GGlobal Parabolic Induction and AbstractAutomorphicity
Gal DorTel-Aviv UniversityFebruary 2021
Abstract
In [Dor20a], the author has constructed a category of abstractly au-tomorphic representations for GL p q over a function field F . This is asymmetric monoidal Abelian category, constructed with the goal of hav-ing the irreducible automorphic representations as its simple objects. Thegoal of this paper is to systematically study this category.We will prove several structural theorems about this category. We willshow that it admits an adjoint pair p r aut , i aut q of automorphic parabolicrestriction and induction functors, respectively. This will allow us to showthat the category of abstractly automorphic representations decomposesinto cuspidal and Eisenstein components, in analogy with the Bernsteindecomposition of the category of p -adic representations.Moreover, along the way, we will give a new perspective on the inter-twining operator of GL p q (and on the functional equation for Eisensteinseries), as a form of self-duality of the functor of parabolic induction. Wewill also illustrate how the role of analytic continuation in this theorycan be thought of as trivializing a twist by a certain line bundle, whichcorresponds to an L-function via the results of [Dor20b]. If one choosesto keep the twist as a part of the theory, then one avoids the need foranalytic continuation. Contents
A Introduction 2
A.1 Summary of Results for GL p q A.1.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 4A.1.2 Symmetric Monoidal Structure . . . . . . . . . . . . . . . . . . 6A.1.3 Automorphic Parabolic Induction and Restriction . . . . . . . . 7A.1.4 Bernstein Decomposition . . . . . . . . . . . . . . . . . . . . . . 10
A.2 Speculation for GL p n q a r X i v : . [ m a t h . N T ] F e b .3 Structure of the Paper 13 B Local Theory 13
B.1 Introduction 13B.2 Prerequisites 15
B.2.1 Representation-Theoretical Prerequisites . . . . . . . . . . . . . 15B.2.2 Categorical Prerequisites . . . . . . . . . . . . . . . . . . . . . . 18
B.3 Defining the Multiplication 20
B.3.1 Universal Property of the Multiplication . . . . . . . . . . . . . 22
B.4 Normalized Parabolic Induction 23B.5 Unitality 24
B.5.1 Proof of Unitality . . . . . . . . . . . . . . . . . . . . . . . . . . 26
B.6 Associativity 30B.7 Intertwining Operator 33
B.7.1 Intertwining as Duality . . . . . . . . . . . . . . . . . . . . . . . 33B.7.2 Construction of the Intertwining Operator . . . . . . . . . . . . 34
C Global Theory 40
C.1 Introduction 40C.2 Globalization 42C.3 Parabolic Induction and Automorphicity 44C.4 Decomposition Into Components 47
C.4.1 Cuspidal Components . . . . . . . . . . . . . . . . . . . . . . . 48C.4.2 Eisenstein Components . . . . . . . . . . . . . . . . . . . . . . . 50
C.5 Corollaries 52
C.5.1 Irreducible Abstractly Automorphic Objects . . . . . . . . . . . 53C.5.2 Finite-Length of Automorphic Parabolic Restriction . . . . . . 542 art A
Introduction
Let GL n be the general linear group over a function field F of characteristic ‰ n ?”. The usual definition is thatan irreducible representation of GL n p A q is automorphic if and only if it can begiven as a subquotient of the space of smooth functions on the automorphicquotient GL n p F qz GL n p A q .In [Dor20a], the author has proposed a definition for the case of GL p q whichis no longer restricted to irreducible representations. This was done by definingthe Abelian symmetric monoidal category Mod aut p GL p A qq , the category of abstractly automorphic representations . The category Mod aut p GL p A qq admitsa forgetful functor Mod aut p GL p A qq Ñ Mod p GL p A qq to the usual category of smooth GL p A q -modules. This functor is fully-faithful,and its essential image contains all of the irreducible automorphic (in the usualsense) representations of GL p A q . The paper [Dor20a] proposes that the cate-gory of abstractly automorphic representations Mod aut p GL p A qq is a good set-ting for the study of automorphic phenomena.Unfortunately, due to space constraints, the paper [Dor20a] did not go intodetails about the structural theory of Mod aut p GL p A qq . The main goal of thispaper is to remedy this situation, by thoroughly investigating the properties ofMod aut p GL p A qq , and showing that it has many of the desirable properties onewould want from a “category of automorphic representations”. Essentially, thispaper is a continuation of the research in [Dor20a].Along the way, we will be able to bring to bear some highly non-trivialcategorical tools, and use them to answer questions of automorphic nature,such as questions about intertwining operators and functional equations forEisenstein series. Hopefully, this will demonstrate how this formalism enablesthe use of more categorical methods in the study of automorphic representations.Before giving a more thorough introduction, let us briefly sketch an examplefor the favorable properties and some of the advantages of the constructionMod aut p GL p A qq .There is an analogy between the local theory of p -adic representations andthe global theory of automorphic representations. Both theories admit notionsof induction via parabolic subgroups (parabolic induction in the local theoryand Eisenstein series in the global theory). The notion of “parabolic induc-tion” naturally yields a notion of “cuspidality”, which is associated with nicegrowth properties (this notion is called supercuspidality in the local theory, andcuspidality in the global theory).However, this analogy always occurs in one lower level of categorification inthe global case. Parabolic restriction and induction are functors, while constantterms and Eisenstein series are functions.3ith the framework of abstractly automorphic representations, this analogybecomes much more precise. That is, it turns out that Mod aut p GL p A qq decom-poses as a category into a cuspidal and an Eisenstein part. Instead of being merefunctions, terms such as constant terms and Eisenstein series translate into apair of adjoint functors, which we refer to as automorphic parabolic inductionand restriction . Automorphic parabolic restriction annihilates the cuspidal partof the category, and automorphic parabolic induction generates the Eisensteinpart. The end result is a decomposition analogous to the Bernstein decompo-sition in the local case. The only additional subtlety occurs at the so-called“anomalous spectrum” (see Warning A.1.19 below).The structure of this introduction is as follows. Our goal in this paper is todissect the structure of the category Mod aut p GL p A qq in detail. As a result, wewill start by giving detailed, clean exposition of its structure, stated withoutproofs. This will be done in Section A.1, where we will give a summary of theresults we intend to prove about Mod aut p GL p A qq . In Section A.2, we speculateon how these ideas might possibly be generalized to other general linear groups.Finally, in Section A.3, we will give an overview of the structure of the body ofthis paper. A.1 Summary of Results for GL p q In this subsection, we will summarize the properties of the category of ab-stractly automorphic representations Mod aut p GL p A qq . This subsection is a listof results, stated without proofs. Some of the proofs have already appeared in[Dor20a], and the rest will appear in the body of this paper.The goal of this chapter is to cleanly state all of the structural theoremswe intend to prove about Mod aut p GL p A qq in one place. For the sake of or-ganization, we further sub-divide this subsection into topics as follows. Afterintroducing the basic properties of Mod aut p GL p A qq as a plain category in Sub-section A.1.1, we describe its symmetric monoidal structure in Subsection A.1.2,and its relation with automorphic parabolic induction and restriction (which arerelated to Eisenstein series) in Subsection A.1.3. In Subsection A.1.4 we showhow one can use these automorphic parabolic induction and restriction functorsto describe cuspidal and Eisenstein automorphic representations on the samefooting as the local supercuspidal and principle series representations. A.1.1 Basic Properties
The first statement we make in this subsection is asserting the existence of acomplete and co-complete Abelian category Mod aut p GL p A qq , equipped with acolimit-preserving symmetric monoidal structure denoted by (cid:13) (cid:5) .However, by itself, this statement is fairly empty. One must relate the cat-egory Mod aut p GL p A qq to known categories in order to have a sensible mathe-matical statement. Over the rest of this subsection, we will describe various con-structions and properties that the category Mod aut p GL p A qq possesses, which4ill make this statement meaningful. Theorem A.1.1 (Claim 4.25, Proposition 4.29 and Remark 4.30 of [Dor20a]) . There is a canonical realization functor ι : Mod aut p GL p A qq Ñ Mod p GL p A qq to the category of smooth GL p A q -modules, satisfying the following:1. The functor ι is fully faithful.2. The functor ι respects all limits and colimits. In particular, it is exact.3. The essential image of ι is closed under taking subquotients and contra-gradients.4. The space S G “ S p GL p F qz GL p A qq of smooth and compactly supportedfunctions on GL p F qz GL p A q lies in the essential image of ι . The above is still not enough to uniquely specify Mod aut p GL p A qq and ι .However, the specific construction enjoys many more properties, which we de-scribe below. Warning A.1.2.
