# Test vector for local zeta integral with ramified character

Suppose $$\pi$$ is an unramified principal series representation of $${\rm GL}_2(F)$$, where $$F$$ is a non-archimedean local field with integers $$\mathfrak{o}$$. Let $$W$$ be a function in its Whittaker model. If $$\chi$$ is a quasi-character of $$F^\times$$ then we define its local zeta integral as $$Z(W, s, \chi, g) = \int_{F^\times} W((\begin{smallmatrix} y & 0 \\ 0 & 1 \end{smallmatrix})g) \chi(y) |y|^{s-1/2}\, d^\times y,$$ see e.g. (6.28) of Gelbart's book. Then for $$W= W_0$$ the unique $$K = {\rm GL}_2(\mathfrak{o})$$-invariant Whittaker function with $$W_0(1)=1$$, we have $$Z(W_0,s,\chi,1) = L(s, \pi \otimes \chi),$$ see e.g. prop 6.17b of Gelbart. Note that if $$\chi$$ is ramified, then we have simply $$L(s,\pi \otimes \chi)=1.$$ Let's write $$y \in F^\times$$ as $$p^n x,$$ where $$x \in \mathfrak{o}^\times$$, and $$p$$ is a uniformizer, and $$\chi$$ as $$\chi(p^nx) = |p^n|^{s'} \chi^*(x),$$ where $$\chi^*$$ is a character of $$\mathfrak{o}^\times$$. Then we have
$$Z(W,s, \chi, 1) = \sum_{n \in \mathbb{Z}} W((\begin{smallmatrix} p^n & 0 \\ 0 & 1 \end{smallmatrix}))|p^n|^{s-1/2+s'} \int_{x \in \mathfrak{o}^\times} \chi^*(x)\,dx,$$ which is $$0$$ if $$\chi$$ is ramified. Contradiction.

Am I correct to interpret the above to mean that the $$W_0$$ that I have chosen above is not the correct choice of test vector for the zeta integral when $$\chi$$ is ramified? If so, what is the correct test vector?

• Are you sure you have quoted Gelbart's Prop 6.17 correctly? It's clear that the local zeta integral with the spherical test vector will be 0 if $\chi$ is ramified. The correct test vector is the new vector of $\pi \otimes \chi$, which is not the same as the image in $\pi \otimes \chi$ of the new vector of $\pi$. – David Loeffler Mar 16 at 0:08
• Gelbart Says: "If $\pi_v$ is class one then $W(\pi_v)$ contains exactly one $K_v$ invariant function $W^0_v(g)$ such that $W^0_v(e)=1$ and for this $W^0_v(g)$ [second display of op] obtains." – Sandpiper Mar 17 at 16:43
• If one needs to take the new vector for $\pi \otimes \chi$, then the $\chi$ in the definition of $Z(W, \chi, s, 1)$ is pretty redundant, no? Also, if I'm not mistaken, the new vector for $\pi \otimes \chi$ in the Kirillov model is $1_{\mathfrak{o}^\times}$, for which $Z(W, \chi, s, 1)$ is still 0. If I take the new vector for $\pi \otimes \chi$ and use that in a zeta integral of the form $Z(W,s,g)$, then this zeta integral equals $L(s, \pi \otimes \chi)$, of course. Is that what Gelbart means here? – Sandpiper Mar 17 at 17:24
• @Sandpiper, I think Gelbart is actually mistaken here; what he has written is only correct if $\chi$ is unramified. – Peter Humphries Mar 17 at 19:28
• @Sandpiper What I meant, more precisely, is that the correct test vector is the vector in $\pi$ whose image in $\pi \otimes \chi$ is the new vector of $\pi \otimes \chi$. – David Loeffler Mar 17 at 23:27

## 2 Answers

[Expanding my comment to an answer]

The new vector of $$\pi$$ is not the correct test vector to use when $$\chi$$ is ramified; and if Gelbart really claims this holds for all $$\chi$$, then he is wrong.

You can decompose the Kirillov model of $$\pi$$ as a (countably infinite) direct sum of eigenspaces for the action of $$\mathfrak{o}^\times$$. It's easy to check that any function that is in an eigenspace other than the $$\chi^{-1}$$-eigenspace will be sent to 0 by $$Z(-, s, \chi, 1)$$. Conversely, a Schwartz function on $$F$$ which lies in the $$\chi^{-1}$$-eigenspace for $$\mathfrak{o}^\times$$ had better vanish at 0 and hence have compact support in $$F^\times$$; so this eigenspace is precisely the linear combinations of the functions $$\chi^{-1} \cdot 1_{\varpi^n \mathfrak{o}^\times}$$, for $$n \in \mathbf{Z}$$ (exercise). So the space of functions $$\{ Z(W, s, \chi, 1) \}$$ for varying $$W$$ is exactly the space of polynomials in $$q^{-s}$$, and the function with Kirillov model $$\chi^{-1} 1_{\varpi^n \mathfrak{o}^\times}$$ will be a test vector for any $$n$$, although $$n = 0$$ is surely the most obvious choice. (Note that the vector with Kirillov function $$\chi^{-1} 1_{\mathfrak{o}^\times}$$ maps to the new vector of $$\pi \otimes \chi$$.)

Yes. The issue is that $$Z(-, s, \chi, 1)$$ is a linear functional which transforms by $$\chi$$ on $$\mathfrak o^\times$$ (say in the Kirillov model). So if you plug in a function which is $$\mathfrak o^\times$$-invariant and $$\chi$$ is ramified, you must get 0.

• Ok, fair enough, but I shouldn't have written the question so that it could admit a yes/no answer. I've edited the original question to ask for what the right test vector should be. – Sandpiper Mar 18 at 9:35
• @Sandpiper You can of course do what David Loeffler suggests in the comments to get a test vector, though I'm not sure if it's the "correct" one. E.g., maybe you want a certain translate of the new vector---see my paper with File and Pitale or Vatsal's recent preprint on the arXiv for the case of ramified test vectors for toric periods. – Kimball Mar 18 at 13:19
• @Kimball It is the correct one. – David Loeffler Mar 18 at 13:40
• @DavidLoeffler Are you saying that vector is of the form $\pi(diag(a,1))W_0$ where $W_0$ is the newvector in $\pi$? I could be misremembering, but I think this is not true. – Kimball Mar 18 at 15:28
• @Kimball No, I am not saying that, it is obviously false; I am saying something else. (The new vector of $\pi \times \chi$ is not the twist of the new vector of $\pi$.) See my detailed answer. – David Loeffler Mar 18 at 23:39