Spherical distribution vectors

In this paper we consider a locally compact second countable unimodular group $G$ and a closed unimodular subgroup $H$. Let $\rho$ be a finite dimensional unitary representation of $H$ with closed image. For the unitary representation of $G$ obtained by inducing $\rho$ from $H$ to $G$ a decomposition in Hilbert subspaces of a certain space of distributions is given. It is shown that the representations relevant for this decomposition are determined by so-called $(\rho,H)$ spherical distributions, which leads to a description of the decomposition on the level of these distributions. \u

The AKNS-hierarchy

We present here an overview for the Encyclopaedia of Mathematics of the various forms and properties of this system of equations together with its geometric and Lie algebraic background

Spaces of boundary values related to a multipoint version of the KP-hierarchy

In this paper one considers a finite number of points in the complex plane and various spaces of boundary values on circles surrounding these points. To this geometric configuration one associates a Grassmann manifolds that is shown to yield solutions of a multipoint version of the linearization of the $KP$-hierarchy. These Grassmann manifolds are built in such a way that the determinant line bundle and its dual over them still make sense. The same holds for the so-called $\tau$-functions, determinants of certain Fredholm operators that measure the failure of equivariance at lifting the commuting flows of ths hierarchy to these bundles. Solutions of the linearization are described by wave functions of a certain type. They are perturbations of the trivial solution with the leading term of the perturbation determining the type. One concludes with showing that, if a plane in the Grassmann manifold yields wave functions of different types, they are connected by a differential operator in the coordinates of the flows

An irreducible smooth non-admissible representation

It is shown for the group of k-rational points of an affine algebraic group G with k a finite extension of Qp that the topological irreducibility of unitary representations of G does not have to be equivalent to the algebraic irreducibility of the representation on the smooth vectors. We give for a specific G an example of an irreducible smooth representation, which is not admissible

Hilbert flag varieties and their Kähler structure

In this paper we introduce the infinite-dimensional flag varieties associated with integrable systems of the $KdV$- and $Toda$-type and we discuss the structure of these manifolds. As an example we treat the Fubini-Study metric on the projective space associated with a separable complex Hilbert space and we conclude by showing that all flag varieties introduced before possess a K\"{a}hler structure

Spherical distribution vectors

In this paper we consider a locally compact second countable unimodular group $G$ and a closed unimodular subgroup $H$. Let $\rho$ be a finite dimensional unitary representation of $H$ with closed image. For the unitary representation of $G$ obtained by inducing $\rho$ from $H$ to $G$ a decomposition in Hilbert subspaces of a certain space of distributions is given. It is shown that the representations relevant for this decomposition are determined by so-called $(\rho,H)$ spherical distributions, which leads to a description of the decomposition on the level of these distributions

Holomorphic line bundles over Hilbert flag varieties

In this contribution we present a geometric realization of an infinite dimensional analogue of the irreducible representations of the unitary group. This requires a detailed analysis of the structure of the flag varieties involved and the line bundles over it. These constructions are of importance in quantum field theory and in the framework of integrable systems. As an application, it is shown how they occur in the latter context