Note that despite the fact that the essential image of ι isclosed under subquotients, it is not closed under extensions.Observe that Theorem A.1.1 implies that ι takes irreducible objects to irre-ducible objects, and moreover that any irreducible automorphic representationfrom Mod p GL p A qq lies in the essential image of ι . We will prove that theconverse also holds: Theorem A.1.3 (Theorem C.5.1) . The realization ι p M q of an irreducible object M P Mod aut p GL p A qq is an irreducible automorphic representation, in the senseof being a subquotient of the contragradient Ą S G of S G “ S p GL p F qz GL p A qq . In particular, the functor ι induces a bijection between irreducible objects ofMod aut p GL p A qq and irreducible automorphic representations in Mod p GL p A qq in the classical sense.Recall that the category Mod p GL p A qq has a large center Z acting on it. Claim A.1.4 (Follows from Lemma 2.14 of [Dor20a]) . The center Z has aunique action on Mod aut p GL p A qq which is compatible with the realization func-tor ι : Mod aut p GL p A qq Ñ Mod p GL p A qq . In particular, objects in Mod aut p GL p A qq of finite length have decomposi-tions into generalized Hecke eigenspaces. Remark
A.1.5 . The following claim is beyond the scope of this paper, and willnot be proven here. Observe that the category Mod aut p GL p A qq is enriched overVect “ Vect C , the category of complex vector spaces. However, it is possible to5how that it actually has a rational structure . This means that Mod aut p GL p A qq is given by base change from a category Mod aut Q p GL p A qq enriched over thecategory of rational vector spaces Vect Q :Mod aut p GL p A qq “ Vect C b Vect Q Mod aut Q p GL p A qq . A.1.2 Symmetric Monoidal Structure
As mentioned above, the category Mod aut p GL p A qq of abstractly automorphicrepresentations carries a symmetric monoidal structure, which we denote by (cid:13) (cid:5) .This notation was chosen deliberately in order to emphasize that this monoidalstructure is not compatible with any standard symmetric monoidal structureof Mod p GL p A qq . In this subsection, we describe some of the properties of thissymmetric monoidal structure.First, we claim that: Claim A.1.6 (Lemma 2.14 of [Dor20a]) . The symmetric monoidal structure (cid:13) (cid:5) is compatible with the action of the center Z on Mod aut p GL p A qq . In fact, we can also make the following statement. Recall that an irreducibleautomorphic representation V P Mod p GL p A qq is called generic if it admits aWhittaker model. Claim A.1.7 (Example 3.54 of [Dor20a]) . Let M P Mod aut p GL p A qq be anirreducible object. Then the realization ι p M q is a generic automorphic represen-tation if and only if M admits a non-zero map from the unit of the symmetricmonoidal structure (cid:13) (cid:5) . Moreover, such a map induces an isomorphism: M – M (cid:13) (cid:5) M. Remark
A.1.8 . Note that Claims A.1.6 and A.1.7 imply that (cid:13) (cid:5) is well-behavedwith respect to the spectral decomposition of objects in Mod aut p GL p A qq . Thisis very different from the behaviour of the usual tensor product b on the categoryMod p GL p A qq , where the product of irreducible objects is typically very far fromirreducible.Additionally, we can describe the realization of the unit of (cid:13) (cid:5) . Recallthat S G “ S p GL p F qz GL p A qq is the space of smooth and compactly sup-ported functions on GL p F qz GL p A q . Let I G Ď S G be the subspace consistingof functions that are orthogonal to all characters of the form χ p det p g qq , for χ : A ˆ { F ˆ Ñ C ˆ a grossencharacter.Then we claim that: Claim A.1.9 (Follows from Definition 4.26 of [Dor20a]) . The realization of theunit of the symmetric monoidal structure (cid:13) (cid:5) is canonically isomorphic to I G . .1.3 Automorphic Parabolic Induction and Restriction Let T be the torus GL ˆ GL . Denote by Mod aut p T p A qq the category of smoothrepresentations of the Abelian group T p A q{ T p F q . This category admits a canon-ical realization functor ι : Mod aut p T p A qq Ñ Mod p T p A qq into the category of smooth T p A q -modules, given by restriction along the map T p A q Ñ T p A q{ T p F q . Moreover, Mod aut p T p A qq admits a symmetric monoidal structure b T p A q{ T p F q given by the relative tensor product over T p A q{ T p F q . Let us abuse notationand denote this symmetric monoidal structure by (cid:13) (cid:5) as well.It is easy to see that Mod aut p T p A qq , along with ι and (cid:13) (cid:5) , satisfies the appro-priate analogues of Theorems A.1.1 and A.1.3, as well as Claims A.1.4, A.1.6,A.1.7 and A.1.9.Therefore, it is natural to ask about the compatibility of Mod aut p T p A qq withMod aut p GL p A qq . Indeed, it turns out that the two structures are intimatelyrelated.Fix a parabolic subgroup P Ď GL , with Levi given by T . Let i : Mod p T p A qq Ñ Mod p GL p A qq be the usual (un-normalized) parabolic induction functor with respect to P .Our first claim is that this functor respects abstract automorphicity: Theorem A.1.10 (Theorem C.2.1 and Theorem C.3.3) . There is an essentiallyunique functor i aut completing the diagram: Mod aut p T p A qq i aut (cid:15) (cid:15) ι (cid:47) (cid:47) Mod p T p A qq i (cid:15) (cid:15) Mod aut p GL p A qq ι (cid:47) (cid:47) Mod p GL p A qq . The functor i aut is non-unital right-lax with respect to the symmetric monoidalstructure (cid:13) (cid:5) .Recollection A.1.11 . Recall that a non-unital right lax functor between the twomonoidal categories C and D is a functor F : C Ñ D equipped with a naturaltransformation (referred to as the multiplication on F ): F p A q b D F p B q Ñ F p A b C B q , satisfying an appropriate associativity property. Essentially, this means that F sends non-unital algebras to non-unital algebras.Similarly, there is a notion of a unital right-lax functor , which is also equippedwith a unit map D Ñ F p C q satisfying appropriate compatibility conditions.A unital right-lax functor sends unital algebras to unital algebras.The reader can find out more about these notions in, e.g., [nLa21].7 emark A.1.12 . The non-unital right-lax structure on the functor i aut is non-canonical. Rather, it turns out that there is a twist p i aut of i aut , which is canoni-cally non-unital right-lax. The twist p i aut is isomorphic i aut , but the isomorphismbetween them is non-canonical.We will give a more detailed explanation of this phenomenon in Section B.4below. For now, we can give an informal summary. It turns out that the differ-ence between i aut and p i aut is related to the analytic continuation process usuallyused in defining the intertwining operator. Specifically, p i aut is a twist of i aut bya (non-canonically) trivial line bundle, which has a canonical trivialization afteranalytic continuation.In particular, p i aut enjoys a canonical description of the intertwining operatorwhich does not require analytic continuation. That is, the role of the analyticcontinuation in the usual theory is merely to trivialize the twist giving p i aut from i aut .This issue is also related to the L-function that appears in the formula forthe functional equation for Eisenstein series. Specifically, it turns out that thespace by which the functor i aut needs to be twisted is precisely the space of zetaintegrals for this L-function, in the sense of [Dor20b]. See also Remark B.3.2. Remark
A.1.13 . One can also obtain a treatment of the intertwining opera-tor that does not require analytic continuation by taking Fourier transforms(essentially, Kirillov models). See, e.g., Chapter 2, Section 3, Subsection 3 of[GGPS69].
