Decomposition Rules for the Ring of Representations of Non-Archimedean GL_n

Let R be the Grothendieck ring of complex smooth finite-length representations of the sequence of p-Adic groups {GLn(F)}∞ n=0, with multiplication defined through parabolic induction. We study the problem of the decomposition of products of irreducible representations in R. We obtain a necessary condition on irreducible factors of a given product by introducing a width invariant. Width 1 representations form the previously studied class of ladder representations. We later focus on the case of a product of two ladder representations, for which we establish that all irreducible factors appear with multiplicity one. Finally, we propose a general rule for the composition series of a product of two ladder representations and prove its validity for cases in which the irreducible factors correspond to smooth Schubert varieties.