  We investigate the representation theory of the rational and trigonometric
Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or
cylindrical) skew diagrams.
  We introduce and study standard tableaux and plane partitions on periodic
diagrams, and in particular, compute some generating functions concerning plane
partitions, where Kostka polynomials and their level restricted generalization
appear.
  On representation side, we study representations of Cherednik algebras which
admit weight decomposition with respect to a certain commutative subalgebra.
All the irreducible representations of this class are constructed
combinatorially through standard tableaux on periodic diagrams, and this
realization as "tableaux representations" provides a new combinatorial approach
to the investigation of these representations.
  As consequences, we describe the decomposition of a tableaux representation
as a representation of the degenerate affine Hecke algebra, which is a
subalgebra of the Cherednik algebra, and also describe the spectral
decomposition of the spherical subspace (the invariant subspace under the
action of the Weyl group) of a tableaux representation with respect to the
center of the degenerate affine Hecke algebra, In particular, the computation
of the character of the spherical subspace is reduced to the computation of the
generating function for the set of column strict plane partitions, and we
obtain an expression of the characters in terms of Kostka polynomials as
announced in math.QA/0508274.
