Category decomposition of \(\operatorname{Rep}_k( \mathrm{SL}_n(F))\). (English) Zbl 1506.22013
Summary: Let \(F\) be a non-archimedean local field with residual characteristic \(p\), and \(k\) an algebraically closed field of characteristic \(\ell \neq p\). We establish a category decomposition of \(\operatorname{Rep}_k( \mathrm{SL}_n(F))\) with respect to the \(\mathrm{GL}_n(F)\)-inertially equivalent supercuspidal classes of \(\mathrm{SL}_n(F)\), and we establish a block decomposition of the supercuspidal subcategory of \(\operatorname{Rep}_k( \mathrm{SL}_n(F))\). Finally we give an example to show that in general a block of \(\mathrm{SL}_n(F)\) is not defined with respect to a unique inertially equivalent supercuspidal class of \(\mathrm{SL}_n(F)\), which is different from the case when \(\ell = 0\).
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 20C20 | Modular representations and characters |
Keywords:
modulo \(\ell\) representations of \(p\)-adic groups; special linear groups; block decomposition; Bernstein decompositionReferences:
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