Congruences of parahoric group schemes. (English) Zbl 1460.22006

Ganapathy, Radhika

Algebra Number Theory 13, No. 6, 1475-1499 (2019).

Summary: Let \(F\) be a nonarchimedean local field and let \(T\) be a torus over \(F\). With \(\mathcal{T}^{NR}\) denoting the Néron-Raynaud model of \(T\), a result of C.-L. Chai and J.-K. Yu [Ann. Math. (2) 154, No. 2, 347–382 (2001; Zbl 1098.14014)] asserts that the model \(\mathcal{T}^{NR}\times_{\mathfrak{O}_F}\mathfrak{O}_F/\mathfrak{p}_F^m\) is canonically determined by \((\mathrm{Tr}_l(F),\Lambda)\) for \(l\gg m\), where \(\mathrm{Tr}_l(F)=(\mathfrak{O}_F/\mathfrak{p}_F^l,\mathfrak{p}_F/\mathfrak{p}_F^{l+1},\varepsilon)\) with \(\epsilon\) denoting the natural projection of \(\mathfrak{p}_F/\mathfrak{p}_F^{l+1}\) on \(\mathfrak{p}_F/\mathfrak{p}_F^l\), and \(\Lambda:=X_*(T)\).
In this article we prove an analogous result for parahoric group schemes attached to facets in the Bruhat-Tits building of a connected reductive group over \(F\).

MSC:

22E50	Representations of Lie and linear algebraic groups over local fields
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

parahoric; close local fields

Citations:

Zbl 1098.14014

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