Tame cuspidal representations in non-defining characteristics. (English) Zbl 1523.22019

Let \(F\) be a nonarchimedean local field of residual characteristic \(p\neq 2\), and let \(G\) be a connected reductive group that splits over a tamely ramified extensions of \(F\). When \(p\) does not divide the order of the absolute Weyl group of \(G\), the complex smooth irreducible supercuspidal representations of \(G(F)\) have been constructed exhaustively, thanks to the works of Yu, Kim, and the author. The current paper applies a similar construction as in Yu’s work, to construct smooth irreducible cuspidal representations of \(G(F)\) over an algebraically closed field \(R\) of characteristic \(\ell\neq p\). However there are a few differences from the complex case. For example, \(R\)-representations of finite reductive groups over \(\mathbb{F}_p\) may not be semisimple in general, so extra works need to be done in order to prove the irreducibility. Also the theory of types cannot be applied directly in the proof of exhaustion theorem for the \(\ell\)-modular case. These difficulties are settled in the current paper.
As mentioned, the construction of irreducible cuspidal \(R\)-representations is similar to Yu’s construction. Roughly speaking, the inducing data consist of a sequence of twisted Levi subgroups \(G=G_1\supset G_2\cdots\supset G_{n+1}\), some \(R\)-valued characters \(\phi_i\) of \(G_{i+1}\), and an irreducible \(R\)-representation \(\rho\) of some finite reductive quotient, subject to certain conditions. Following the works of Yu and the author, a compact induction \(\text{c-ind}^{G(F)}_{\widetilde{K}}\widetilde\rho\) constructed from these data, which uses the mod-\(\ell\) Weil-Heisenberg representation instead of the complex one, gives a smooth irreducible cuspidal \(R\)-representation. Similar to the complex case one has the exhaustion theorem: when \(p\) does not divide the order of the absolute Weyl group of \(G\), every such representation arises from this construction.


22E50 Representations of Lie and linear algebraic groups over local fields
20G05 Representation theory for linear algebraic groups
20C20 Modular representations and characters
20G25 Linear algebraic groups over local fields and their integers
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