何计算其手征代数特征标的通用程序，可简洁重推Virasoro等代数的特征标。
Special unipotent representations and the coadjoint orbit method
本文针对线性实约化群或元辛群，基于相关循环对特殊幂幺表示进行几何分类，通过变形量子化给出其哈里什 - 钱德拉模的统一构造，证明相关模有不可约循环且酉可化，突出卢茨蒂格猜想应用。
In this work we generalize the concept of product by generators to the class of solvable Lie algebras. We analyze the number of invariants by the coadjoint representation by means of Maurer-Cartan equations and give some applications to product structures on Lie algebras.
The main result of this article is an application of the theory of invariant convex cones of Lie algebras to the study of unitary representations of Lie supergroups. It also includes an exposition of recent results of the second author on the classification of irreducible unitary representations of nilpotent Lie supergroups using the method of coadjoint orbits.
We discuss groups corresponding to Kohno Lie algebra of infinitesimal braids and actions of such groups. We construct homomorphisms of Lie braid groups to the group of symplectomorphisms of the space of point configurations in $R^3$ and to groups of symplectomorphisms of coadjoint orbits of SU(n).
In this paper, first we show that there is a Hom-Lie algebra structure on the set of $(σ,σ)$-derivations of a commutative algebra. Then we construct dual representations of a representation of a Hom-Lie algebra. We introduce the notions of a Manin triple for Hom-Lie algebras and a purely Hom-Lie bialgebra. Using the coadjoint representation, we show that there is a one-to-one correspondence between Manin triples for Hom-Lie algebras and purely Hom-Lie bialgebras. Finally, we study coboundary purely Hom-Lie bialgebras and construct solutions of the classical Hom-Yang-Baxter equations in some sp…
We present some recent results on smooth vectors for unitary irreducible representations of nilpotent Lie groups. Applications to the Weyl-Pedersen calculus of pseudo-differential operators with symbols on the coadjoint orbits are also discussed.
The purpose of this paper is to prove Duflo's conjecture for $(G,π, AN)$ where $G$ is a simple Lie group of Hermitian type and $π$ is a discrete series of $G$ and $AN$ is the maximal exponential solvable subgroup for an Iwasawa decomposition $G=KAN$. This is essentially reduced from the following general theorem we prove in this paper: let $G$ be a connected semisimple Lie group . Then a strongly elliptic $G$-coadjoint orbit $\mathcal{O}$ is holomorphic if and only if $\text{p}(\mathcal{O})$ is an open $AN$-coadjoint orbit, where $\text{p} : \mathfrak{g}^* \longrightarrow (\mathfrak{a}\oplus\m…
Let p be a maximal truncated parabolic subalgebra of a simple Lie Algebra. It was shown in many cases that the Poisson centre Y(p) is a polynomial algebra. We construct a slice for the coadjoint action of p, thus extending a theorem of Kostant. The role of the principal sl_2-triple is played by an adapted pair.
We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups.
For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl-Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.
We prove that the canonical dimension of an admissible Banach space or a locally analytic representation of an arbitrary semisimple p-adic Lie group is either zero or at least half the dimension of a non-zero coadjoint orbit. This extends the results of Ardakov-Wadsley and Schmidt in the split semisimple case.
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