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SIGMA 4 (2008), 002, 57 pages arXiv:0801.0822
https://doi.org/10.3842/SIGMA.2008.002
E-Orbit Functions
Anatoliy U. Klimyk a and Jiri Patera b
a) Bogolyubov Institute for Theoretical Physics,
14-b Metrologichna Str., Kyiv 03680, Ukraine
b) Centre de Recherches Mathématiques,
Université de Montréal, C.P.6128-Centre ville, Montréal,
H3C 3J7, Québec, Canada
Received December 20, 2007; Published online January 05, 2008
Abstract
We review and further develop the theory of E-orbit
functions. They are functions on the Euclidean space En
obtained from the multivariate exponential function by symmetrization
by means of an even part We of a Weyl group W, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described.
They are closely related to symmetric and antisymmetric orbit
functions which are received from exponential functions by symmetrization
and antisymmetrization procedure by means of a Weyl group W.
The E-orbit functions, determined by integral parameters, are invariant
with respect to even part Weaff of the
affine Weyl group corresponding to W.
The E-orbit functions determine a
symmetrized Fourier transform, where these functions serve as a kernel
of the transform. They also determine a transform on
a finite set of points of the fundamental domain Fe
of the group Weaff (the discrete E-orbit
function transform).
Key words:
E-orbit functions; orbits; products of orbits; symmetric orbit functions; E-orbit function transform; finite E-orbit function transform; finite Fourier transforms.
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