Regular poles for the p-adic group $GSp_4$

Abstract

We compute the regular poles of the L-factors of the admissible and irreducible representations of the group GSp4 , which admit a nonsplit Bessel functional and have a Jacquet module length of at most 2 with respect to the unipotent radical of the Siegel parabolic, over a non-Archimedean local field of odd characteristic. We also compute the L -factors of the generic representations of GSp4 .

Regular poles for the p-adic group $GSp_4$-II

“…5) After determining the asymptotic behavior of φ u we showed that for some choice of u ∈ V Π we have that C , C 1 , and C 2 above are nonzero. In Proposition 5.8 and Proposition 5.11 of [2] it was proved that this is a consequence of the existence of the homomorphisms from the constituents of the Jacquet module to the character Λ, which depends on the Bessel existence conditions. By using these results in Theorem 5.9 of [2] and Theorem 5.16 of [2] we computed the regular poles of each representations that were considered.…”

“…3) If the length of the Jacquet module is one, then by Proposition 3.2 of [2], for |x| sufficiently small we have φ u (x) = Cχ(x)…”

“…By Lemma 3.4 of [2], a regular pole is the pole of CL(s, χ). 5) After determining the asymptotic behavior of φ u we showed that for some choice of u ∈ V Π we have that C , C 1 , and C 2 above are nonzero.…”

“…2) If the length of the Jacquet module is zero, then by Proposition 3.1 of [2], φ u has a compact support in k * and there is no regular pole.…”

“…By using Iwasawa decomposition in Proposition 2.5 of [2] it was proved that the regular poles of the L-factors are the poles of the meromorphic continuation of the integrals…”