A construction method for induced types and its application to \(G_2\)
Let \(G\) be a reductive \(p\)-adic group, and let \(\mathfrak R (G)\) be the category of all complex smooth representations of \(G\). Then \(\mathfrak R (G)\) can be decomposed into the product of the Bernstein components \(\mathfrak R^s (G)\), where the parameter \(s\) corresponds to the \(G\)-inertial equivalence class \([M,\sigma]_G\) of an irreducible supercuspidal representation \(\sigma\) of a Levi subgroup \(M\) of \(G\). Let \(J\) be an open compact subgroup of \(G\), and let \(\tau\) be a smooth irreducible representation of \(J\) such that \((J, \tau)\) is a covering pair of \((J_M, \tau_M)\) relative to each parabolic subgroup of \(G\) with Levi factor \(M\). It was proved by \textit{C. Bushnell} and \textit{P. Kutzko} [Proc. Lond. Math. Soc. 77, 582-634 (1998)] that the pair \((J, \tau)\) is an \([M,\sigma]_G\)-type in \(G\). Let \(F\) be a nonarchimedean local field of residue characteristic different from 2 and 3. In this paper the author describes the method of covering pair construction for certain reductive groups, applies this method to the group \(G= G_2 (F)\) of Chevalley type \(G_2\) over \(F\) and the Levi subgroup \(M \cong GL_2 (F)\) attached to the long simple root of \(G_2\), and constructs a sequence of covering pairs for each pair of the form \((J_M, \tau_M)\).
scientific article; zbMATH DE number 1270882
scientific article