Almost unramified automorphic representations for split groups over \(\mathbb F_{q}(t)\). (Q1810560)

Let \({\mathbb F}_q\) be a finite field, \({\mathbb F}={\mathbb F}_q(t)\) and \(\mathbb{A}\) be the adèles of \({\mathbb F}\). This paper concerns some automorphic representations of a split semisimple group \(G\) defined over \({\mathbb F}_q\). Define \(K_v\) to be the pre-image of \(B({\mathbb F}_q)\) under \(G(O_v) \to G({\mathbb F}_q)\) if \(v = \infty\) or 0 and \(K=G(O_v)\) otherwise. Let \({\mathbb K} = \prod_v K_v\). Let \(M = L^2(G({\mathbb F})\backslash G(\mathbb{A})/ \mathbb{K})\). Denote by \(\bigotimes_v H_v\) the convolution algebra of compactly supported measures on \(G({\mathbb A})\) which are left and right invariant under \({\mathbb K}\). The main theorem of this article describes the local constituents of the irreducible representations of \(\bigotimes_v H_v\) that occur in the discrete part of \(M\). These representations lie in the residual discrete spectrum coming from the residues of Eisenstein series. A small error in the paper is corrected in [\textit{A. Prasad}, Erratum to: ``Almost unramified automorphic representations for split groups over \(\mathbb F_q(t)\)'', J. Algebra 280, No. 1, 412--413 (2004)].