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Let $F$ be a global number field, i.e. a finite extension of the field of rational numbers. Let $\sigma$, $\pi$ be automorphic representations of $\mathrm{GL}_n(F)$ and $\mathrm{GL}_{n+1}(F)$ respectively. Let $v$ be a finite place of $F$. We consider the local $L$-factor (defined by Jacquet-Piateskii-Shapiro-Shalika) $L(s, \sigma_v \times \pi_v)$.

The local $L$-factors are defined starting with the local JPSS zeta integral $$ Z(s,W, W^{\prime}) = \int_{N_n \backslash \mathrm{GL}_n(F)} W \left( \begin{bmatrix} h & 0 \\ 0 & 1 \end{bmatrix} \right) W^{\prime}(h) |\det(h)|^{s-\frac{1}{2}} \mathrm{d}h $$ for Whittaker functions $W \in \mathcal{W}_{\psi}(\pi)$ and $W^{\prime} \in \mathcal{W}_{\psi}(\sigma)$.

General question: existence of test vectors: Can one construct Whittaker functions $W_0 \in \mathcal{W}_{\psi}(\pi), W_0^{\prime} \in \mathcal{W}_{\psi}(\sigma)$ such that $$ Z(s,W_0, W_0^{\prime}) = L(s, \sigma_v \times \pi_v)? \quad (\star) $$

Answer in the unramified cases: We know the existence of test vectors when the local representations $\sigma_v$ and $\pi_v$ are both unramified.

The test vectors are new vectors, or another name is "essential Whittaker functions". This result may date back to Jacquet-Piateskii-Shapiro-Shalika's article "Conducteur des représentations génériques du groupe linéaire" (1981), especially the theorem on page 208.

My question: what about the case where $\pi_v$ or $\sigma_v$ is ramified (or both are ramified)? Is this problem still open or are there any known results?

Some search:

  • Recently I found the article "Test vectors for Rankin–Selberg L-functions" by Booker-Krishnamurthy-Lee, where the authors construct test vectors for the naive local Rankin-Selberg L-factors through a process of unipotent averaging. However, it seems to me that the naive local Rankin-Selberg L-factors differ from the JPSS local $L$-factors that we want precisely at places where $\pi_v$ or $\sigma_v$ is ramified, missing the local JPSS factor associated to the ramified isobaric summands.
  • In the newly posted article, the author claimed in the proof of Proposition 2.10 that the existence of test vectors follows "by the definition of Rankin–Selberg L-factors" and the original article of JPSS is cited. But I cannot see how this follows directly from the definition. It seems to me that from the definition, we can only say that there exist families of finitely many Whittaker functions $\{W_i\}$ and $\{W_i^{\prime}\}$ such that $$ L(s, \sigma_v \times \pi_v) = \sum_{i} Z(s, W_i, W_i^{\prime}) $$ (e.g. page 7 of Cogdell's note)

Besides the above references, unfortunately, I found no other results of this type for ramified places.

Thank you all for your attention and sorry if the question is too naive to be on this site.

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  • $\begingroup$ Did you read : arxiv.org/abs/1501.07587 (Test vectors for local cuspidal Rankin-Selberg integrals of GL(n), and reduction modulo ℓ ) by Robert Kurinczuk, Nadir Matringe) ? $\endgroup$ Apr 24 at 12:09
  • $\begingroup$ @PaulBroussous Ah, thank you so much! Unfortunately, I have missed this paper. :( $\endgroup$
    – Hetong Xu
    Apr 24 at 14:03

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Are you asking for a proof of existence, or an explicit construction? These are very different things!

It is immediate from the definition that there exists a finite family $(W_i, W_i')_{i \in I}$ with $\sum_i Z(W_i, W_i', s) = L(\pi \times \sigma, s)$, as you quote from Cogdell. It does not follow formally from the definitions that one can always take $\#I = 1$ (although I admit that I do not know of a counterexample off-hand). [*]

However, it is not immediate from the definitions that one can explicitly write down such a family $(W_i, W_i')$ for a given $\pi$ and $\sigma$, and I do not know of any good way of doing this (despite having spent many years intensively researching the arithmetic of Rankin-Selberg L-functions and their generalisations). If the $GL_n$ representation is unramified (but the $GL_{n+1}$ representation is arbitrary), you can take the local new vectors in both reps; that's about as far as explicit recipes can take you.

In some sense the "moral" of the theory is that it's a fruitless pursuit to try to find recipes for test vectors in all cases, and instead, when applying this theory you should build the rest of the machinery flexibly enough that you can allow the test vectors to be whatever they need to be.

[*] If Yifeng claims this in his preprint and doesn't supply a proof, then that is arguably a minor gap in his arguments; but it is a trivially fixable one, since one can just apply his arguments $\#I$ many times and add the results – this is a standard trick in p-adic L-function theory.

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  • $\begingroup$ although I admit that I do not know of a counterexample off-hand - My vague recollection is that Stade concluded you can't always take #I = 1 in the archimedean case. $\endgroup$
    – Kimball
    Apr 24 at 12:41
  • $\begingroup$ Thank you so much! Actually, similar to what Prof. Yifeng Liu did in Proposition 2.10 of his paper, I am hoping to choose good test vectors such that the local JPSS zeta integral $Z(s, W, W^{\prime})$ can be computed explicitly. The best thing that I can hope for is to find $W$ and $W^{\prime}$ wisely such that the local integral is precisely the local $L$-factor. This is the motivation of the post. Therefore, though one can apply his argument $\# I$ times and add the result to get the local $L$-factor, it seems that we cannot get the evaluation of one individual local zeta integral. $\endgroup$
    – Hetong Xu
    Apr 24 at 13:58
  • $\begingroup$ And thank you for pointing out the "moral" of the theory, which I have always been confused about when learning constructions of various $p$-adic $L$-functions. :) $\endgroup$
    – Hetong Xu
    Apr 24 at 14:01
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    $\begingroup$ @Kimball This is not quite accurate. There is a conjecture stated in a survey of Bump that no $K$-finite test vectors exist for archimedean $\mathrm{GL}_n \times \mathrm{GL}_m$ Rankin-Selberg integrals whenever $n \geq m + 2$. For $m \in \{n,n-1\}$, $K$-finite test vectors ought to exist. $\endgroup$ Apr 25 at 14:05
  • $\begingroup$ In any case, as David mentioned in his answer, this question is in general open. See Remark 3.19 of this paper of mine for a brief survey of what is known: arxiv.org/abs/2008.12406. If you are just looking for explicit choices of $W,W'$ for which $Z(s,W,W')$ can be computed explicitly, then Booker-Krishnamurthy-Lee's weak test vector, which gives the naive local $L$-function, should do the job. $\endgroup$ Apr 25 at 14:07

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