Decomposition rules for the ring of representations of non-Archimedean $GL_n$

Maxim Gurevich

Published 2016-09-15Version 1

Let $\mathcal{R}$ be the Grothendieck ring of complex smooth finite-length representations of the groups $\{GL_n(F)\}_{n=0}^\infty$ ($F$ a fixed $p$-adic field) taken together, with multiplication defined in the sense of parabolic induction. We study the problem of the decomposition of a product of irreducible representations in $\mathcal{R}$. We introduce a width invariant for elements of the ring. By showing that the invariant gives an increasing ring filtration, we obtain a necessary condition on irreducible factors of a given product. Irreducible representations of width $1$ 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 conjecture 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.