Block decomposition of the category of \(\ell\)-modular smooth representations of \(\mathrm{GL}_n(\mathrm{F})\) and its inner forms (Q2811190)

Let \(F\) be a non-archimedean local field with residual characteristic \(p\). Let \(D\) be a central division algebra over \(F\) and \(G = \mathrm{GL}_m(D)\) for some \(m\geq 1\). The subject is the structure of the category \({\mathcal R}_R(G)\) of smooth representations of \(G\) with coefficients in an algebraically closed field \(R\) of characteristic \(\neq p\).NEWLINENEWLINEThe irreducible smooth \(R\)-representations of \(G\) were classified in [\textit{A. Mínguez} and \textit{V. Sécherre}, Duke Math. J. 163, No. 4, 795--885 (2014; Zbl 1293.22005); Proc. Lond. Math. Soc. (3) 109, No. 4, 823--891 (2014; Zbl 1302.22013)] by means of an extension of the theory of types. In fact, there are two extensions of Bushnell and Kutzko's semisimple types for \(\mathrm{GL}_m(F), R = {\mathbb C}\): first the extension to \(\mathrm{GL}_m(D), R = {\mathbb C}\) in preceding articles by Sécherre and Stevens and next in [loc. cit.] to the modular case \((R = \overline {{\mathbb F}}_l, l \neq p)\).NEWLINENEWLINELet \(\pi\) be an irreducible smooth representation of \(G\). We know from [loc. cit.] that there is a unique \(G\)-conjugacy class of supercuspidal pairs \((M,\rho)\) of \(G\) such that \(\pi\) is isomorphic to a subquotient of the parabolic induction \(i_M^G(\rho)\). This class is called the supercuspidal support of \(\pi\). For any inertial class \(\Omega\) of supercuspidal pairs of \(G\), let \(\mathrm{Irr}(\Omega)\) denote the set of classes of irreducible representations of \(G\) with supercuspidal support contained in \(\Omega\). The sets \(\mathrm{Irr}(\Omega)\) form a partition of the set of all classes of irreducible smooth representations of \(G\).NEWLINENEWLINENow consider the full subcategory \({\mathcal R}_R(\Omega)\) of \({\mathcal R}_R(G)\) whose objects are the representations \(\pi\) such that all irreducible subquotients of \(\pi\) have there supercuspidal support in \(\Omega\). In the present article it is proved that the subcategories \({\mathcal R}_R(\Omega)\) form a decomposition of \({\mathcal R}_R(G)\) into blocks. The proof is given via the theory of supertypes.NEWLINENEWLINELet \((J,\lambda)\) be a semisimple supertype of \(G\). The compact open subgroup \(J\) of \(G\) has a finite quotient \(J/J^1\) isomorphic to a product of \(\mathrm{GL}_{n_i}(k_i)\), where the \(k_i\) are extensions of the residue field \(k_F\) of \(F\). Using the known decomposition of the category of representations of \(J/J^1\) the authors construct a full subcategory \({\mathcal R}(J,\lambda)\) of \({\mathcal R}(G)\). The category \({\mathcal R}(G)\) is a product of all subcategories \({\mathcal R}(J,\lambda)\), when the \((J,\lambda)\) are taken modulo equivalence. Also, there is a bijection between inertial classes \(\Omega\) of supercuspidal pairs of \(G\) and equivalence classes of semisimple supertypes \((J,\lambda)\) of \(G\), such that the set \(\mathrm{Irr}(\Omega)\) is identified with the set of classes of irreducible subquotients of \(\mathrm{ind}_J^G(\lambda)\). The block decomposition of \({\mathcal R}(G)\) is now obtained by proving: for corresponding \(\Omega\) and \((J,\lambda)\) we have \({\mathcal R}(\Omega) = {\mathcal R}(J,\lambda)\) and: the categories \({\mathcal R}(J,\lambda)\) are indecomposable.