Extension of Whittaker functions and test vectors
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Abstract
We show that certain products of Whittaker functions and Schwartz functions on a general linear group extend to Whittaker functions on a larger general linear group. This generalizes results of Cogdell and PiatetskiShapiro (Representation Theory, Number Theory, and Invariant Theory. Progress in Mathematics, Birkhäuser/Springer, Cham, 2017) and Jacquet et al. (Am J Math 105(2):367–464, 1983). As a consequence, we prove that the Rankin–Selberg Lfactor of the product of a discrete series representation and the Zelevinsky dual of a discrete series representation is given by a single Rankin–Selberg integral.
Keywords
Automorphic Lfunctions Rankin–Selberg method Whittaker models Representations of padic groupsMathematics Subject Classification
11F70 11F661 Introduction
In their seminal work, Jacquet–PiatetskiShapiro–Shalika developed the Rankin–Selberg method for automorphic representations, treating the local theory in [14]. The local Euler factors, or Lfactors, are defined as greatest common divisors of families of local Rankin–Selberg integrals. As a consequence of the definition, each Lfactor can be written as a finite sum of Rankin–Selberg integrals, but it is not clear whether one can find test vectors expressing the Lfactor as a single Rankin–Selberg integral. When the local representations are cuspidal, this is possible by an explicit computation [16].
In this article, we prove a local result on the extension of Whittaker and Schwartz functions (stated precisely at the end of this introduction). We show that this result simultaneously generalizes results of [6] and [14], both of which have proved useful in the theory of integral representations of Lfactors. As a consequence of our result we answer the test vector question in the affirmative for the Lfactor of the product of a discrete series representation and the Zelevinsky dual of a discrete series representation.
Let F be a locally compact nonarchimedean local field, \(\psi \) be a nontrivial character of F, and put \(G_n={\text {GL}}_n(F)\). Let \(\pi \) and \(\pi '\) be irreducible smooth representations of \(G_n\) and \(G_m\) with \(n\geqslant m\), and let \({\mathcal {S}}(\pi )\) and \({\mathcal {S}}(\pi ')\) be the standard modules whose irreducible quotients are \(\pi \) and \(\pi '\) respectively. If \(n>m\), the Lfactor \(L(s,\pi ,\pi ')\) attached to the pair \((\pi ,\pi ')\) by Jacquet et al. [14] is a finite sum \(\sum _i I(s,W_i,W'_i)\) of RankinSelberg integrals for Whittaker functions \(W_i\in W({\mathcal {S}}(\pi ),\psi )\) and \(W'_i\in W({\mathcal {S}}(\pi '),\psi ^{1})\) in the Whittaker models of the standard modules. If \(n=m\), it is a finite sum \(\sum _i I(s,W_i,W'_i,\phi _i)\) of Rankin–Selberg integrals with \(W_i,W'_i\) as before and \(\phi _i\in {\mathcal {C}}_c^\infty (F^n)\) a Schwartz function. A tuple \((W,W')\) or \((W,W',\phi )\) whose Rankin–Selberg integral equals the Lfactor is called a test vector. The systematic study of test vectors for local Rankin–Selberg integrals was initiated in the doctoral thesis [15], where many interesting partial results were obtained.
Let \(\delta \) and \(\delta '\) be discrete series representations of \(G_n\) and \(G_m\) respectively, and \({}^{t}\delta '\) denote the Zelevinsky dual of \(\delta '\). In this article we prove the following test vector result:
Theorem
 i.If \(n>m\)$$\begin{aligned} L(s,\delta ,{}^{t}\delta ')=I(s,W,W'); \end{aligned}$$
 ii.If \(n=m\), in addition there exists a Schwartz function \(\phi \) on \(F^n\), such thatMoreover one can always choose \(\phi ={\mathbf {1}}_{({{\mathfrak {p}}}^f)^{n1} \times (1+{{\mathfrak {p}}}^f)}\) for f large enough.$$\begin{aligned} L(s,\delta ,{}^{t}\delta ')=I(s,W,W',\phi ). \end{aligned}$$
In fact, \(L(s,\delta ,{}^{t}\delta ')=L(s,\delta ,\delta ')\), so this also shows that the Euler factor \(L(s,\delta ,\delta ')\) is given by a single integral \(I(s,W,W')\) or \(I(s,W,W',\phi )\); however with \(W'\) inside \(W({\mathcal {S}}({}^{t}\delta '),\psi ^{1})\) rather than inside \(W(\delta ',\psi ^{1})\). Indeed, in this case, this makes the test vector question simpler as \(W({\mathcal {S}}({}^{t}\delta '),\psi ^{1})\) contains \(W(\delta ',\psi ^{1})\) as a proper subspace. An easy, yet already instructive, example is given in Sect. 4.2, where we take \(\delta \) and \(\delta '\) to be Steinberg representations. In the general case, we do not address the question of finding explicit test vectors, which would require other techniques, for example Bushnell–Kutzko type theory in the spirit of [16, 20]. The techniques of this paper are entirely different to [16]; here we use Bernstein and Zelevinsky’s theory of derivatives, in particular Cogdell–PiatetskiShapiro’s interpretation of derivatives [6], to reduce to the cuspidal case.
