Let $\mathcal{ R}$ be the Grothendieck ring of complex smooth finite-length representations of the sequence of p-adic groups $\{ GL_n(F)\}_{ n=0}^\infty$, with multiplication defined through parabolic induction. We study the problem of the decomposition of products of irreducible representations in $\mathcal{ R}$. 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.
Categories: math.RT math.QA