Ibrahim Nonkane: Decomposition of modules over the Weyl algebra

This thesis is concerned with various results using the Weyl algebra Aₙ(K).

In the first paper, we use properties of certain Aₙ(K)-modules to construct Noetherian operators attached to a primary ideal J in the polynomial ring K[x₁,...,xₙ], where K is a field of characteristic 0.

In the second paper, we consider the direct image of an irreducible Aₙ(K)-module under a finite map π: X = spec B → Y = spec A. We study the decomposition in the case of the invariants of the symmetric group, B = ℂ[x₁,...,xₙ] ⊃ A = ℂ[x₁,...,xₙ]^Sₙ . We first describe the generators of the simple components of π₊(B) and give their multiplicities. Secondly, we describe another basis of each irreducible module after localization. Finally using Brauer's characterization of characters we give a partial generalization to arbitrary finite extensions.