Theorem 3.10 Let \(n >k \geqslant 1\). For any Schwartz function \(\phi \) on \(F^{nk}\) and any \(W_0\in W(\tau ^{(k)},\psi )\), the map \(\mathrm {det}(g)^{k/2}W_0(g)\phi (\eta _{nk} g)\) extends to a function in \(\tau \subseteq {\text {Ind}}_{N_n}^{P_n}(\psi )\).
In particular, this applies to the case where \(\tau \) is a submodule of the space of restrictions to \(P_n\) of functions in the Whittaker model of a representation of Whittaker type of \(G_n\).
2 Notation and preliminaries
Let G be a locally compact totally disconnected group. By a representation of G we mean a smooth representation on a complex vector space. We denote by \({\mathfrak {S}}(G)\) the category of (smooth) representations of G, and by \({\mathfrak {R}}(G)\) the category of finite length representations of G. When practical, we use the same notation for the collection of objects in a category and the underlying category; so, for example, \(\pi \in {\mathfrak {S}}(G)\) will mean \(\pi \) is an object of \({\mathfrak {S}}(G)\).
We let \(G_0\) denote the trivial group. For \(G=G_n\) we set \({\mathfrak {S}}(n)={\mathfrak {S}}(G_n)\), \({\mathfrak {R}}(n)={\mathfrak {R}}(G_n)\), \({\mathfrak {S}}=\bigsqcup _{n=0}^\infty {\mathfrak {S}}(n)\) and \({\mathfrak {R}}=\bigsqcup _{n=0}^\infty {\mathfrak {R}}(n)\). We also denote by \({\text {Irr}}(n)\) the collection of irreducible representations of \(G_n\), and set \({\text {Irr}}=\bigsqcup _{n=0}^\infty {\text {Irr}}(n)\).
For H a closed subgroup of a locally compact totally disconnected group G, we use the notation \({\text {Ind}}_H^G:{\mathfrak {S}}(H)\rightarrow {\mathfrak {S}}(G)\) for the functor of normalized induction. For representations \(\pi _i\) of \(G_{n_i}\), \(i=1,\dots ,t\) we denote by \(\pi _1\times \cdots \times \pi _t\) the representation of \(G_{n_1+\cdots +n_t}\) obtained from \(\pi _1\otimes \cdots \otimes \pi _t\) by normalized parabolic induction for the standard parabolic of type \((n_1,\cdots ,n_t)\). For a representation \(\pi \) and a character \(\chi \) of \(G_n\) let \(\chi \pi \) be the representation on the space of \(\pi \) given by \((\chi \pi )(g)=\chi (g)\pi (g)\) for \(g\in G_n\).
Definition 2.1
We say that a representation \(\pi \in {\mathfrak {R}}(P_n)\) is of Whittaker type if \({\text {Hom}}_{N_n}(\pi ,\psi )\) is of dimension one. We say that a representation \(\pi \in {\mathfrak {R}}(G_n)\) is of Whittaker type if \(\pi \mid _{P_n}\) is of Whittaker type, i.e. if \({\text {Hom}}_{N_n}(\pi ,\psi )\) is of dimension one.