Remark
A.1.14 . In the course of proving Theorem A.1.10, we will also obtainthe functional equation for Eisenstein series as an almost immediate corollary.This relative simplicity is a benefit of our approach. See Remark C.3.9.We refer to i aut as automorphic parabolic induction . Moreover, it turns outthat while the usual parabolic restriction functor r : Mod p GL p A qq Ñ Mod p T p A qq is very badly behaved with respect to automorphic representations, one can stillhave a good notion of automorphic parabolic restriction: Theorem A.1.15 (Remark C.3.5 and Theorem C.5.4) . The functor i aut : Mod aut p T p A qq Ñ Mod aut p GL p A qq has a left adjoint r aut , referred to as automorphic parabolic restriction .Moreover, the functor r aut sends objects of finite length to objects of finitelength.Remark A.1.16 . Note that unlike r aut , the usual parabolic restriction functor r does not send objects of finite length to objects of finite length. For example, anirreducible automorphic representation may be principle series at every place,in which case its “usual” parabolic restriction has infinite length.It is natural to expect that the functor r aut kill abstractly automorphicrepresentations that are “cuspidal”. Indeed:8 laim A.1.17 (Corollary C.4.3) . If M P Mod aut p GL p A qq is irreducible, andits realization ι p M q is a cuspidal automorphic representation, then r aut p M q “ .Remark A.1.18 . Note that the analogue of Claim A.1.17 fails to hold for theusual parabolic restriction functor r . Indeed, there are cuspidal irreducible auto-morphic representations which are principle series at every place, and thus theirusual parabolic restriction is non-zero. This illustrates the favorable propertiesof the automorphic parabolic restriction r aut . Warning A.1.19.
While the functor r aut kills all abstractly automorphic rep-resentations whose realization is cuspidal, the converse is not true. The prob-lem is that there are irreducible automorphic representations that appear assubquotients in parabolic induction from automorphic characters, but never assub-representations. Abstractly automorphic representations with such realiza-tions are killed by r aut , despite clearly being “Eisenstein” in nature.These examples occur in the so-called anomalous , or non-isobaric , spectrumof GL p q . Specifically, if we let triv T P Mod aut p T p A qq be the trivial T p A q{ T p F q -module, then the induction: V “ i aut p triv T q is infinitely reducible. The object V contains a unique irreducible sub-object,whose realization is the trivial G -module triv G . The remaining irreducible sub-quotients have realizations that are trivial in almost all places, but Steinberg inthe rest. It is possible to show that the quotient of V by its unique irreduciblesub-object is killed by the automorphic parabolic restriction functor r aut . Warning A.1.20.
Warning A.1.19 causes an unfortunate clash of notationaround the term “cuspidal”.The situation mentioned in Warning A.1.19, where some representationsare subquotients of parabolically induced ones but are still killed by parabolicrestriction, is not unique to the abstractly automorphic setting. For example,when one considers the representations of p -adic groups with coefficients in F (cid:96) , (cid:96) ‰ p , a similar situation occurs.In such settings, it is customary to use the term “cuspidal” to refer to anyrepresentation killed by parabolic restriction, and reserve the term “supercus-pidal” to refer to representations that have nothing at all to do with parabolicinduction. See [KS15] for a randomly chosen modern example, and compare thelanguage of [Vig89] and [Vig01]. Unfortunately, it is not customary to refer toanomalous representations, as in Warning A.1.19, as “cuspidal”. Moreover, it isalso customary for cuspidal automorphic representations to not be referred toas “supercuspidal”.This paper is aimed at primarily at researchers of automorphic represen-tation theory. Therefore, the author has chosen to adopt the convention ofthis theory over its more esoteric counterparts. In particular, in this text, theterm “cuspidal” is used to refer to representations that are wholly unrelated toparabolic induction, while the term “anomalous” is used to refer to representa-tions killed by parabolic restriction yet are not cuspidal.9 .1.4 Bernstein Decomposition In this subsection, we will describe a few additional properties of the categoryMod aut p GL p A qq . Specifically, we will discuss its decomposition into compo-nents, in a way analogous to the local Bernstein decomposition.In the local theory of smooth GL -modules over local fields, one decom-poses the category of smooth GL -modules into components. Each componentis either supercuspidal, or induced from pairs of characters of GL . This de-composition is done via the parabolic induction and restriction functors i, r ,and supercuspidal components are killed by r . This is called the Bernsteindecomposition.It turns out that a similar decomposition holds for the global categoryMod aut p GL p A qq . Let C p GL p A qq “ C Eis p GL p A qq ž C cusp p GL p A qq be the disjoint union of the set C Eis p GL p A qq with the set C cusp p GL p A qq . Here, C Eis p GL p A qq is the set of unordered pairs of characters A ˆ { F ˆ Ñ C ˆ up tocontinuous twists |¨| s b |¨| s . The set C cusp p GL p A qq consists of irreduciblecuspidal automorphic representations up to continuous twist | det p¨q| s . Usingthe functors i aut , r aut from Subsection A.1.3 above, we will show: Theorem A.1.21 (Theorem C.4.12) . There is a canonical decomposition
Mod aut p GL p A qq “ ź c P C p GL p A qq Mod aut c p GL p A qq compatible with the symmetric monoidal product (cid:13) (cid:5) .Moreover, the restriction of r aut to Mod aut c p GL p A qq with c P C cusp p GL p A qq is . In fact, note that an irreducible M P Mod aut p GL p A qq always lies in a sin-gle component Mod aut c p GL p A qq for some c P C p GL p A qq . The decompositionabove has the property that if the realization of M is an irreducible cuspidalautomorphic representation, then c P C cusp p GL p A qq is the component corre-sponding to ι p M q . Likewise, if the realization of M is a subquotient of theEisenstein series parabolically induced from χ b χ , then c P C Eis p GL p A qq corresponds to χ b χ . Remark
A.1.22 . As mentioned in Warning A.1.19, due to the anomalous spec-trum, there are some subquotients of parabolic inductions i aut p χ b χ q whichcannot be expressed as sub-modules. In particular, for these specific representa-tions, we have that r aut is equal to 0 despite the fact that these representationsbelong to Eisenstein components c P C Eis p GL p A qq .A similar issue appears in the kind of Bernstein decomposition that occursin the case of p -adic representations with coefficients in characteristic (cid:96) ‰ p , asin [Vig89]. 10 .2 Speculation for GL p n q The goal of this section is to speculate about possible generalizations of theresults of this paper, stated in Section A.1, to other general linear groups G “ GL n , with n ą
