This paper concerns a correspondence of automorphic representions between the groups \(G_2({\mathbb{A}})\) and \(SL_3({\mathbb{A}})\), where \({\mathbb{A}}\) denotes the ring of adeles of a number field \(F\).
One main result of the paper is to prove an identity between “geometric terms” of the relative Kuznetsov trace formulas on these groups, \[ I_{H,G}(f,\psi)=I_{M_e,M}(f',\psi_M), \] where \(f\) and \(f'\) are “matching” in some sense, and at least one local component of \(f\) must be a matrix coefficient of a supercuspidal. Here \(G=G_2\), \(H\cong SL(3)\), \(M\cong GL(2)\) is a maximal Levi of \(G_2\), \(M_e\cong GL(1)\times GL(1)\), \(\psi\) is a fixed character of \(U(F)\setminus U({\mathbb{A}})\) (\(U\subset G_2\) is a standard maximal unipotent group), and \(\psi_M\) is a corresponding character of \(U_M\).
The other main result of the paper is to prove an identity between “spectral terms”, \[ I_{H,G}(\Pi,f,\psi)=I_{M_e,M}(\pi,f',\psi_M), \] where \(f\) and \(f'\) are “matching” in some sense, \(\pi\) is any cuspidal representation of \(M\) with at least one supercuspidal component, and where \(\Pi\) is a “corresponding” irreducible cuspidal representation of \(G\).
The authors show that \(\Pi\) belongs to the residual spectrum as well, thanks to the “local supercuspidality” assumption on \(f\). This assumption on \(f\) forces many of the terms of the trace formula to vanish, making the comparisons easier, but the authors remark that they believe that the main results hold even without the assumption.
The paper depends most notably on the papers of H. Jacquet [Proc. Indian Acad. Sci., Math. Sci. (Ramanujan Birth Centenary Volume) 97, 117-155 (1987; Zbl 0659.10031)], H. Jacquet and S. Rallis [Pac. J. Math. 154, 265-283 (1992; Zbl 0778.11032)], and D. Jiang [\(G_2\)-periods and residual representations. (English) J. Reine Angew. Math. 497, 17-46 (1998; Zbl 0931.11014].