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We consider the structure of the Goldman Lie algebra for the closed torus, and show that it is finitely generated over the rationals. We also consider other traditional Lie algebra structures and determine that the Goldman Lie algebra for…

In the paper "The Second cohomology of nilpotent orbits in classical Lie algebras, Kyoto J. Math. 60 (2020), no. 2, 717-799" by I. Biswas, P. Chatterjee and C. Maity homotopy types of nilpotent orbits are explicitly described in the case of…

A group $G$ is said to be totally $k$-closed for a positive integer $k$ if, in each of its faithful permutation representations on a set $\Omega^k$, $G$ is the largest subgroup of the symmetric group $\operatorname{Sym}(\Omega)$ that…

Let $G_{\mathbb R}$ be the set of real points of a complex linear reductive group and $\hat G_\lambda$ its classes of irreducible admissible representations with infinitesimal integral regular character $\lambda$. In this case each cell of…

We prove an avoidance principle for expanding translates of unipotent orbits for some semisimple homogeneous spaces. In addition, we prove a quantitative isolation result of closed orbits and give an upper bound on the number of closed…

Let $Q$ be an algebraic group with $q=\Lie Q$ and $V$ a $Q$-module. The index of $V$ is the minimal codimension of the $Q$-orbits in the dual space $V^*$. There is a general inequality, due to Vinberg, relating the index of $V$ and the…

This paper gives methods to describe the adjoint orbits of $\mathbf{G}(\mathfrak{o}_r)$ on $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ where $\mathfrak{o}_r=\mathfrak{o}/\mathfrak{p}^r$ ($r\in\mathbb{N}$) is a finite quotient of the…

We present a new proof of a part of the nilpotent orbit theorem for unipotent complex variations of Hodge structure. In our proof, the $L^2$ extension theorem of Ohsawa-Takegoshi type plays an essential role.

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and…

The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain…

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is…

For each $n=1,2,\dots$, let $\mathsf{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^n$ be the affine group over the integers. For every point $x=(x_1,\dots,x_n) \in \mathbb{R}^n$ let $\mathrm{orb}(x)=\{\gamma(x)\in \mathbb{R}^n\mid\gamma\in…

In this paper we consider non-compact non-complex exceptional Lie algebras, and compute the dimensions of the second cohomology groups for most of the nilpotent orbits. For the rest of cases of nilpotent orbits, which are not covered in the…

We consider the conjugation-action of the Borel subgroup of the symplectic or the orthogonal group on the variety of nilpotent complex elements of nilpotency degree $2$ in its Lie algebra. We translate the setup to a…

We prove that any finite W-algebra U(g,e) admits a one-dimensional representation fixed by the action of the component group of the centraliser of e. As a consequence, for any nilpotent orbit O in g there exists a multiplicity-free (and…

In the classification of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determining the…

In this paper, we establish a general criterion for good pairs, namely pairs consisting of a nilpotent orbit and an even good grading in a simple Lie algebra, which guarantees the existence of a quantum Hamiltonian reduction between…

An algorithm is described to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for modules that are constructed as a submodule of a…

We present a general algorithm for constructing a free resolution for unit groups of orders in semisimple rational algebras. The approach is based on computing a contractible $G$-complex employing the theory of minimal classes of quadratic…

Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for…