On the Archimedean local zeta integrals and L-factors

This thesis mainly comprises two parts. Both parts are related with local archimedean zeta integrals. In Part I, we consider the Rankin-Selberg type L-function for U(1, 1). By Howe’s reductive dual pairs and duality correspondence theory for archimedean case, we may choose a vector in the subspace of joint harmonics(we will discuss this later). Then we can compute the matrix coefficient of the Weil representation of U(1, 1) with trivial splitting character. Combining the matrix coefficient of the discrete series of U(1, 1), we give the explicit computation of the local zeta integral for the chosen vector. Finally, we will get the formula of the local zeta integral for any vector in that subspace. In Part II, we consider the zeta integral constructed by Godement-Jacquet for any irreducible admissible representation π0 of GL(m,H). By Langlands classification, the classification of irreducible tempered representations and the classification of unitary representations of GL(m,H), we see that π0 is the unique irreducible quotient of some representation π induced from the essentially square integrable representations on the minimal parabolic subgroup of GL(m,H). Then using Jacquet’s method we show that the L-factor of π0 is the same as that of π