Note on prehomogeneous vector spaces

Let \(V\) be a complex prehomogeneous vector space under the action of a linear algebraic group \(G\) . Assume the poset of orbit closures in the Zariski topology \(\{\overline{Gx}:x\in V\}\) coincides with a (partial) flag \(V_0=0<V_1<\dots <V_k=V\) in \(V\) . Then for any Borel subgroup \(B\) of \(G\) , the poset \(\{\overline{B x}:x\in V\}\) coincides with a full flag in \(V\) .     

Relative invariants of some 2-simple prehomogeneous vector spaces

In this paper, we shall construct explicitly irreducible relative invariants of two 2-simple prehomogeneous vector spaces. Together with a preprint by the same authors, this completes the list of all relative invariants of regular 2-simple prehomogeneous vector spaces of type I.

     

On parameterizations of rational orbits of some forms of prehomogeneous vector spaces

In this paper, we consider the structure of rational orbit of some prehomogeneous vector spaces motivated by Wright and Yukie’s work. After we give a refinement of Wright and Yukie’s construction, we apply it to the inner forms of the D ₅ and E ₇ types. Parameterizations of reducible algebras for the cubic cases are included. Financial support is provided by Research Fellowships for Young Scientists of Japan Society for the Promotion of Science.     

Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms

The theory of zeta functions associated with prehomogeneous vector spaces (p.v. for short) provides us a unified approach to functional equations of a large class of zeta functions. However the general theory does not include zeta functions related to automorphic forms such as the HeckeL-functions and the standardL-functions of automorphic forms on GL(n), even though they can naturally be considered to be associated with p.v.’s. Our aim is to generalize the theory to zeta functions whose coefficients involve periods of automorphic forms, which include the zeta functions mentioned above. In this paper, we generalize the theory to p.v.’s with symmetric structure ofK _ε-type and prove the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character. In another paper, we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary representations of compact groups. Dedicated to the memory of Professor K G Ramanathan     

Micro-local analysis for the Metropolis algorithm

We prove sharp rates of convergence to stationarity for a simple case of the Metropolis algorithm: the placement of a single disc of radius h randomly into the interval [ − 1 − h, 1 + h], with h > 0 small. We find good approximations for the top eigenvalues and eigenvectors. The analysis gives rigorous proof for the careful numerical work (in Exp. Math. 13, 207–213). The micro-local techniques employed offer promise for the analysis of more realistic problems.     

Some results of functional analysis for ultrametric spaces and valued vector spaces

We prove a Hahn-Banach type theorem and a generalization of Baire's theorem for ultrametric spaces (with totally ordered value sets). Some applications to valued vector spaces with value groups of arbitrary rank are given (Principle of Uniform Boundedness, Open Mapping Theorem).Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday     

Harmonic analysis for vector fields on hyperbolic spaces

We acknowledge the support by the Swiss National Science Foundation     

Monotonic analysis over ordered topological vector spaces: IV

In this paper, we present an extension for non-negative increasing and co-radiant (ICR) functions over a topological vector space. We characterize the essential results of abstract convexity such as support set, subdifferential and polarity of these functions. We also give some characterizations of a certain kind of polarity and separation property for non-convex radiant and co-radiant sets.     

Frames for vector spaces and affine spaces

A finite frame for a finite dimensional Hilbert space is simply a spanning sequence. We show that the linear functionals given by the dual frame vectors do not depend on the inner product, and thus it is possible to extend the frame expansion (and other elements of frame theory) to any finite spanning sequence for a vector space. The corresponding coordinate functionals generalise the dual basis (the case when the vectors are linearly independent), and are characterised by the fact that the associated Gramian matrix is an orthogonal projection. Existing generalisations of the frame expansion to Banach spaces involve an analogue of the frame bounds and frame operator.The potential applications of our results are considerable. Whenever there is a natural spanning set for a vector space, computations can be done directly with it, in an efficient and stable way. We illustrate this with a diverse range of examples, including multivariate spline spaces, generalised barycentric coordinates, and vector spaces over the rationals, such as the cyclotomic fields.