Let \(F\) be a number field, and let \({\mathbb A}_F\) be its ring of adeles. If \(\pi\) is a cuspidal representation of \(\text{GL}_2({\mathbb A}_F)\), \(\text{Sym}^n \pi\) is known to be automorphic for \(n=2,3,4\). The \(n=2\) case is the work of Gelbart and Jacquet from 1978, whereas the other cases are established relatively recently by Kim-Shahidi and Kim [H. H. Kim and F. Shahidi, Ann. Math. (2) 155, No. 3, 837–893 (2002; Zbl 1040.11036); H. H. Kim, J. Am. Math. Soc. 16, No. 1, 139–183 (2003; Zbl 1018.11024)]. In this paper, the author neatly summarizes the key features of the proof of automorphy of \(\text{Sym}^3 \pi\), and then discusses a possible method to prove the automorphy of higher symmetric power lifts, with a focus on the \(n=5\) case.
The automorphy of \(\text{Sym}^3\) is a corollary to the automorphy of the tensor product lift \(\boxtimes : \text{GL}(2) \times \text{GL}(3) \to \text{GL}(6),\) since \(\pi \boxtimes \text{Sym}^2 \pi = \text{Sym}^3 \pi \boxplus (\pi \otimes \omega_\pi).\) If \(\pi_1\) and \(\pi_2\) are cusp forms on \(\text{GL}(2)\) and \(\text{GL}(3)\), respectively, the candidate \(\pi_1 \boxtimes \pi_2\) for the tensor product lift is constructed by local means by appealing to the local Langlands correspondence. The automorphy of \(\pi_1 \boxtimes \pi_2\) is proved by the converse theorems machinery of Cogdell and Piatetski-Shapiro. The necessary analytic properties, in order to invoke this machinery, for the triple \(L\)-functions \(L(s,(\pi_1 \boxtimes \pi_2)\times \sigma)\) with \(\sigma\) certain cusp forms on \(\text{GL}(n)\), \(n=1,2,3,4\), are obtained by the Langlands-Shahidi method, applied to \((\text{SL}_5,\text{SL}_2 \times \text{SL}_3)\), \((D_5,\text{SL}_2\times \text{SL}_3 \times \text{SL}_2)\), \((E_6, \text{SL}_2\times \text{SL}_3 \times \text{SL}_3),\) and \((E_7, \text{SL}_2\times \text{SL}_3 \times \text{SL}_4)\).
Since, \(\pi \boxtimes \text{Sym}^4 \pi = \text{Sym}^5 \pi \boxplus (\text{Sym}^3 \pi \otimes \omega_\pi),\) the automorphy of \(\text{Sym}^5\) can be reduced to the automorphy of the tensor product lift \(\boxtimes : \text{GL}(2) \times \text{GL}(5) \to \text{GL}(10).\) In this case, the converse theorem machinery demands nice properties for \(L(s,(\pi_1 \boxtimes \pi_2)\times \sigma)\), for \(\sigma\) living on \(\text{GL}(n),n=1,\dots,8\). When \(n=1,2,3\), the classical Langlands-Shahidi method can be applied to \(\text{SL}_7,D_7,E_8\) to study analytic properties of these triple \(L\)-functions. However for \(n \geq 4\), infinite dimensional groups will have to be considered.
The author makes several insightful remarks on the potential difficulties that can arise in generalizing the Langlands-Shahidi method to these groups. He shows that a straightforward generalization of the theory of Eisenstein series as in [H. Garland, Algebraic groups and arithmetic. Proceedings of the international conference, Mumbai, India, 2001. New Delhi: Narosa Publishing House. Studies in Mathematics. Tata Institute of Fundamental Research 17, 275–319 (2004; Zbl 1157.11314)] will not lead to any new \(L\)-functions. Many interesting new features of the adjoint action of \({^L}M\) on \({^L}{\mathfrak n}\) are exhibited. The induction set-up which makes the main theorem of the author’s [Ann. Math. (2) 132, No. 2, 273–330 (1990; Zbl 0780.22005)], breaks down in the infinite dimensional case. The author also remarks that possibly tools other than Eisenstein series may be required to understand the spectrum since the groups are quite complicated, for instance they are no longer locally compact.