The spin \(L\)-function on the symplectic group GSp(6)
type: Item
instance of: scholarly article
title: The spin \(L\)-function on the symplectic group GSp(6)
zbMATH Open document ID: 0911.22013
DOI: 10.1007/BF02760921
author: San Cao Vo
published in: Israel Journal of Mathematics
publication date: 1999-04-28T00:00:00Z
review text: 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 deduce…
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F46
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 1123735
zbMATH Keywords: automorphic form
zbMATH Keywords: spin \(L\)-function
zbMATH Keywords: local functional equation
zbMATH Keywords: Jacquet functors
zbMATH Keywords: local zeta integrals
zbMATH Keywords: Whittaker functions
reviewed by: Solomon Friedberg
MaRDI profile type: Publication
cites work: REPRESENTATIONS OF THE GROUPGL(n,F) WHEREFIS A NON-ARCHIMEDEAN LOCAL FIELD
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cites work: Rankin-Selberg convolutions for 𝑆𝑂_{2𝑙+1}×𝐺𝐿_{𝑛}: local theory
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description: scientific article; zbMATH DE number 1123735