In either case, a finite length representation \(\pi \) is of Whittaker type if and only if \(\pi ^{(n)}\simeq {\mathbb {C}}\). In particular, if \(\pi _1\) and \(\pi _2\) are both representations of Whittaker type of \(G_{n_1}\) and \(G_{n_2}\) respectively, then the representation \(\pi _1\times \pi _2\) is also of Whittaker type according to [3, Corollary 4.14, (c)]. Let \(\pi \) be a representation of Whittaker type of \(P_n\) (resp. \(G_n\)). By Frobenius reciprocity, there is a unique up to scalar nonzero intertwining operator from \(\pi \) to \({\text {Ind}}_{N_n}^{P_n}(\psi )\) (resp. \({\text {Ind}}_{N_n}^{G_n}(\psi )\)), and we denote by \(W(\pi ,\psi )\) the image of \(\pi \) and call it the Whittaker model of \(\pi \), though it is not always a model of \(\pi \), i.e. in general \(W(\pi ,\psi )\) is a quotient of \(\pi \), but not isomorphic to it.
An irreducible representation of Whittaker type of \(G_n\) is called generic. In fact, by [8], the generic representations of \(G_n\) are those irreducible representations \(\pi \) such that \({\text {Hom}}_{N_n}(\pi ,\psi )\ne 0\). By exactness of the nth derivative functor, a representation of \(G_n\) of Whittaker type has a unique generic subquotient. For the group \(G_n\), it follows from [3, Lemma 4.5] and [23, Sect. 9], that if \(\delta _1,\dots ,\delta _t\) are irreducible essentially square integrable, which we call discrete series, representations, then \(\delta _1\times \cdots \times \delta _t\) is a representation of Whittaker type.

\(=~{{\text {cusp}}}\): cuspidal,

\(=~{{\text {disc}}}\): essentially square integrable/discrete series,

\(=~{{\text {gen}}}\): generic,

\(=~{{\text {stand}}}\): standard module,

\(=~{{\text {stand}}}{{\text {cusp}}}\): cuspidally induced standard module,

\(=~{{\text {Whitt}}}\): Whittaker type,

\(=~{{\text {cent}}}\): with central character.
We denote by \({\mathcal {C}}_c^\infty (F^n)\) the space of smooth functions on \(F^n\) with compact support.
3 Whittaker models and derivatives
Here we recall some useful but not very wellknown facts from [7] and [6] about Whittaker models and the interpretation of derivatives in the space of Whittaker functions. Then we push the techniques of [6] in the spirit of [18] to obtain our first main result, which is Theorem 3.10 about extending Whittaker functions to larger linear groups.
3.1 The Whittaker model of representations of Whittaker type
Here we highlight a result of [7] on Whittaker models of representations of Whittaker type. We shall only need it in order to prove that our first main result Theorem 3.10 extends [14, Proposition 9.1]. However, the result was not known to us until recently, and it is a quite striking result on representations of Whittaker type. It says that the Whittaker model of such a representation is a submodule of the Whittaker model of a standard module. Notice that by the main result of [12], the Whittaker model of a standard module is isomorphic to this standard module and it moreover has a Kirillov model. The result follows from [7, Lemma 3.2.4] and [7, Lemma 4.3.9]. The setting of [7] being much more general and the result being stated there with a different terminology, we give a largely selfcontained proof here, which in any case uses the ideas of [7].
Proposition 3.1
Let \(\pi \in {\text {Irr}}_{{\text {gen}}}(n)\), then it is the unique irreducible submodule of a cuspidally induced standard module \({\mathcal {S}}_c(\pi )\in {\mathfrak {R}}_{{{\text {stand}}}{{\text {cusp}}}}(n)\). All cuspidally induced standard modules containing \(\pi \) are isomorphic.
Proof
By [23, Theorem 6.1], with the same notations, \(\pi \) is of the form \(\langle a\rangle \) for a a sequence of cuspidal segments satisfying the nonpreceding ordering condition (see [23] for the precise statement). However, the representation \(\pi \) being generic, its highest derivative is \(\pi ^{(n)}\simeq {\mathbb {C}}\). By [23, Theorem 6.1], this implies that the cuspidal segments occurring in a are all cuspidal representations, and this exactly says that \(\pi \) is the unique irreducible submodule of a cuspidally induced standard module. It is also a consequence of [23, Theorem 6.1] that all cuspidally induced standard modules containing \(\pi \) are isomorphic \(\square \)
If \(\pi \) is of Whittaker type, we set \({\mathcal {S}}_c(\pi )={\mathcal {S}}_c(\pi ^{{\text {gen}}})\), we will soon show that \(W(\pi ,\psi )\subseteq W({\mathcal {S}}_c(\pi ),\psi )\), as a consequence of the following proposition.
