Related papers: Representations associated to small nilpotent orbi…

We discuss kernels on complact classical groups $G=\U(n)$, $\OO(2n)$, $\Sp(n)$ defined by the formula $K(z,u)=|\det(1-zu^*)|^s$. We obtain the explicit Plancherel formula for these kernels and the interval of positive-definiteness. We also…

The Coulomb branches of certain 3-dimensional N=4 quiver gauge theories are closures of nilpotent orbits of classical or exceptional algebras. The monopole formula, as Hilbert series of the associated Coulomb branch chiral ring, has been…

We deduce the periodicity 8 for the type of $Pin$ and $Spin$ representations of the orthogonal groups $O(n)$ from simple combinatorial properties of the finite Clifford groups generated by the gamma matrices. We also include the case of…

Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)\to \mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-theory classes not…

The decomposition of the spinor bundle of the spin Grassmann manifolds $G_{m,n}=SO(m+n)/SO(m)\times SO(n)$ into irreducible representations of $\mathfrak{so}(m)\oplus\mathfrak{so}(n)$ is presented. A universal construction is developed and…

We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov's method of orbits for nilpotent groups over local fields. When the semisimple…

Let G be a connected reductive group over a field of characteristic zero, and consider an orthogonal representation of G. We give a simple criterion for whether the representation lifts to the spin group, in terms of the highest weights of…

In this article we analyze the quotients of the maximal compact subalgebras of the split real Kac-Moody algebras of the En series resulting from the generalized spin representations introduced in part 1. It turns out that these quotients…

The first part of this thesis studies the notion of a "quantum representation", introduced by J.-M. Souriau in order to provide a polarization-free characterization of the Lie group representations attached to coadjoint orbits. When the…

We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation on a smooth vector bundle, i.e., one whose kernel consists only of ineffective arrows. As an application,…

In this paper, we study the moduli space of unipotent Weil-Deligne representations valued in a split reductive group $G$ and characterise which irreducible components are smooth. We apply the smoothness results proved to show that a certain…

We determine the equivariant real structures on nilpotent orbits and the normalizations of their closures for the adjoint action of a complex semisimple algebraic group on its Lie algebra.

The Weierstrass representation for spheres in $\R^3$ and, in particular, effective construction of immersions from data of spectral theory origin is discussed. These data are related to Dirac operators on a plane and on an infinite cylinder…

Let $F$ be a non-archimedean local field. In this paper we explore genericity of irreducible smooth representations of $GL_n(F)$ by restriction to a maximal compact subgroup $K$ of $GL_n(F)$. Let $(J, \lambda)$ be a Bushnell--Kutzko type…

Let G be a connected real reductive algebraic group, and let K be a maximal compact subgroup of G. We prove that the conjugation orbit space Hom(Z^{2d},K)/K is a strong deformation retract of the space Hom(Z^{2d},G)/G of equivalence classes…

A nilpotent orbit $O$ of a complex semisimple Lie algebra $\mathfrak{g}$ has finite fundamental group. Associated with an etale cover of $O$, we have a finite cover of the closure $\bar{O}$ of $O$. In this article we consider the finite…

Let $\FRAK{g}$ be a classical simple Lie superalgebra. To every nilpotent orbit $\cal O$ in $\FRAK{g}_0$ we associate a Clifford algebra over the field of rational functions on $\cal O$. We find the rank, $k(\cal O)$ of the bilinear form…

Let $G$ be a connected semisimple Lie group with its maximal compact subgroup $K$ being simply-connected. We show that the twisted equivariant $KK$-theory $KK^{\bullet}_{G}(G/K, \tau_G^G)$ of $G$ has a ring structure induced from the…

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…

The spin of particles on a non-commutative geometry is investigated within the framework of the representation theory of the q-deformed Poincare algebra. An overview of the q-Lorentz algebra is given, including its representation theory…