On the symmetric square: orbital integrals (Q1086367)

Stable and unstable orbital integrals on the group \(G=\text{PGL}(3)\) are matched with stable orbital integrals on the groups \(H=\text{SL}(2)\) and \(H_ 1=\text{PGL}(2)\). In particular, the stable orbital integral of the unit element in the Hecke algebra of \(G\) is matched with the stable orbital integral of the unit element in the Hecke algebra of \(H\). These results are fundamental for the study of the symmetric square lifting of automorphic forms from \(H\) to \(G\).