Lemma 3.2
Let \(\pi \in {\mathfrak {R}}_{{\text {Whitt}}}(n)\), if \(\tau \in {\text {Irr}}(n)\) admits a nontrivial extension by \({\mathcal {S}}_c(\pi )\), then \(\tau =\pi ^{{\text {gen}}}\).
Proof
We denote by \({\text {Res}}_{P_n}\) the map from \({\text {Ind}}_{N_n}^{G_n}(\psi )\) to \({\text {Ind}}_{N_n}^{P_n}(\psi )\) which is the restriction of functions to \(P_n\). By the main result of [12], if \(\pi \in {\mathfrak {R}}_{{{\text {stand}}}}(n)\), then \(\pi \simeq W(\pi ,\psi )\) and \({\text {Res}}_{P_n}\) is injective on \(W(\pi ,\psi )\). We will now show that the last part of this result remains true for \(\pi \in {\mathfrak {R}}_{{\text {Whitt}}}\), i.e. that if \(\pi \in {\mathfrak {R}}_{{\text {Whitt}}}\), then \(W(\pi ,\psi )\) has a Kirillov model.
Proposition 3.3
[7, Lemma 3.2.4, Lemma 4.3.9] Let \(\pi \in {\mathfrak {R}}_{{\text {Whitt}}}(n)\) with \(n\geqslant 2\), then \(W(\pi ,\psi )\subseteq W({\mathcal {S}}_c(\pi ),\psi )\), in particular \({\text {Res}}_{P_n}\) is injective on \(W(\pi ,\psi )\).
Proof
Consider \(W({\mathcal {S}}_c(\pi ),\psi )\subseteq W({\mathcal {S}}_c(\pi ),\psi )+W(\pi ,\psi )\subseteq {\text {Ind}}_{N_n}^{G_n}(\psi )\). If \(W(\pi ,\psi )\) was not contained in \(W({\mathcal {S}}_c(\pi ),\psi )\), considering a JordanHölder sequence of \(\frac{W({\mathcal {S}}_c(\pi ),\psi )+W(\pi ,\psi )}{W({\mathcal {S}}_c(\pi ),\psi )}\) , there would exist a \(G_n\)module V such that \(W({\mathcal {S}}_c(\pi ),\psi )\subseteq V \subseteq W({\mathcal {S}}_c(\pi ),\psi )+W(\pi ,\psi )\), and such that \(\tau =\frac{V}{W({\mathcal {S}}_c(\pi ),\psi )}\) is irreducible. If the extension V of \(\tau \) by \({\mathcal {S}}_c(\pi )\) was trivial, then \(\tau \subseteq V \subseteq {\text {Ind}}_{N_n}^{G_n}(\psi )\) would be generic. If nontrivial, by Lemma 3.2, this would also imply that \(\tau \) is generic. This is absurd as \(\tau \) would be a generic subquotient of \(W(\pi ,\psi )\) different from \(W(\pi ^{{{\text {gen}}}},\psi )\), because \(W(\pi ^{{{\text {gen}}}},\psi )\) is contained in \(W({\mathcal {S}}_c(\pi ),\psi )\), contradicting multiplicity 1 for \(W(\pi ,\psi )\). As \({\mathcal {S}}_c(\pi )\) is a standard module, the restriction map \({\text {Res}}_{P_n}\) is injective of on \(W({\mathcal {S}}_c(\pi ),\psi )\), hence on \(W(\pi ,\psi )\). \(\square \)
3.2 The Cogdell–Piatetski–Shapiro interpretation of derivatives
We first recall two important results from [6, Sect. 1]. Then, thanks to those results and those of Sect. 3.1, we can interpret [14, Proposition 9.1] as the inclusion of the Whittaker model of a representation of Whittaker type \(\pi _2\) in the appropriate derivative of the Whittaker model of \(\pi _1\times \pi _2\), for \(\pi _1\) another representation of Whittaker type. Notice that according to [3, Proposition 3.2, (f)], the map \(\Phi ^\) sends the representation \({\text {Ind}}_{N_n}^{P_n}(\psi )\) surjectively onto \({\text {Ind}}_{N_{n1}}^{P_{n1}}(\psi )\). The first result from [6] that we shall need is the following observation, which is an immediate consequence of the proof of [6, Proposition 1.1].
