Almost unramified discrete spectrum for split groups over \(\mathbb{F}_q(t)\)
Let \(G\) be a split adjoint group defined over the finite field \({\mathbb F}_q\). Let \(F={\mathbb F}_q(t)\) and let \(\mathbb{A}\) be the adèles of \(F\). The author considers some automorphic representations of \(G\) in the discrete part of \(L^2(G(F)\backslash G(\mathbb{A}))\). For each place \(v\) of \(F\), denote by \(F_v\) the corresponding local field and by \({\mathcal O}_v\) its ring of integers. Let \(I_v\) denote the Iwahori subgroup in \(G(F_v)\). The main results describe the local constituents at 0 and \(\infty\) of the irreducible representations that occur in \(L^2(G(F)\backslash G(\mathbb{A}))\) that have vectors invariant under the compact open subgroup \[ K = I_\infty \times I_0 \times \prod_{v \neq 0,\infty} G({\mathcal O}_v). \] Results are given in terms of isomorphisms of modules over Hecke algebras, using the Iwahori-Matsumoto involution.
scientific article; zbMATH DE number 1820906