An \(L\)-function of degree 27 for spin\(_9\) (Q1417920)

Let \(\mathbb A\) be the ring of adeles of a global field and \(\pi\) a generic cuspidal representation of \(\text{Spin}_9(\mathbb A)\), whose \(L\)-group is \(\text{PGSp}_8(\mathbb C)\). Let \(\rho\) denote the (27-dimensional) second fundamental representation of \(\text{PGSp}_8(\mathbb C)\). This paper studies a Rankin-Selberg integral for \(L_S(\pi,\rho,s)\). It is shown to be Eulerian and the unramified computation is carried out for the local integrals obtained from the factorization of the global integral. The method employed is to embed \(\text{Spin}_9\) into the exceptional group \(F_4\) and to use an Eisenstein series there. This same Eisenstein series is used in the authors' earlier work on the adjoint \(L\)-function of \(\text{GL}_4\) [J. Reine Angew. Math. 505, 119--172 (1998; Zbl 1068.11035)]. This phenomenon fits in the philosophy introduced in \textit{D. Ginzburg} and \textit{S. Rallis} [Int. Math. Res. Not. 1994, No. 5, 201--208 (1994; Zbl 0821.11034)], since the adjoint representation of \(\text{GL}_4(\mathbb C)\) and the second fundamental representation of \(\text{PGSp}_8(\mathbb C)\) have the same structure of invariants. Details have been omitted from several proofs in the paper; a fuller version can be found on the web at \url{http://math.stanford.edu/~bump}.