Fourier coefficients of Eisenstein series of the exceptional group of type \(G_2\)
Theorem 1 here asserts that the Fourier coefficients \[ E_\psi(g,s;f_s) =\int_{N(F)\setminus N(\mathbb A)}E(ng,s;f_s)\psi(n)dn \] of the Eisenstein series \(E(g,s;f_s)=\sum_{\gamma\in P(F)\setminus G(F)}f_s(\gamma g)\) attached to a factorizable element \(f_s\) in the induced representation \(\text{ind}_{P(\mathbb A)}^{G(\mathbb A)}(\delta_P^s)\), where \(G\) is the exceptional group of type \(G_2\) over a number field \(F\) and \(P=MN\) is the maximal parabolic associated to the short simple root \(\beta\) (thus \(M=GL(2)\) and \(N\) is a five dimensional Heisenberg group), are factorizable. Theorem 2 asserts that after multiplication by the normalizing factor \(\zeta(3s)\zeta(6s-2)\zeta(9s-3)\), the unramified factors are equal to those of \(\zeta_K(3s-1)/\zeta_F(3s-1)\), where \(K\) is a three dimensional semisimple commutative algebra over \(F\) determined by the nontrivial character \(\psi\). When \(K/F\) is a cubic extension generated by a root of the irreducible \(x^3+bx-a\) (\(b\not=0\)), it is assumed that \(F\) contains a third root of unity. This gives a new proof that \(\zeta_K(s)/\zeta_F(s)\) is holomorphic for cubic \(K/F\) (a previous proof -- based on a trace formula argument -- is in an unpublished Columbia thesis of P. Kumar). The holomorphy for any \(K/F\) follows from Artin's conjecture.
scientific article; zbMATH DE number 1136640