Proposition 3.4
For \(n\geqslant 3\), the map \(\Phi ^{}:{\text {Ind}}_{N_n}^{P_n}(\psi )\rightarrow {\text {Ind}}_{N_{n1}}^{P_{n1}}(\psi )\) identifies with the twisted restriction map \(\nu ^{\frac{1}{2}}{\text {Res}}_{P_{n1}}\), where \({\text {Res}}_{P_{n1}}\) is the restriction of functions to \(P_{n1}\).
The second result is deeper. It is a consequence of the proof of [6, Proposition 1.6] (see [18, Corollary 2.1] for the precise statement and its proof).
Proposition 3.5
In fact, we shall use the above result later. For the moment we rather need the following, which is part of the proof of [18, Corollary 2.1], and is a kind of converse to Proposition 3.5.
Proposition 3.6
Notice that by [3, Proposition 4.13, (a) and (c)] (which is more precise than the more commonly used [3, Corollary 4.14] describing the subquotients of the Bernstein–Zelevinsky filtration), if \(\pi _1\) and \(\pi _2\) are representations of \(G_{n_1}\) and \(G_{n_2}\) such that \(\pi _1^{(n_1)}\simeq {\mathbb {C}}\), then \(\pi _2\subseteq (\pi _1\times \pi _2)^{(n_1)}\). Here we give another result of this type, which by [12] is equivalent to it if \(\pi _1\) and \(\pi _1\times \pi _2\) are standard modules. It is essentially a reformulation of [14, Proposition 9.1] using the above interpretation of derivatives.
Proposition 3.7
Proof
3.3 Extension of Whittaker functions
In this section we prove one of the main results of the paper, which is simultaneously a generalization of one part of [6, Corollary of Proposition 1.7] and of [14, Proposition 9.1]. Both these technical results have proved very useful in the study of RankinSelberg Lfactors. Our generalization will be used in Sect. 4.3 to prove the existence of test vectors for the Lfactors that we are interested in.
Proposition 3.8
Proof
After fixing a Haar measure dx on \(F^r\), we denote by \({\widehat{\phi }}\) the Fourier transform of \(\phi \in {\mathcal {C}}_c^\infty (F^r)\) with respect to dx and the character \(\psi \otimes \dots \otimes \psi \) of \(F^n\).
Corollary 3.9
Proof
We thus obtain the following important result. As we said before, it is obviously a generalization of one part of [6, Corollary of Proposition 1.7], but it is also a generalization of [14, Proposition 9.1] thanks to Proposition 3.7 and Proposition 3.3.
Theorem 3.10
Proof
4 Test vectors for Lfactors of pairs of discrete series
The aim of the second part of this paper is to show that \(L(s,{\text {St}}_l(\rho ),{\text {Sp}}_k(\rho '))=L(s,{\text {St}}_l(\rho ),{\text {St}}_k(\rho '))\) is given by a single RankinSelberg integral. We first recall their definitions.
4.1 Lfactors for pairs of discrete series
All results of this section are fundamental facts from [14]. We normalize the Haar measure on \(G_n\) to give volume 1 to \(K_n\), and on any closed subgroup H of \(G_n\) we normalize the Haar measure to give volume 1 to \(H\cap K_n\). If H is unimodular, this then defines a unique nonzero right invariant measure on \(H\backslash G_n\). We consider \(\pi \in {\mathfrak {R}}_{{\text {Whitt}}}(n)\) and \(\pi '\in {\mathfrak {R}}_{{\text {Whitt}}}(m)\) with \(n\geqslant m\geqslant 1\).
