Let G be a connected linear semisimple Lie group with Lie algebra g, and let K_C --> Aut(p_C) be the complexified isotropy representation at the identity coset of the corresponding symmetric space G/K. Suppose that O is a nilpotent G-orbit in g and O* is the nilpotent K_C-orbit in p_C associated to O by the Kostant-Sekiguchi correspondence. We show that the complexity of O* as a K_C variety measures the failure of the Poisson algebra of smooth K-invariant functions on O to be commutative.
For any Hermitian Lie group $G$ of tube type we give a geometric quantization procedure of certain $K_{\mathbb{C}}$-orbits in $\mathfrak{p}_{\mathbb{C}}^*$ to obtain all scalar type highest weight representations. Here $K_{\mathbb{C}}$ is the complexification of a maximal compact subgroup $K\subseteq G$ with corresponding Cartan decomposition $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ of the Lie algebra of $G$. We explicitly realize every such representation $π$ on a Fock space consisting of square integrable holomorphic functions on its associated variety $Ass(π)\subseteq\mathfrak{p}_{\mathbb{C…
Given a complex projective algebraic variety, write H(X) for its cohomology with complex coefficients and IH(X) for its Intersection cohomology. We first show that, under some fairly general conditions, the canonical map H(X)\to IH(X) is injective. Now let Gr = G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. We prove, using the general result above, a conjecture of D. Peterson describing the cohomology algebra H(X) in terms of the centralizer of the principal nilpotent in the Langlands dual of Lie(G). In the last sectio…
In this paper we obtain various results about the geometry of nilpotent orbits. In particular, we obtain a better understanding of the Kostant-Sekiguchi correspondence and Kronheimer's instanton flow. We utilize the moment map of Ness and the SL(2)-orbit theorem from Hodge theory. The results of this paper are used in the proof of the Barbarsch-Vogan conjecture in math.RT/0005305 (Ann. of Math. (2) 151 (2000), no. 3, 1071-1118).
In this paper we study finite W-algebras for basic classical superalgebras and Q(n) associated to the regular even nilpotent coadjoint orbits. We prove that this algebra satisfies the Amitsur-Levitzki identity and therefore all its irreducible representations are finite-dimensional. In the case of Q(n) we give an explicit description of the W-algebra in terms of generators and relation and realize it as a quotient of the super-Yangian of Q(1).
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 $\m…
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.
Let g be a complex semisimple Lie algebra and let G' be the Langlands dual group. We give a description of the cohomology algebra of an arbitrary spherical Schubert variety in the loop Grassmannian for G' as a quotient of the form Sym(g^e)/J. Here, J is an appropriate ideal in the symmetric algebra of g^e, the centralizer of a principal nilpotent in g. We also discuss a `topological' proof of Kostant's famous result on the structure of the polynomial algebra on g.
In two 2006 papers, Kostant and Wallach constructed a complexified Gelfand-Zeitlin integrable system for the Lie algebra $\fgl(n+1,\C)$ and introduced the strongly regular elements, which are the points where the Gelfand-Zeitlin flow is Lagrangian. Later Colarusso studied the nilfibre, which consists of strongly regular elements such that each $i\times i$ submatrix in the upper left corner is nilpotent. In this paper, we prove that every Borel subalgebra contains strongly regular elements and determine the Borel subalgebras containing elements of the nilfibre by using the theory of $K_{i}=GL(i…
In the case of complex symplectic and orthogonal groups, we find $(\mathfrak{g}, K)-$modules with the property that their $K-$structure matches the structure of regular functions on the closures of nilpotent orbits. This establishes a version of the Orbit Method of Kirrilov-Kostant-Souriau as proposed by Vogan. In the process we give another proof of the classification of nilpotent orbits with normal closure in the Lie algebra of a classical group first established by Kraft-Procesi.
Let us fix a complex simple Lie algebra and its non-compact real form. This paper focuses on non-zero adjoint nilpotent orbits in the complex simple Lie algebra meeting the real form. We show that the poset consisting of such nilpotent orbits equipped with the closure ordering has the minimum. Furthermore, we determine such the minimum orbit in terms of the Dynkin--Kostant classification even in the cases where the orbit does not coincide with the minimal nilpotent orbit in the complex simple Lie algebra. We also prove that the intersection of the orbit and the real form is the union of all mi…
Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F) that picks out a G(F)-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F). This generalizes an earlier result obtained by DeBacker and one of the authors under stronger hypotheses. We then show that if F is p-adic, then the characteristic function of this set behaves well with respect to endoscopic transfer.
