Abstract
AbstractWe use coefficient systems on the affine Bruhat–Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.
Highlights
Let F be a non-Archimedean local field, possibly of non-zero characteristic, and let G be a reductive algebraic group over F, briefly called a reductive p-adic group
Let π be an admissible representation of G on a complex vector space V
We need acyclicity for finite subcomplexes of the building because this provides chain complexes of finite-dimensional vector spaces, which are used in [11] to express the character of V as a sum over contributions of polysimplices in the building. We use this formula to find for each regular semisimple element γ and each vertex x in the building a number r such that the character is constant on Ux(r)γ; the constant r depends on the distance between x and a subset of the building corresponding to the maximal torus containing γ, on theregularity of γ, and on the level of the representation V, that is, on the smallest e ∈ N such that V is generated by the Uy(e)-invariants for all vertices y
Summary
Let F be a non-Archimedean local field, possibly of non-zero characteristic, and let G be a reductive algebraic group over F, briefly called a reductive p-adic group. We need acyclicity for finite subcomplexes of the building because this provides chain complexes of finite-dimensional vector spaces, which are used in [11] to express the character of V as a sum over contributions of polysimplices in the building We use this formula to find for each regular semisimple element γ and each vertex x in the building a number r such that the character is constant on Ux(r)γ; the constant r depends on the distance between x and a subset of the building corresponding to the maximal torus containing γ, on the (ir)regularity of γ, and on the level of the representation V , that is, on the smallest e ∈ N such that V is generated by the Uy(e)-invariants for all vertices y.
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