On the Local Theory of Certain Global Zeta Integrals and Related Problems

Abstract

In this thesis, we are going to develop some local theory for the global integral constructed by D. Ginzburg \cite{Gin} in 1991, which represents the partial $L$-function of the adjoint representation of $\GL_3$. We are going to show the absolute convergence of local integrals and establish their meromorphic continuations. To accomplish this, we have to use the asymptotic expansion of the Whittaker functions. Such asymptotic expansions are developed in N. Matringe's work \cite{Mat1} in 2011 when the local field $F$ is non-archimedean. When $F$ is archimedean, we will follow the method suggested by D. Soudry \cite{Sou} to obtain the asymptotic expansion of Whittaker functions for an irreducible admissible generic representation $\pi$ of $\GL_3(F)$. When $F$ is non-archimedean, we can also show that the local integrals form a fractional ideal, hence they admit a g.c.d, that defines the local adjoint $L$-function for $\GL_3$. When $F$ is archimedean, by applying the asymptotic expansion, we are able to show that the local integrals are continuous bilinear forms on the projective tensor product of the representation spaces. Following the Bruhat Theory, we can obtain the Uniqueness Theorem, which leads to the functional equation of the local archimedean integrals and the definition of local gamma factors $\Gamma(s,\pi, \Ad,\psi)$. To compute $\Gamma(s,\pi, \Ad,\psi)$ explicitly, we will provide a useful lemma which helps us to exchange the order of Mellin transform and Fourier transform. With the help of such a lemma, we are finally able to compute the local Gamma factors explicitly. To complete the local theory for Ginzburg's global integral, we have to establish the local functional equation at the non-archimedean places where the local representation $\pi$ of $\GL_3$ is ramified. The existence of the g.c.d. will be proved in this thesis, yet it still remains to show that the g.c.d. provides the local adjoint $L$-function for $\GL_3$, according to local Langlands correspondence. At the archimedean places, we have to show that the local integrals must be a holomorphic multiple of the local adjoint $L$-function. Moreover, we also expect that the local $L$-function can be written as a finite linear combination of the local integrals $Z(W_v,f_s)$ with $W_v, f_s$ chosen to be $K$-finite. We will consider these unsolved problems in a future work.