Tempered representations of \(p\)-adic groups: special idempotents and topology
In the present paper, P. Schneider et E.-W. Zink carry on their investigation of the category \(\mathcal M^t(G)\) of isomophism classes of tempered representations of a connected reductive \(p\)-adic group \(G\), in the same way as the study of smooth representations of \(G\) due to J. Bernstein, and to \textit{C. J. Bushnell} and \textit{P. C. Kutzko} [Proc. Lond. Math. Soc. (3) 77, No. 3, 582--634 (1998; Zbl 0911.22014)]. The abelian category \(\mathcal M^t(G)\), equivalently viewed as the category of all non-degenerate modules over the algebra \(\mathcal S(G)\) of Schwartz functions on \(G\), decomposes into a direct product of component categories \(\mathcal M^t(\Theta)\) where \(\Theta\) runs over the connected components of the Harish-Chandra spectrum \(\Omega^t(G)\), that are the unitary inertial classes of discrete pairs [\textit{P. Schneider} and \textit{E.-W. Zink}, Geom. Funct. Anal. 17, 2018--2065 (2008; Zbl 1144.22012)]. On the other hand, each idempotent \(e\) of \(\mathcal S(G)\) defines the full subcategory \(\mathcal M^t_{e}(G)\) of all non-degenerate \(\mathcal S(G)\)-modules \(V\) generated by \(eV\). The idempotent \(e\) is special in \(\mathcal S(G)\) if the category \(\mathcal M^t_{e}(G)\) is abelian. At first, the authors prove that for each special idempotent \(e\) of \(\mathcal S(G)\), there exists a finite set \(\theta\) of component categories, called the support of \(e\), such that the category \(\mathcal M^t_{e}(G)\) is the direct product of the component categories \(\mathcal M^t(\Theta)\) where \(\Theta\) runs over \(\theta\). Conversely, they describe any finite direct product of components, in particular any component \(\mathcal M^t(\Theta)\), as a subcategory defined by a special idempotent \(e\) of \(\mathcal S(G)\). The strategy of proofs is the one of Bushnell and Kutzko [loc. cit.] but the arguments are of analytic nature. Then, in order to characterize the special idempotents of \(\mathcal S(G)\) belonging to the Hecke algebra \(\mathcal H(G)\), the authors define a preorder on the set of connected components of \(\Omega^t(G)\) that allows them to give a necessary condition on an idempotent of \(\mathcal H(G)\) to be special in \(\mathcal S(G)\). The rest of the paper is devoted to the study of this preorder, restricted to the support of an idempotent associated to an Iwahori subgroup of \(G\). In this case, and under more or less restrictive conditions on \(G\), partial orders are defined from the Kazhdan-Lusztig classification or Bernstein-Zelevinsky classification and the authors prove that their preorder is strictly coarser or finer than the reverse of those.
scientific article; zbMATH DE number 6668631
scientific article