  Infinitesimal Cherednik algebras, first introduced in [EGG], are continuous
analogues of rational Cherednik algebras, and in the case of gl_n, are
deformations of universal enveloping algebras of the Lie algebras sl_{n+1}.
Despite these connections, infinitesimal Cherednik algebras are not
widely-studied, and basic questions of intrinsic algebraic and representation
theoretical nature remain open. In the first half of this paper, we construct
the complete center of H_\zeta(gl_n) for the case of n=2 and give one
particular generator of the center, the Casimir operator, for general n. We
find the action of this Casimir operator on the highest weight modules to prove
the formula for the Shapovalov determinant, providing a criterion for the
irreducibility of Verma modules. We classify all irreducible finite dimensional
representations and compute their characters. In the second half, we
investigate Poisson-analogues of the infinitesimal Cherednik algebras and use
them to gain insight on the center of H_\zeta(gl_n). Finally, we investigate
H_\zeta(sp_{2n}) and extend various results from the theory of H_\zeta(gl_n),
such as a generalization of Kostant's theorem.