By definition, following the authors of [14] again, if \(\pi \) and \(\pi '\) belong respectively to \({\text {Irr}}(n)\) and \({\text {Irr}}(m)\), then the Whittaker models \(W({\mathcal {S}}(\pi ),\psi )\) and \(W({\mathcal {S}}(\pi '),\psi ^{1})\) are uniquely determined by \(\pi \) and \(\pi '\). We set \(L(s,\pi ,\pi ')=L(s,{\mathcal {S}}(\pi ),{\mathcal {S}}(\pi '))\) where \({\mathcal {S}}(\pi )\) and \({\mathcal {S}}(\pi ')\) are the standard modules over \(\pi \) and \(\pi '\).
Proposition 4.1
Remark 4.2
If \(\rho \) and \(\rho '\) are not equal up to an unramified twist, then it is not new that the RankinSelberg Lfactor \(L(s,{\text {St}}_l(\rho ),{\mathcal {S}}_k(\rho '))=1\) is given by a single integral: this follows from the proof of [14, Theorem 2.7]. If they are equal up to unramified twist, it is enough to show the test vector result for \(\rho '=\rho ^\vee \).
Following Remark 4.2, henceforth we can and will assume that \(\rho '=\rho ^\vee \). We will also suppose that \(\psi \) has conductor 0, i.e. is trivial on \({\mathfrak {o}}\), but not on \({{\mathfrak {p}}}^{1}\).
Let r be the integer such that \(\rho \in {\text {Irr}}_{{{\text {cusp}}}}(r)\). If \(l=k\) we will find \(W\in W({\text {St}}_l(\rho ),\psi )\), \(W'\in W({\mathcal {S}}_k(\rho ^\vee ),\psi ^{1})\), and \(\phi \in {\mathcal {C}}_c^\infty (F^n)\) such that \(I_{lr}(s,W,W',\phi )=L(s,{\text {St}}_l(\rho ),{\text {St}}_k(\rho ^\vee ))\), whereas if \(l>k\), we will find \(W\in W({\text {St}}_l(\rho ),\psi )\) and \(W'\in W({\mathcal {S}}_k(\rho ^\vee ),\psi ^{1})\) such that \(I_{lr,kr}(s,W,W')=L(s,{\text {St}}_l(\rho ),{\text {St}}_k(\rho ^\vee ))\). Hence it is fair to say that we find test vectors for \(L(s,{\text {St}}_l(\rho ),{\text {Sp}}_k(\rho ^\vee ))\) rather then \(L(s,{\text {St}}_l(\rho ),{\text {St}}_k(\rho ^\vee ))\) although the Lfactors are equal.
Taking \(W'\in W({\mathcal {S}}_k(\rho ^\vee ),\psi ^{1})\) makes things simpler, as the space \(W({\mathcal {S}}_k(\rho ^\vee ),\psi ^{1})\) contains \(W({\text {St}}_k(\rho ^\vee ),\psi ^{1})\) as a proper subspace. We believe that it is possible to take \(W'\in W({\text {St}}_k(\rho ^\vee ),\psi ^{1})\), but it seems much more difficult to us. To justify and motivate the fact that allowing to take \(W\in W({\mathcal {S}}_k(\rho ),\psi ^{1})\) simplifies matters, we start with the toy example \(\rho ={\mathbf {1}}\).
4.2 The case of Steinberg representations
If \(l>k\), then Corollary 3.3 of [18] immediately gives the existence of test vectors.
Proposition 4.3
Proposition 4.4
4.3 The general case
In this section, for \(l\geqslant k\geqslant 1\), we find test vectors for \(L(s,{\text {St}}_l(\rho ),{\text {Sp}}_k(\rho ^{\vee }))\) by reducing the problem to the known case of pairs of cuspidal representations ( [16, Theorem 9.1]) thanks to Theorem 3.10.
Proposition 4.5
Set \(n=lr\) and \(m=kr\), we are now in position to prove the second and last main result of the paper.
Theorem 4.6
Proof
Notes
Acknowledgements
This work was started during a research visit of the second author (N.M.) to Imperial College London and he would like to thank them for their hospitality. The visit was supported by the LMS (Research in Pairs Grant) and by GDRI: Representation Theory (20162020). The authors thank David Helm, Gil Moss, Dipendra Prasad and Shaun Stevens for fruitful conversations. Most importantly, we thank the referee for pointing out a mistake in a computation of a previous version, his precise reading and accurate corrections, and his very useful suggestions concerning the presentation of the paper. The second author was supported by the Grant ANR13BS010012 FERPLAY.
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