Some consequences of Arthur's conjectures for special orthogonal even groups

Abstract

In this paper we construct explicitly a square integrable residual automorphic representation of the special orthogonal group $SO_{2n}$, through Eisenstein series. We show that this representation comes from an elliptic Arthur parameter $\psi$ and appears in the space $L^2(SO_{2n}(\mathbb{Q})\backslash SO_{2n}(\mathbb{A}_{\mathbb{Q}}))$ with multiplicity one. Next, we consider parameters whose Hecke matrices, at the unramified places, have eigenvalues bigger (in absolute value), than those of the parameter constructed before. The main result is, that these parameters cannot be cuspidal. We establish bounds for the eigenvalues of Hecke operators, as consequences of Arthur's conjectures for $SO_{2n}$. Next, we calculate the character and the twisted characters for the representations that we constructed. Finally, we find the composition of the global and local Arthur's packets associated to our parameter $\psi$. All the results in this paper are true if we replace $\mathbb{Q}$ by any number field $F$.