The spin \(L\)-function on the symplectic group GSp(6)
Let \(\pi\) be an automorphic representation of \(GSp_6\) (\(6\times 6\) matrices) over the adeles of a global field. There is an irreducible 8-dimensional ``spin'' representation of the connected \(L\)-group \(\text{GSpin}(7,{\mathbb{C}})\), and a corresponding Langlands \(L\)-function \(L_S(s,\pi,Spin)\) defined for \(Re(s)>>0\) as an Euler product over all but a finite set \(S\) of bad places, whose Euler factor at a finite place \(p\notin S\) is of degree \(8\) in \(Np^{-s}\), where \(N\) denotes the absolute norm. In this paper, the author studies this \(L\)-function when \(\pi\) is irreducible, smooth, generic and cuspidal. He shows that the global spin \(L\)-function \(L_S(s,\pi,Spin)\) has meromorphic continuation to the entire complex \(s\)-plane and is entire except for possible poles at \(s=0,1\) which are necessarily simple. The author's starting point is the Rankin-Selberg integral for \(L(s,\pi,Spin)\) in this case, obtained by Bump and Ginzburg [\textit{D. Bump} and \textit{D. Ginzburg}, Int. Math. Res. Not. 1992, No. 8, 153-160 (1992; Zbl 0808.22005)]. This is an integral which factors into local zeta integrals, which in turn give the local spin \(L\)-function at all but a finite number of places. Let \(\pi\) be a representation (generic, etc.) of \(GSp_6\) over a non-archimedean local field \(F\), and let \(\rho\) be the unramified principal series representation of \(GL(2,F)\) induced from a power of the modular function of the Borel subgroup. The author's first main result is the uniqueness of a bilinear form on \(V_\pi\times V_\rho\) which transforms under a character of the subgroup of upper triangular unipotents in \((2,2,2)\) block form and under the diagonally-embedded \(GL(2,F)\). This is proved by a consideration of Jacquet functors and their adjoint functors, analogously to Bernstein-Zelevinsky; the author uses a (rather complicated) descent through a series of parabolic subgroups. Once he has established this uniqueness, the author deduces that the local zeta integrals which arise in the Bump-Ginzburg integral define rational functions of \(Np^{-s}\), hence have meromorphic continuation to all complex \(s\). In proving this he makes use of the continuation principle of Bernstein in his famous (unpublished) letter to Piatetski-Shapiro. Next, the author establishes the local functional equation for these zeta integrals, and he shows that for good choice of the data the local zeta integral does not vanish (in fact, can be made identically 1). Then the author turns to the archimedean places, establishing continuation and nonvanishing results there. In this he uses estimates for Whittaker functions due to Jacquet, Piatetski-Shapiro and Shalika and to Jacquet-Shalika. Stitching this local information together with the Bump-Ginzburg integral, he arrives at his global result.
scientific article; zbMATH DE number 1123735
scientific article