A filtration on rings of representations of non-Archimedean GLₙ

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Let F be a p-adic field. Let R be the Grothendieck ring of complex smooth finite-length representations of the groups {GLₙ(F)}ₙ₌₀∞ taken together, with multiplication defined in the sense of parabolic induction. We introduce a width invariant for elements of R and show that it gives an increasing filtration on the ring. Irreducible representations of width 1 are precisely those known as ladder representations. We thus obtain a necessary condition on irreducible factors of a product of two ladder representations. For such a product we further establish a multiplicity-one phenomenon, which was previously observed in special cases.