Integral representation of adjoint L-factor for GL(2)

Question

My question is about a local computation in the paper of Gelbart and Jacquet, "A relation between automorphic representations of GL2 and GL3", from 1978.

Let $\sigma$ be an irreducible smooth complex representation of $GL_2(F)$, where $F$ is a non-archimedean local field (characteristic 0 if it helps), and $\chi$ a smooth character of $F^\times$.

Jacquet has defined a Rankin--Selberg $L$-factor $L(\sigma \otimes \sigma^\vee \otimes \chi, s)$, which is always of the form 1/(polynomial in $q^{-s}$) where $q$ is the size of the residue field. Gelbart and Jacquet define the adjoint L-factor by $$ L(Ad(\sigma), \chi, s) := \frac{L(\sigma \otimes \sigma^\vee \otimes \chi, s)}{L(\chi, s)}.$$

In the Gelbart--Jacquet paper, they write down an integral $I(s, f, \Phi, \Psi, W)$ on the metaplectic group $Mp_2(F)$, which depends on $s$ and various choices of auxiliary data ($W$ is a vector in the Whittaker model of $\sigma$, $\Phi$ and $\Psi$ are Schwartz functions on $F$, etc).

If either:

• $\sigma$ is unramified (or a twist of an unramified representation);
• or $\sigma$ is ramified, but $\chi$ is much more ramified than $\sigma$ is, so both $L(\sigma \otimes \sigma^\vee \otimes \chi, s)$ and $L(\chi,s )$ are identically 1,

then they show that the auxiliary data can be chosen in such a way that $I(s, \dots) = L(\operatorname{Ad}(\sigma), \chi, s)$.

Does this hold more generally? Is it true, for arbitrary $\sigma$ and $\chi$, that we can choose a finite collection of quadruples $(f_i, \Phi_i, \Psi_i, W_i)$, $i=1 \dots r$, such that $$\sum_{i=1}^r I(s, f_i, \Phi_i, \Psi_i, W_i) = L(\operatorname{Ad} \sigma, \chi, s)?$$

(EDIT: I didn't make it plain originally that I was willing to allow a finite collection of test data, rather than just one. This form of the statement is true, essentially by definition, for the integral representations of the standard L-function on $GL_n$, and the Rankin--Selberg L-function on $GL_m \times GL_n$, so I'm looking for a generalisation of that to the adjoint L-function.)

Answer 1

I'm not familiar with Gelbart and Jacquet's integral representation you mention (so this is more of a long comment than an answer). One can see some techniques from the local Rankin--Selberg GLn\times GLm convolutions setting in:

• the 2010 dissertation of Kyung-Mi Kim (called Test Vectors of Rankin--Selberg convolutions of general linear groups and available online);

• and in my work with Nadir Matringe last year (arXiv.1501.07587v3) which treats the cuspidal case and gives the existence of data such that a single Rankin-Selberg integral realises the L-factor (we give explicit data using Bushnell--Kutzko type theory when the L-factor is non-trivial (hence in particular n=m); that there exists such data when the L-factor is trivial is known from the work of Jacquet--Piatetski-Shapiro--Shalika as we mention in our introduction.)

You can find more details in the introductions to both of the above. In the Rankin--Selberg case, we expect that an amalgamation of these techniques would answer the general question (giving a finite number of explicit vectors such that the sum of the corresponding integrals realises the L-factor), however determining under which conditions there is a single integral I'm not sure.