Related papers: Nilpotent centralizers and Springer isomorphisms

For any connected reductive group $G$ over $\mathbb{C}$, we revisit Goresky-Kottwitz-MacPherson's description of the torus equivariant Borel-Moore homology of affine Springer fibers $\mathrm{Sp}_\gamma\subset \mathrm{Gr}_G$, where…

Springer resolution of the set of nilpotent elements in a semisimple Lie algebra plays a central role in geometric representation theory. A new structure on this variety has arisen in several representation theoretic constructions, such as…

A linear mapping $\phi$ from an algebra $\mathcal{A}$ into its bimodule $\mathcal M$ is called a centralizable mapping at $G\in\mathcal{A}$ if $\phi(AB)=\phi(A)B=A\phi(B)$ for each $A$ and $B$ in $\mathcal{A}$ with $AB=G$. In this paper, we…

Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is the fixed point subspace of an element of G. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then…

We define several versions of a class of varieties $X_{\mathfrak{g}}$ attached to a complex reductive Lie algebra $\mathfrak{g}$, generalizing the Hilbert scheme of points on the plane. These include trigonometric and elliptic versions…

We study general nilpotent algebras. The results obtained are new even for the classical algebras, such as associative or Lie algebras. We single out certain generic properties of finite-dimensional algebras, mostly over infinite fields.…

We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer of…

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$ and $\mathcal{Z(G)}$ be the center of $\mathcal{G}$. Suppose that ${\mathfrak q}\colon \mathcal{G}\times \mathcal{G}\longrightarrow \mathcal{G}$ is an…

This paper proves a number of flatness results for centralizers of sections of a reductive group scheme over a general base scheme. To this end, we establish relative versions of the Jordan decomposition. Using our results, we obtain a…

In this paper, by establishing an explicit and combinatorial description of the centralizer of a distinguished nilpotent pair in a classical simple Lie algebra, we solve in the classical case Panyushev's Conjecture which says that…

Let $G$ be a simple simply-connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}={\rm Lie}(G)$. We discuss various properties of nilpotent orbits in $\mathfrak{g}$, which have previously…

Let K be any field, and let G be a semisimple group over K. Suppose the characteristic of K is positive and is very good for G. We describe all group scheme homomorphisms phi:SL(2) --> G whose image is geometrically G-completely reducible…

Let $\mathcal P(S)$ be the semigroup obtained by equipping the family of all non-empty subsets of a (multiplicatively written) semigroup $S$ with the operation of setwise multiplication induced by $S$ itself. We call a subsemigroup $P$ of…

Let $G$ be a simple algebraic group over the complex field $\mathbb C$, $P$ a parabolic subgroup containing $B$ its Borel subgroup, $P'$ its derived group and $\mathfrak m$ the Lie algebra of its nilradical. The nilfibre $\mathscr N$ for…

For a classical group $G$ over a field $F$ together with a finite-order automorphism $\theta$ that acts compatibly on $F$, we describe the fixed point subgroup of $\theta$ on $G$ and the eigenspaces of $\theta$ on the Lie algebra…

The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety's cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action.…

A Cayley graph over a group $G$ is said to be central if its connection set is a normal subset of $G$. We prove that every central Cayley graph over a simple group $G$ has at most two pairwise nonequivalent Cayley representations over $G$…

For $G$ a connected, reductive group over an algebraically closed field $k$ of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of $\mathfrak{g}:=\mathrm{Lie}(G)$ and the unipotent variety of $G$…

Suppose that $\mathcal{C}$ is a class of groups consisting only of periodic groups and $\mathfrak{P}(\mathcal{C})^{\prime}$ is the set of prime numbers each of which does not divide the order of any element of a $\mathcal{C}$-group. A…

Let $G$ be a finite group. For each $m>1$ we define the symmetric canonical subset $S=S(m)$ of the Cartesian power $G^m$ and we consider the family of Cayley graphs $\mathscr{G}_m(G)=Cay(G^m,S)$. We describe properties of these graphs and…