  In the paper, properties of antisymmetric orbit functions are reviewed and
further developed. Antisymmetric orbit functions on the Euclidean space $E_n$
are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a
Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such
functions are described. These functions are closely related to irreducible
characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign,
values of antisymmetric orbit functions are repeated on copies of the
fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl
group) in the entire Euclidean space $E_n$. Antisymmetric orbit functions are
solutions of the corresponding Laplace equation in $E_n$, vanishing on the
boundary of the fundamental domain $F$. Antisymmetric orbit functions determine
a so-called antisymmetrized Fourier transform which is closely related to
expansions of central functions in characters of irreducible representations of
the group $G$. They also determine a transform on a finite set of points of $F$
(the discrete antisymmetric orbit function transform). Symmetric and
antisymmetric multivariate exponential, sine and cosine discrete transforms are
given.