2. Other than Remark A.2.1 at the very end of this section, wewill not consider more general reductive groups. Unlike Section A.1, nothinghere is proven. Instead, this section merely contains speculation based on theauthor’s experience in proving the results of Section A.1. In fact, there mighteven be non-trivial obstructions for some of the speculation here! Hence, we askthat the reader treat the speculation of this section with the caution appropriatefor educated guesses.The text below is essentially a re-formulation of Section A.1, only stated ingreater generality. As a result, we will be much more succinct.Let F be a function field of characteristic ‰
2. Our guess is that for each G “ GL n p A q there should be a complete and co-complete Abelian symmetricmonoidal category Mod aut p G q , equipped with a colimit-preserving symmetricmonoidal structure denoted by (cid:13) (cid:5) . Of course, this claim is vacuous by itself.Instead, we conjecture that for every G “ GL n p A q there is such a category, withrealization functors and compatibility properties as follows:1. There is a realization functor ι : Mod aut p G q Ñ Mod p G q to the category ofsmooth G -modules, satisfying the following:(a) The functor ι is fully faithful.(b) The functor ι respects all limits and colimits. In particular, it isexact.(c) The essential image of ι is closed under taking subquotients and con-tragradients.(d) The space S G “ S p G p F qz G p A qq of smooth and compactly supportedfunctions on G p F qz G p A q lies in the essential image of ι .Note that this implies that ι sends irreducible objects to irreducible ob-jects, and that all irreducible automorphic representations lie in the es-sential image of ι .2. The realization ι p M q of an irreducible object M P Mod aut p GL n p A qq is anirreducible automorphic representation, in the sense of being a subquotientof the contragradient Ą S G of S G .3. Let Z p G q denote the center of the category Mod p G q . Then Z p G q hasa unique action on Mod aut p G q which is compatible with the realizationfunctor ι : Mod aut p G q Ñ Mod p G q .4. The symmetric monoidal structure (cid:13) (cid:5) is compatible with the action of thecenter Z p G q on Mod aut p G q . 11. Let M P Mod aut p G q be an irreducible object. Then the realization ι p M q isa generic automorphic representation if and only if M admits a non-zeromap from the unit of the symmetric monoidal structure (cid:13) (cid:5) . Moreover,such a map induces an isomorphism: M – M (cid:13) (cid:5) M.
6. Let I G be the subspace of S G “ S p G p F qz G p A qq which is orthogonal toall automorphic forms of residue of Eisenstein series type. Then the real-ization of the unit of the symmetric monoidal structure (cid:13) (cid:5) is canonicallyisomorphic to I G .7. Let P Ď G be a parabolic subgroup, and let M be the corresponding Levi.Then M – ś GL n i . We note that the category of smooth M -modules isthe Lurie tensor productMod p M q “ â Mod p GL n i p A qq , and let Mod aut p M q “ â Mod aut p GL n i p A qq also be the Lurie tensor product. We define the realization functor ι : Mod aut p M q Ñ Mod p M q to be the tensor product of the realization functors for each GL n i . Let i : Mod p M q Ñ Mod p G q be the parabolic induction functor along P . Then there is an essentiallyunique functor i aut completing the diagram:Mod aut p M q i aut (cid:15) (cid:15) ι (cid:47) (cid:47) Mod p M q i (cid:15) (cid:15) Mod aut p G q ι (cid:47) (cid:47) Mod p G q .
8. The functor i aut (or one of its twists) is non-unital right-lax with respectto the symmetric monoidal structure (cid:13) (cid:5) .9. The functor i aut : Mod aut p M q Ñ Mod aut p G q has a right adjoint r aut , sat-isfying the following:(a) The functor r aut sends objects of finite length to finite length.(b) If M P Mod aut p G q is irreducible, and its realization ι p M q is a cuspidalautomorphic representation, then r aut p M q “ aut p G q admits a decompositionMod aut p G q “ ź c P C p G q Mod aut c p G q compatible with the symmetric monoidal product (cid:13) (cid:5) , for an appropriateset of components C p G q indexed by cuspidal automorphic representationsof G and its subgroups. Remark
A.2.1 . The above speculation is probably not quite true for more gen-eral reductive groups. For example, in an upcoming paper, the author intendsto investigate the case of quaternion groups B . In this case, the categoryMod aut p B p A q ˆ q turns out to not quite be symmetric monoidal, but instead bea self-dual module category for Mod aut p GL p A qq . Instead, the author can spec-ulate that there is a coloured operad encoding associative bi-linear operationsbetween the various categories Mod aut p G q for more general reductive groups.That is, the author expects that instead of the bi-linear operation (cid:13) (cid:5) : Mod aut p GL p A qq b Mod aut p GL p A qq Ñ Mod aut p GL p A qq , one has more sophisticated bi-linear operations relating abstractly automorphicrepresentations for several different groups. These operations should possessvarious associativity constraints, and include various functors induced by thetheta correspondence. A.3 Structure of the Paper
The structure of this paper is as follows. In Part B, we will establish theprerequisite local results for the global ones that will follow. Specifically, we willfocus on describing the interaction of the parabolic induction functor with thesymmetric monoidal structure (cid:13) (cid:5) . Our main result will be Theorem B.6.1, whichstates that the parabolic induction functor (when appropriately normalized) isnon-unital right-lax.In Part C, we will take the local theory described above and use it to inves-tigate the category Mod aut p G q . We will show that parabolic induction respectsthe property of being abstractly automorphic in Theorem C.3.3, and show thedecomposition of Mod aut p G q into components in Theorem C.4.12. Part B
Local Theory
B.1 Introduction
In this part, we will analyze the relationship of the parabolic induction functorwith the symmetric monoidal structure (cid:13) (cid:5) , focusing solely on the local case.13pecifically, let F be a local field, let G “ GL p F q be the general lineargroup over F , let U “ U p F q Ď G be the subgroup of upper triangular unipotentmatrices, and let T “ T p F q Ď G be the subgroup of diagonal matrices. Thisdata defines the (un-normalized) parabolic induction functor i : Mod p T q Ñ Mod p G q between smooth T -modules and smooth G -modules. The left adjoint to i isdenoted by r : Mod p G q Ñ Mod p T q .The functors r, i play an important role in the study of the category Mod p G q (see, e.g., [BZ76]). We think of Mod p T q as having the symmetric monoidal struc-ture b T given by the relative tensor product over T , and of Mod p G q as havingthe monoidal structure (cid:13) (cid:5) of [Dor20a]. It is natural to try to understand theinteraction of the adjoint pair p r, i q with these symmetric monoidal structures.It will turn out that in order to have good monoidal properties, we must firstre-normalize the functor i : Mod p T q Ñ Mod p G q into a new functor p i : Mod p T q Ñ Mod p G q , which is a twist of i . This re-normalization is essentially an incarnationof the L-function that appears in the study of the intertwining operator forGL p q .Once we have done so, we will be able to show our main result for this part:the functor p i : Mod p T q Ñ Mod p G q is non-unital right-lax. Remark
B.1.1 . Recall the notion of a non-unital right-lax functor from Recol-lection A.1.11.In our case, we will use this property in Part C because it implies that anyobject in the image of such a right-lax functor F is automatically enhanced withthe structure of a (non-unital) F p C q -module.After showing that p i is canonically non-unital right-lax, we will then showthat p i is “almost” unital right-lax, in the following sense. It turns out that therestriction p i r η ´ s : Mod p T qr η ´ s Ñ Mod p G q to an appropriate localization of Mod p T q (corresponding to formally invertingthe L-function appearing in the normalization factor) is unital right-lax.As an added bonus, we will construct the intertwining operator as a con-sequence of the multiplicative structure on p i . See Section B.7 for a preciseformulation.The structure of this part is as follows. In Section B.2, we formally introduceour notation as well as the functors r, i and their basic properties. In Section B.3,we will define the data that gives the multiplication on i , as well as observe theneed for the normalization factor. In Section B.4, we introduce the normalizedparabolic induction functor p i with its multiplication map. Note that at thispoint we will not yet know that this multiplication is associative.In Section B.5, we will discuss the unitality of the multiplication on p i . We willshow that p i cannot be unital unless some specific map η is inverted in Mod p T q .Additionally, in Section B.6, we will finish the proof that p i is non-unital right-laxby showing that its multiplication is associative.14inally, in Section B.7, we will show that the multiplication on p i induces aself-duality, which turns out to be the intertwining operator. This will let us givea relatively clean description of this intertwining operator and its properties, interms of a morphism of Frobenius algebras. In particular, we completely avoidthe process of analytic continuation. B.2 Prerequisites
We will follow the notation of [Dor20a]. We begin by recalling this notation inSubsection B.2.1 below, which will introduce notation for the parabolic induc-tion and restriction functors as well.Afterwards, we will give some category-theoretical reminders in Subsec-tion B.2.2, specifically about the notions of Frobenius algebras and categories.This subsection will only be useful in Section B.7, where we use these notionsto analyze the intertwining operator. Since the rest of the text has little to nodependence on these categorical notions, readers might want to skip Subsec-tion B.2.2 on a first reading.
B.2.1 Representation-Theoretical Prerequisites
For this part, we let F be a non-Archimedean local field of characteristic notequal to 2. Denote by O Ď F the ring of integers, and let q be the size of itsresidue field. Let G “ GL p F q , and select the following subgroups. We set U “ "ˆ ˚ ˙* , the group of upper triangular unipotent matrices. The subgroup U Ď G isnormalized by the torus T “ "ˆ ˚ ˚ ˙* Ď G. Denote the category of smooth left G -modules by Mod p G q “ Mod L p G q , andlet I RL : Mod L p G q Ñ Mod R p G q denote the equivalence between smooth left and right G -modules given by I RL p π qp g q ¨ v “ π p g T q ¨ v for v P V and p π, V q P Mod L p G q . Denote the inverse of I RL by I LR .For an algebraic variety X , we let S p X q “ S p X p F qq be the space of locallyconstant functions with compact support. In particular, if X carries a G -action,then S p X q is an object of Mod p G q .We fix a non-trivial additive character e : F Ñ C ˆ , and use it to define θ : U Ñ C ˆ ˆ b ˙ ÞÑ e p b q .
15e denote the functor of θ -co-invariants by Φ ´ : Mod p G q Ñ Vect.We give the category Mod p G q a symmetric monoidal structure as follows.Let Y “ S p M p F q ˆ F ˆ q be the S ˙ G -module defined in [Dor20a], and (cid:13) (cid:5) the corresponding symmetric monoidal structure on Mod p G q : A (cid:13) (cid:5) B “ ` I RL p A q b I RL p B q ˘ b G ˆ G Y. Here, the tensor product is taken over the first and third actions of G on Y ,with the G action on the result A (cid:13) (cid:5) B coming from the remaining G -action on Y . Note that the order of the specific G -actions used to define (cid:13) (cid:5) is irrelevant,due to the symmetry under S .We identify the unit of the symmetric monoidal structure (cid:13) (cid:5) with (cid:5) “ ! f : G Ñ C ˇˇˇ f is smooth andcompactly supported mod U , f p ug q “ θ p u q f p g q ) , where G acts by right-translation. Remark
B.2.1 . For the sake of being self-contained, we explicitly recall some ofthe actions on Y .First, the actions of the first and third copies of G on Y are given by: p g , , g q ¨ Ψ p x, y q “ Ψ ` g ´ xg ´ T , y ¨ det p g g T q ˘ for g , g P G , y P F ˆ , x P M p F q and Ψ p x, y q P Y “ S p M p F q ˆ F ˆ q . Thesetwo actions are interchanged by the action of the permutation p , q P S , whichis given by: Ψ p x, y q ÞÑ Ψ p x T , y q . Moreover, the remaining action of G on Y is given by: ˆ , ˆ a b ˙ , ˙ ¨ Ψ p x, y q “ | a | ¨ e p by det p x qq ¨ Ψ p x, ay q and ˆ , ˆ ´ ˙ , ˙ ¨ Ψ p x, y q “ | y | ż M p F q Ψ p z, y q ¨ e p´ y ¨ (cid:104) x, z (cid:105) q d z. Here, the bi-linear pairing (cid:104) x, z (cid:105) on M p F q is given by tr p x q tr p z q ´ tr p xz q , andthe measure d z on M p F q is normalized such that e is self-dual.Recall that the functor Φ ´ : Mod p G q Ñ Vect acts as a trace functor, in thesense that the pairing Φ ´ p A (cid:13) (cid:5) B q is naturally isomorphic to the bi-functor I RL p A q b G B , where b G is the usualrelative tensor product over G . This bi-functor is a self-duality of Mod p G q .This turns Mod p G q into a Frobenius algebra in the (higher categorical) sense of C -linear presentable categories. See also Subsection B.2.2 below.16imilarly, letting Mod p T q denote the category of smooth T -modules, we giveMod p T q a symmetric monoidal structure as follows. The product is given bythe relative tensor product over T : A b T B, with unit T given by S p T q “ S p F ˆ ˆ F ˆ q .Let us describe the parabolic induction and restriction functors, which arethe main objects of research in this paper. Definition B.2.2.
Let i : Mod p T q Ñ Mod p G q be the functor given by: i p V q “ " f : G Ñ V ˇˇˇˇ f is smooth , f ˆˆ a bd ˙ g ˙ “ π p a, d q ¨ f p g q * for p π, V q P Mod p T q . We refer to i : Mod p T q Ñ Mod p G q as the parabolic induc-tion functor . Remark
B.2.3 . We can also define the functor i : Mod p T q Ñ Mod p G q via the G, T -bi-module i p T q , as i p V q “ i p T q b T V. Note that the bi-module i p T q is given by the object S p G q { U p , ´ q „ ÝÑ S p U z G q , (1)where V ÞÑ V p , ´ q denotes the twist of the T -action by p a, d q ÞÑ | a { d | , andwhere S p G q { U denotes the co-invariants of S p G q under the right action of U .The isomorphism between the two spaces in Equation (1) is given by: f p g q ÞÑ ż U f p ug q d u. Indeed, this map respects the T -action: ˇˇˇ ad ˇˇˇ f ˆˆ a d ˙ g ˙ ÞÑ ˇˇˇ ad ˇˇˇ ż U f ˆˆ a d ˙ ug ˙ d u “ ż U f ˆ u ˆ a d ˙ g ˙ d u. Remark
B.2.4 . Observe that the composition Φ ´ ˝ i : Mod p T q Ñ Vect is iso-morphic to the forgetful functor:Φ ´ ˝ i p V q – V, via the isomorphism: f p g q ÞÑ ż f ˆˆ ´ ˙ ˆ v ˙˙ e p´ v q d v. This can be thought of as a trace map on i .17 efinition B.2.5. We denote the left adjoint of the functor i : Mod p T q Ñ Mod p G q by r : Mod p G q Ñ Mod p T q , and refer to r as the parabolic restrictionfunctor . Remark
B.2.6 . The functor r : Mod p G q Ñ Mod p T q is given by the formula r p V q “ U z V, which carries a residual T -action because T normalizes U . Here, the notation U z V refers to the co-invariants of V under the left action of U . Remark
B.2.7 . Because of Remark B.2.3, we also have the following fact, for all A P Mod p T q and B P Mod p G q :Φ ´ p i p A q (cid:13) (cid:5) B q “ I RL p i p A qq b G B – A b T ´ U T z B p , ´ q ¯ . This is essentially a duality between parabolic induction with respect to U andparabolic induction with respect to U T . That is, the functors of parabolicinduction with respect to U and the twist by p , ´ q of the parabolic restrictionwith respect to U T are dual under the pairing Φ ´ p´ (cid:13) (cid:5) ´q .This can be re-stated more cleanly using the following notation for the nor-malized action of the Weyl group on Mod p T q : Definition B.2.8.
For a T -module p π, V q , we denote by V p w q the followingtwist of V : p π, V q ÞÑ p π , V p w qq π p a, d q ¨ v “ ˇˇˇ ad ˇˇˇ π p d, a q ¨ v. Remark
B.2.9 . Remark B.2.7 boils down to the claim that the functors i p V q and i p V p w qq are dual in the sense of Definition B.2.13 below.Concretely, if F : Mod p T q Ñ Mod p G q and F : Mod p T q Ñ Vect are C -linearcolimit preserving functors, then there is a one-to-one correspondence betweennatural morphisms:Φ ´ p i p A q (cid:13) (cid:5) F p B qq “ I RL p i p A qq b G F p B q Ñ F p A b T B q and natural morphisms F p A q Ñ i p F p A qp w qq . Note that F p A q automatically carries a T -module structure. B.2.2 Categorical Prerequisites
In this subsection, we provide category-theoretical background on the notion ofFrobenius algebras, as well as relative Frobenius algebras. Since most of therest of the text will not be dependent on this subsection, readers may wish toskip it on a first reading. 18 efinition B.2.10.
Let C be a symmetric monoidal category. We say that apairing (cid:104) ¨ , ¨ (cid:105) : A b C A Ñ C is a duality between two objects A, A P C if the following composition is anisomorphism for all B, C P C :Hom p B, C b C A q Ñ Hom p B b C A, C b C A b C A q (cid:104) ¨ , ¨ (cid:105) ÝÝÑ
Hom p B b C A, C q . For an object A , a pair p A , (cid:104) ¨ , ¨ (cid:105) q as above is called duality data . If A hasduality data, then it is said to be dualizable . The duality data for a dualizableobject A is essentially unique, and we denote by A _ the dual object A of A . Definition B.2.11.
Recall that a
Frobenius algebra in a symmetric monoidalcategory C is an algebra object A P C , with a trace map λ : A Ñ C , satisfyingthat the composition of λ with the multiplication of A induces a self-duality: A b C A Ñ C of A .Additionally, let P r L be the category of locally presentable categories, withcolimit preserving functors between them. The category P r L is a natural settingfor the Adjoint Functor Theorem, in the sense that the local presentabilitycondition guarantees that all functors in P r L admit right adjoints.The category P r L is a symmetric monoidal category, with unit given by thecategory of sets. The monoidal structure of P r L is the Lurie tensor product ,defined by the universal property that a functor C b C Ñ C is a bi-functor which respects colimits separately in each variable.Note that an algebra object in P r L is just a symmetric monoidal categorywhere the symmetric monoidal structure respects colimits separately in eachvariable. Definition B.2.12. A Frobenius algebra in categories is a Frobenius algebra in P r L .There are also relative versions of the above notions. Given any symmetricmonoidal category C , it is possible to talk about C -linear symmetric monoidalcategories , and C -linear Frobenius algebras in categories in the obvious manner.The case where C “ Vect, the category of C -vector spaces, will be especiallyimportant to us.If we are given a category C which is itself a Frobenius algebra in categories,then it makes sense to define another kind of duality on it. Definition B.2.13.
Let C be a C -linear Frobenius algebra in categories, withtrace functor Λ : C Ñ C . We say that a pairing (cid:104) ¨ , ¨ (cid:105) : Λ p A b C A q Ñ C
19s a relative duality between two objects A P C and A P C if the followingcomposition is an isomorphism for all B P C and C P C :Hom C ` B, C b A ˘ Ñ Hom C ` B b A, C b A b A ˘ ÑÑ Hom C ` Λ p B b A q , Λ p C b A b A q ˘ p ♠ q ÝÝÑ p ♠ q ÝÝÑ
Hom C ` Λ p B b A q , C b Λ p A b A q ˘ (cid:104) ¨ , ¨ (cid:105) ÝÝÑ
Hom C p Λ p B b A q , C q . Here, we used the C -linearity of Λ for the transition p ♠ q .Note that in the definition above, we have abused notation and used the samesymbol b for the symmetric monoidal structure of C , the symmetric monoidalstructure of C and the action of C on C . Definition B.2.14.
Let C be a C -linear Frobenius algebra in categories, withtrace functor Λ : C Ñ C . A relative Frobenius algebra in C is an algebra object A P C , with a trace map λ : Λ p A q Ñ C , satisfying that the composition of λ with the multiplication of A induces a relative self-duality:Λ p A b C A q Ñ C of A . Remark
B.2.15 . For the rest of this paper, we will abuse notation and omit theword “relative” from the terms “relative Frobenius algebra” and “relative dual-ity”. Moreover, we will always be working with Vect-linear Frobenius algebrasand dualities.
B.3 Defining the Multiplication
In this section, we will give the construction which induces the data of beingnon-unital right-lax on (a normalization of) the parabolic induction functor i : Mod p T q Ñ Mod p G q . Let L P Mod p T q be defined by: Definition B.3.1.
Let L P Mod p T q be the space of functions S p F ˆ F ˆ q , with T -action given by p a, d q ¨ F p λ, y q “ ˇˇˇ ad ˇˇˇ F p d ´ λ, ady q , λ P F, y P F ˆ . Remark
B.3.2 . In the language of [Dor20b], one can give the space L the follow-ing informal interpretation. Let us think of the spectrum of T as parametrizedby pairs of characters χ p a q| a | s χ p d q| d | s , and let us informally think of T asthe ring of regular functions on this spectrum. One can think of elements of L as functions on the spectrum of T via the Mellin transform. Thought of likethat, the module L is generated by the Hecke L-function L p χ ´ χ , s ´ s ` q .Our main claim for this section is: 20 laim B.3.3. There is a canonical natural map m : i p A q (cid:13) (cid:5) i p B q Ñ i p L b T A b T B q . Up to the presence of the term L , this map is the data needed to give a non-unital right-lax structure on i . Of course, in order to actually have a right-laxstructure, one must show that this data satisfies certain associativity axioms.We will dedicate the rest of this section to proving Claim B.3.3, as the first stepin discussing the right-lax properties of i . Along the way, we will also prove auniqueness result for this multiplicative structure, Proposition B.3.6, which willshow that this choice of multiplication is canonical. Additionally, although themap i will not be unital right-lax, we will later use Proposition B.3.6 to showthat it does become unital after inverting a certain map.This multiplication map of Claim B.3.3 is induced via the following con-struction: Construction
B.3.4 . We let µ : Y Ñ L be given by Ψ p g, y q ÞÑ Ψ ˆˆ λ ˙ , y ˙ . The natural map of Claim B.3.3 is induced by the following claim, usingRemarks B.2.6 and B.2.7:
Claim B.3.5.
The map µ : Y Ñ L given in Construction B.3.4 descends to amap of T ˆ T ˆ T -modules: µ : U T ˆ U ˆ U T z Y pp , ´ q , p , q , p , ´ qq Ñ L . Here, the notation ´pp , ´ q , p , q , p , ´ qq denotes twisting the three residual T -actions, and the three T -actions on L are the same one.Proof. It is easy to verify that µ is U T ˆ U ˆ U T -invariant. Equivariance under T ˆ T ˆ T is also immediate. Proof of Claim B.3.3.
Let A and B be two T -modules. We tensor the map µ : U T ˆ U ˆ U T z Y pp , ´ q , p , q , p , ´ qq Ñ L by A and B , relative to the first and third T -actions, respectively. Using Re-mark B.2.3, we obtain a natural map: r p i p A q (cid:13) (cid:5) i p B qq Ñ L b T A b T B. .3.1 Universal Property of the Multiplication When we later discuss the unitality of the right-lax structure on i , we will needa slightly stronger variant of Claims B.3.3 and B.3.5. Specifically, it turns outthat there is a unique map of the kind constructed in Claim B.3.5. This meansthat the multiplication map on i is more-or-less unique. Proposition B.3.6.
The map of T -modules µ : U T ˆ U ˆ U T z Y pp , ´ q , p , q , p , ´ qq b T ˆ T ˆ T T Ñ L given by forcing the three T actions on U T ˆ U ˆ U T z Y pp , ´ q , p , q , p , ´ qq to bethe same is an isomorphism. The meaning of Proposition B.3.6 is that:
Corollary B.3.7.
Let F : Mod p T q Ñ Mod p T q be any functor. Then any natu-ral map: i p A q (cid:13) (cid:5) i p B q Ñ i ˝ F p A b T B q factors uniquely through the map m : i p A q (cid:13) (cid:5) i p B q Ñ i p L b T A b T B q of Claim B.3.3.Remark B.3.8 . It will follow from Proposition B.7.10 below that, in fact, allfunctors F : Mod p T q Ñ Mod p G q with natural map: i p A q (cid:13) (cid:5) i p B q Ñ F p A b T B q factor uniquely through the map m : i p A q (cid:13) (cid:5) i p B q Ñ i p L b T A b T B q . Proof of Proposition B.3.6.
Note that the co-invariants of Y under the middleaction of U are S p M p F q det “ ˆ F ˆ q . Now, we can break this space into orbits under the action of the parabolicsubgroup of lower triangular matrices. Specifically, we can give S p M p F q det “ ˆ F ˆ q a filtration, with graded parts: "ˆˆ u ˙ ˆ λ ˙ ˆ v ˙ , y ˙* , "ˆˆ u ˙ ˆ λ ˙ , y ˙* Y "ˆˆ λ ˙ ˆ v ˙ , y ˙* , "ˆˆ λ ˙ , y ˙* . The first two graded parts have the wrong equivariance properties under thevarious actions of T . Thus, we are left with the last graded part, which gives µ . 22he reader should note that the map of Claim B.3.3 does not quite make i into a non-unital right-lax functor, due to the presence of the term L . This willbe handled in the next section. B.4 Normalized Parabolic Induction
In Section B.3, we defined a natural map i p A q (cid:13) (cid:5) i p B q Ñ i p L b T A b T B q . In order to have an actual right-lax functor, we need to modify i to get rid ofthe extra factor of L . This will be the focus of this section.The fact that i needs to be normalized is not too surprising. When workingwith parabolic induction, one often finds the need to normalize the inductionvia an L-function. This normalization factor often appears in the functionalequation for Eisenstein series in the global theory, and for the intertwiningoperator in the local theory. It turns out that this normalization factor preciselycorresponds to the object L , in the sense of [Dor20b]. See also Remark B.3.2.Let H om T : Mod p T q op ˆ Mod p T q Ñ Mod p T q be the inner Hom functor ofMod p T q , which is right adjoint to b T . We set: Definition B.4.1.
Define the normalized parabolic induction functor p i : Mod p T q Ñ Mod p G q by p i p V q “ i p H om T p L , V qq . We denote its left adjoint, the normalized parabolic restriction functor , by p r : Mod p G q Ñ Mod p T q . Remark
B.4.2 . The term “normalized parabolic induction” is used in the liter-ature to refer to a different kind of normalization, which we do not use here.By choosing a square root of the cardinality q of the residue field of F , onecan simplify the Weyl twist of Definition B.2.8. Letting χ “ χ b χ : T Ñ C ˆ ,this means that the intertwining operator can be made to relate the parabolicinduction of χ b χ with χ b χ , instead of relating χ with χ p w q .In this text, we will not make use of this kind of normalization, and we donot choose a square root of q . Remark
B.4.3 . By Theorem A.1 of [Dor20b], there is a non-canonical isomor-phism L – T , meaning that the normalized functor is non-canonically isomor-phic to the standard one: i – p i .Claim B.3.3 now immediately gives us: Claim B.4.4.
