Narrow Results

In this paper, we give a pedagogical presentation of the irreducible unitary representations of ${ℂ}^4⋊Spin(4,ℂ)$, that is, of the universal cover of the complexified Poincaré group ${ℂ}^4⋊SO(4,ℂ)$. These representations were first investigated by Roffman in 1967. We provide a modern formulation of his results together with some facts from the general Wigner–Mackey theory which are relevant in this context. Moreover, we discuss different ways to realize these representations and, in the case of a non-zero “complex mass”, we give a detailed construction of a more explicit realization. This explicit realization parallels and extends the one used in the classical Wigner case of ${ℝ}^4⋊{Spin}^0(1,3)$. Our analysis is motivated by the interest in the Euclidean formulation of Fermionic theories.

Let G_ℝ be a real form of a complex, semisimple Lie group G. Assume is an even nilpotent coadjoint G_ℝ-orbit. We prove a limit formula, expressing the canonical measure on as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the negative chamber defined by the parabolic subalgebra associated with .

We construct the Radon–Penrose transform as an intertwining operator of the indefinite unitary group U(p,q), from the Dolbeault cohomology group on the indefinite Grassmann manifold consisting of positive k-planes to the space of holomorphic functions over the bounded symmetric domain. We prove that the Penrose transform is injective, and that its image is exactly the space of global holomorphic solutions to the system of partial differential equations of determinant type of size k + 1.

In this paper, we start the program of the existence of the smooth equivariant geodesics in the equivariant Mabuchi moduli space of Kähler metrics on type II cohomogeneity one compact Kähler manifold. In this paper, we deal with the manifolds $M_n$ obtained by blowing up the diagonal of the product of two copies of a $C{P}^n$.