We give an explicit description of adjoint quotient maps for Jacobson-Witt algebra Wn and special algebra Sn. An analogue of Kostant's differential criterion of regularity is given for Wn. Furthermore, we describe the fiber of adjoint quotient map for Sn and construct the analogs of Kostant's transverse slice. In addition, we analysis the nilpotent elements and generalize some Premet's results to Sn.
We study the center of $U(\mathfrak{n})$, where $\mathfrak{n}$ is the locally nilpotent radical of a splitting Borel subalgebra of a simple complex Lie algebra $\mathfrak{g}=\mathfrak{sl}_{\infty}(\mathbb{C})$, $\mathfrak{so}_{\infty}(\mathbb{C})$, $\mathfrak{sp}_{\infty}(\mathbb{C})$. There are infinitely many isomorphism classes of Lie algebras $\mathfrak{n}$, and we provide explicit generators of the center of $U(\mathfrak{n})$ in all cases. We then fix $\mathfrak{n}$ with "largest possible" center of $U(\mathfrak{n})$ and characterize the centrally generated primitive ideals of $U(\mathfra…
Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves \cite{V2}, \cite{HP1}, \cite{Kdircoh}. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the corresponding nilpotent Lie algebra cohomology in many cases, while in general it has better algebraic behavior and it is more accessible for calculation.
We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group G with Lie algebra g. These slices are associated to (the conjugacy classes of) elements s of the Weyl group W of g. For such slices we prove an analogue of the Kostant cross-section theorem for the action of a unipotent group.
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…
In 1978 Kostant suggested the Whittaker model of the center of the universal enveloping algebra U(g) of a complex simple Lie algebra g. The main result is that the center of U(g) is isomorphic to a commutative subalgebra in U(b), where b is a Borel subalgebra in g. This observation is used in the theory of principal series representations of the corresponding Lie group G and in the proof of complete integrability of the quantum Toda lattice. In this paper we generalize the Kostant's construction to quantum groups. In our construction we use quantum analogues of regular nilpotent elements defin…
This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue of the Kostant partition function, and propose the corresponding two-parameter analogue of the weight multiplicity formula. In a different direction, each principal nilpotent pair gives rise to a harmon…
Let $k$ be an algebraically closed field of any characteristic except 2, and let $G = \GL_n(k)$ be the general linear group, regarded as an algebraic group over $k$. Using an algebro-geometric argument and Dynkin-Kostant theory for $G$ we begin by obtaining a canonical form for nilpotent $\Ad(G)$-orbits in $\glł_n(k)$ which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map $f : (x_{i,j})\mapsto (x_{n+1-j,n+1-i})$), with entries in $\{0,1\}$. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form…
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 …
We calculate the homology of three families of 2-step nilpotent Lie (super)algebras associated with the symplectic, orthogonal, and general linear groups. The symplectic case was considered by Getzler and the main motivation for this work was to complete the calculations started by him. In all three cases, these algebras can be realized as the nilpotent radical of a parabolic subalgebra of a simple Lie algebra, and our first approach relies on a theorem of Kostant, but is otherwise elementary and involves combinatorics of Weyl groups and partitions which may be of independent interest. Our sec…
Let $\mathrm G / F$ be a reductive split $p$-adic group and let $\mathrm U$ be the unipotent radical of a Borel subgroup. We study the cohomology with trivial $\mathbb Z_p$-coefficients of the profinite nilpotent group $N = \mathrm U(\mathcal O_F)$ and its Lie algebra $\mathfrak n$, by extending a classical result of Kostant to our integral $p$-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.
In this paper we determine the precise extent to which the classical sl_2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson--Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer--Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example i…
The strong Macdonald theorems state that, for $L$ reductive and $s$ an odd variable, the cohomology algebras $H^*(L[z]/z^N)$ and $H^*(L[z,s])$ are freely generated, and describe the cohomological, $s$-, and $z$-degrees of the generators. The resulting identity for the $z$-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate $H^*(\mathfrak{p} / z^N \mathfrak{p})$ and $H^*(\mathfrak{p}[s])$ for $\mathfrak{p}$ a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic …
