2022/12/12 by Chan, Kei Yuen
#20C08: 11F70 #22E50 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2212.05919
Let Gn=GLn(F) be the general linear group over a non-Archimedean local field F. We formulate and prove a necessary and sufficient condition on determining when HomGn(π, π') ≠ 0 for irreducible smooth representations π and π' of Gn+1 and Gn respectively. This resolves the problem of the quotient branching law. We also prove that any simple quotient of a Bernstein-Zelevinsky derivative of an irreducible representation can be constructed by a sequence of derivatives of essentially square-integrable representations. This result transferred to affine Hecke algebras of type A gives a generalization of the classical Pieri's rule of symmetric groups. One key new ingredient is a characterization of the layer in the Bernstein-Zelevinsky filtration that contributes to the branching law, obtained by the multiplicity one theorem for standard representations, which also gives a refinement of the branching law. Another key new ingredient is constructions of some branching laws and simple quotients of Bernstein-Zelevinsky derivatives by taking certain highest derivatives.