Related papers: The Model Orbit in $G_2$

Let G be a simple algebraic group over the complex numbers. Let N be the cone of nilpotent elements in the Lie algebra of G. Let K_{G x C^*}(N) denote the Grothendieck group of the category of G x C^*-equivariant coherent sheaves on N. In…

Suppose that $G$ is a finite solvable group and $V$ is a finite, faithful and completely reducible $G$-module. Let $N$ be a nilpotent subgroup of $G$, then there exits $v \in V$ such that $|\bC_N(v)| \leq (|N|/p)^{1/p}$, where $p$ is the…

In this paper, we construct certain unipotent representations for the real orthogonal group and the metaplectic group in the sense of Vogan. Our construction is based on quantum induction which involves the compositions of even number of…

We study the irreducible complex representations of general linear groups over principal ideal local rings of length two with a fixed finite residue field. We construct a canonical correspondence between the irreducible representations of…

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…

This paper aims at the following results: \begin{enumerate} \item The class of all $*$-regular rings forms a variety. \item A subdirectly irreducible $*$-regular ring $R$ is faithfully representable (i.e. isomorphic to a subring of an…

We give a description of all the cuspidal representations of $\mathrm{GL}_4(\mathfrak{o}_2)$, where $\mathfrak{o}_2$ is a finite ring coming from the ring of integers in a local field, modulo the square of its maximal ideal $\mathfrak{p}$.…

Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical…

We study D-modules and related invariants on the space of 2 x 2 x n hypermatrices for n >= 3, which has finitely many orbits under the action of G = GL_2 x GL_2 x GL_n. We describe the category of coherent G-equivariant D-modules as the…

In this paper, as a generalization of Kirillov's orbit theory, we explore the relationship between the dressing orbits and irreducible *-representations of the Hopf C*-algebras (A,\Delta) and (\tilde{A}, \tilde{\Delta}) we constructed…

Let $G$ be a simple algebraic group with $\mathfrak g=Lie(G)$ and $\mathcal O\subset\mathfrak g$ a nilpotent orbit. If $H$ is a reductive subgroup of $G$ with $Lie(H)=\mathfrak h$, then $\mathfrak g=\mathfrak h\oplus\mathfrak m$, where…

Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…

Orbits of the Weyl reflection groups attached to the simple Lie groups $A_2, C_2, G_2$ and Coxeter group $H_2$ are considered. For each of the groups products of any two orbits are decomposed into the union of the orbits. Results are…

In this note I give a formula for calculating the number of orbits of irreducible binary forms of degree $n$ over GF$(p)$ under the action of GL$(2,p)$. This formula has applications to the classification of class two groups of exponent $p$…

Let v be the right regular representation of a compact quantum group G. Then (S.L.Woronowicz, "Compact quantum groups") v contains all irreducible representations of G and each irreducible representation enters v with the multiplicity equal…

We define a set of "enhanced" nilpotent quiver representations that generalizes the enhanced nilpotent cone. This set admits an action by an associated algebraic group $K$ with finitely many orbits. We define a combinatorial set that…

Using an algebraic orbifold method, we present non-commutative aspects of $G_2$ structure of seven dimensional real manifolds. We first develop and solve the non commutativity parameter constraint equations defining $G_2$ manifold algebras.…

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…

Let $O$ be a nilpotent orbit of a complex semisimple Lie algebra $\mathfrak{g}$ and let $\pi: X \to \bar{O}$ be the finite covering associated with the universal covering of $O$. In the previous article we have explicitly constructed a…

In the monograph arXiv:2108.03453, we define the notion of a unipotent representation of a complex reductive group. The representations we define include, as a proper subset, all special unipotent representations in the sense of…