Higher \(\mathrm{Ext}\)-groups in the triple product case (Q6100645)

Let \(F\) be a \(p\)-adic field. Let \(L/F\) be a cubic étale extension and \(D/F\) be a quaternion algebra. Let \(G =\mathrm{Res}_{L/F} D^\times\) and \(H=D^\times\). In this paper the author prove that the higher Ext groups vanish \[ \mathrm{Ext}^{i}_{F^\times\setminus H(F)}\bigl(\pi,\mathbb{C}\bigr) =0, \quad i > 0, \] for any generic \(\pi\in\mathrm{Rep}(F^\times\setminus G(F))\). Combining with results from the local trace formula approach, they obtain a multiplicity formula for any irreducible \(\pi\in\mathrm{Rep}(F^\times\setminus G(F))\).