arXiv subjects
Publications and source records attributed to Esther Galina.
In the context of affine complex Kac-Moody algebras, we define the meaning of nilpotent orbits under the adjoint action of the maximal Kac-Moody group. We also give a parameterization of nilpotent orbits of $\mathfrak{sl}_n^{(1)}(\mathbb C)$.
In this note we determine the irreducible square integrable representations of a simple group which admits an admissible restriction to a subgroup $H$ locally isomorphic to $SL_2(\mathbb R).$ We show such representation is holomorphic and we determine the essentially unique $H$ with this property as well as multiplicity formulae.
For any n>3, we give a family of finite dimensional irreducible representations of the braid group B_n. Moreover, we give a subfamily parametrized by 0<m<n of dimension the combinatoric number (n,m). The representation obtained in the case m=1 is equivalent to the Standard representation.
In the theory of Lie groups, the irreducibility of a unitary representation is not preserved in general by restriction to a subgroup. Kirillov's conjecture says that it is preserved for the groups Gl(n,R) or Gl(n,C) when the subgroup is the subgroup of matrices leaving invariant a non zero vector. This conjecture was proved by Barush using a detailed study of nilpotent orbits. In fact, it is not difficult to see that the conjecture is equivalent to the fact that some system of partial differential equations has no singular distributions as solutions. This system of equations is a regular holonomic D-module and we give a proof of the result by an explicit calculation of the roots of the b-functions associated to this D-module.
We give a method to produce representations of the braid group $B_n$ of $n-1$ generators ($n\leq \infty$). Moreover, we give sufficient conditions over a non unitary representation for being of this type. This method produces examples of irreducible representations of finite and infinite dimension.
We give a method to construct new self-adjoint representations of the braid group. In particular, we give a family of irreducible self-adjoint representations of dimension arbitrarily large. Moreover we give sufficient conditions for a representation to be constructed with this method.
We associate to each nilpotent orbit of a real semisimple Lie algebra $g_o$ a weighted Vogan diagram, that is a Dynkin diagram with an involution of the diagram, a subset of painted nodes and a weight for each node. Every nilpotent element of $g_o$ is noticed in some subalgebra of $g_o$. In this paper we characterize the weighted Vogan diagrams associated to orbits of noticed nilpotent elements.
A celebrated theorem of Harich-Chandra asserts that all invariant eigendistributions on a semisimple Lie group are locally integrable functions. We show that this result is a consequence of an algebraic property of a holonomic D-module defined by Kashiwara and Hotta.
The infinite dimensional Clifford Algebra has a maze of irreducible unitary representations. Here we determine their type -real, complex or quaternionic. Some, related to the Fermi-Fock representations, have no real or quetrnionic structures. But there are many on L(2) of the circle that do and which seem to have analytic meaning.