There is a canonical natural map m : p i p A q (cid:13) (cid:5) p i p B q Ñ p i p A b T B q . In order for this to make p i into a non-unital right-lax functor, we still needto verify that this multiplication respects associativity. We will do so in Sec-tion B.6. 23 .5 Unitality Before discussing associativity, let us discuss the unitality of the multiplicationon p i : Mod p T q Ñ Mod p G q . Specifically, it turns out that while this functorcannot be unital right-lax, it can come very close to it. We begin by givinga counter-example showing that p i cannot be unital, and then show that thiscounter-example is essentially the only obstruction to the unitality of p i .The following is the motivating example for this section. Example
B.5.1 . Let triv T p , ´ q be the one-dimensional representation of T ,given by the character p a, d q ÞÑ | a { d | . This representation satisfies that i p triv T p , ´ qq has a one-dimensional quotient triv G , with kernel given by a Steinberg repre-sentation St.Now, the T -module triv T p , ´ q is clearly a unital commutative algebra withrespect to b T . This implies that if p i : Mod p T q Ñ Mod p G q were unital right-lax, then i p triv T p , ´ qq would have been a unital commutativealgebra. However, let us show that this is impossible.Indeed, suppose that i p triv T p , ´ qq was a unital algebra in Mod p G q . Ob-serve that all potential unit maps (cid:5) Ñ i p triv T p , ´ qq factor through St. Thismeans that the following isomorphism, required by unitality: (cid:5) (cid:13) (cid:5) i p triv T p , ´ qq (cid:47) (cid:47) „ (cid:42) (cid:42) i p triv T p , ´ qq (cid:13) (cid:5) i p triv T p , ´ qq (cid:15) (cid:15) i p triv T p , ´ qq factors through St (cid:13) (cid:5) i p triv T p , ´ qq , which is 0. Thus, we obtain a contradiction.Despite Example B.5.1, the functor p i : Mod p T q Ñ Mod p G q is almost unitalright-lax. Essentially, we claim that Example B.5.1 is the only counter-example.The rest of this section is dedicated to clarifying this statement, formalized inProposition B.5.4.The key observation is that there is actually a canonical map (cid:5) Ñ i p T q given as follows. Construction
B.5.2 . Define the map (cid:5) Ñ S p U z G q – i p T q via the formula W p g q ÞÑ ż F W ˜ˆ ´ ˙ ˆ ´ ˙ ´ ˆ b ˙ g ¸ d b. emark B.5.3 . The map of Construction B.5.2 can also be obtained from theisomorphism Φ ´ ˝ i p V q – V of Remark B.2.4, by identifying Hom p (cid:5) , i p T qq with the roughening of Φ ´ ˝ i p T q “ T .Similarly, the presence of the factor ˆ ´ ˙ is in order to be consistentwith the identification of the functors I RL p (cid:5) q b G p´q and Φ ´ p´q used in Re-mark 3.17 of [Dor20a].Note that there is a canonical embedding η : T “ S p F ˆ ˆ F ˆ q ã Ñ L . Ourmain result for this section is the following “almost unitality” result: Proposition B.5.4.
The following diagram commutes: (cid:5) (cid:13) (cid:5) p i p A q (cid:47) (cid:47) “ (cid:15) (cid:15) p i p L q (cid:13) (cid:5) p i p A q m (cid:15) (cid:15) p i p A q η (cid:47) (cid:47) p i p L b T A q . Here, the top horizontal map is induced from the map of Construction B.5.2 viathe identification i p T q “ p i p L q . Before we discuss the proof of Proposition B.5.4, let us describe its conse-quences.
Remark
B.5.5 . Let Mod p T qr η ´ s be the localization of Mod p T q given by invert-ing η . Then p i may be restricted to a functor p i r η ´ s : Mod p T qr η ´ s Ñ Mod p G q .Now, Proposition B.5.4 implies that once we show that the multiplication m : p i p A q (cid:13) (cid:5) p i p B q Ñ p i p A b T B q is associative in Section B.6, then the restriction p i r η ´ s : Mod p T qr η ´ s Ñ Mod p G q will be unital right-lax. This is Corollary B.6.2. Remark
B.5.6 . Note that this bypasses the difficulty posed by Example B.5.1because the image of triv T p , ´ q under the localization mapMod p T q Ñ Mod p T qr η ´ s is 0. In other words, we claim that Example B.5.1 is the only obstruction to theunitality of p i .In terms of [Dor20b], the meaning of inverting η is the following. The space L is essentially the space of zeta integrals defining the L-function L p χ ´ χ , s ´ s ` q specified in Remark B.3.2. The idea is that the L-function in questionhas just one pole when thought of as a function on the spectrum of T , and thecounter-example triv T p , ´ q lies in that pole. In particular, inverting η killsthe counter-example.Also note that the usual process of meromorphic continuation used to definethe intertwining operator (and the L-function L p χ ´ χ , s ´ s ` q ) also invertsthe map η . As we will see below, that is essentially the purpose of the analyticcontinuation. 25 .5.1 Proof of Unitality We dedicate the rest of this section to the proof of the unitality property Propo-sition B.5.4. The reader may wish to skip the remainder of this section on theirfirst read-through.We will prove Proposition B.5.4 by computing the two compositions andcomparing the results. The main difficulty which makes this not immediateis to evaluate the isomorphism (cid:5) (cid:13) (cid:5) p i p A q „ ÝÑ p i p A q . This can be done by usingthe formula given in Remark 3.17 of [Dor20a]. However, that formula can onlybe used after applying the action of a cyclic permutation on the inputs of Y (recall that Y is actually a S ˙ G -module), which is complicated. Instead ofdoing this directly, we will use the following alternative strategy. The idea isto directly compute the composition of this permutation of S with the map µ of Construction B.3.4. We do this by giving a candidate µ for this re-orderingof the inputs of µ , and using the uniqueness principle of Proposition B.3.6 toestablish that these two maps are indeed the same.We begin by constructing the desired re-ordering µ of the arguments of µ : Construction
B.5.7 . We let the map of T ˆ T ˆ T -modules µ : U ˆ U ˆ U T z Y pp´ , q , p´ , q , p , qq Ñ L p´ , q be given by Ψ p g, y q ÞÑ | λ | ż Ψ ˆˆ b ˙ ˆ λ ˙ , y ˙ d b. Definition B.5.8.
Let F : L p w qp , q Ñ L be the partial Fourier transform,normalized as: F p F qp λ, y q “ | y | ż F p α, y q e p´ αλy q d α. Denote by τ : S ˙ G Ñ End p Y q the action on Y , so that τ pp , , qq is theoperation cyclically permuting the three G -actions on Y , and τ p w, w, w q denotesthe combined application of the matrix w “ ˆ ´ ˙ at all three G -actions.Our claim is that µ can indeed recover the values of µ in the following sense: Claim B.5.9.
The diagram: Y µ (cid:15) (cid:15) τ pp , , qq (cid:47) (cid:47) Y τ p w,w,w q (cid:47) (cid:47) Y µ (cid:15) (cid:15) L L F (cid:111) (cid:111) commutes. Before proving Claim B.5.9, let us begin by showing how it implies Propo-sition B.5.4. 26 roof of Proposition B.5.4.
It is sufficient to consider the case A “ L . We willprove the claim by computing and comparing both compositions.We begin by identifying the domain of both compositions with the space Y of co-invariants of Y , taken with respect to the character θ of U with respectto the first action, and taken with respect to the trivial character of U T withrespect to the third action. The space Y is isomorphic to the space Y , wherethe θ -co-invariants are taken with respect to the second action, and the U T -co-invariants are taken with respect to the first action. That is, we have anisomorphism: Y (cid:47) (cid:47) Y given by τ pp , , qq ´ . We will compute the two compositions when they arepre-composed with this isomorphism, and composed with the co-unit corre-sponding to the adjunction p r, i q . That is, we work with the following diagram,where the maps between the second and third rows are not maps of G -modules,but merely maps of T -modules: Y τ pp , , qq ´ (cid:47) (cid:47) (cid:5) (cid:13) (cid:5) i p T q (cid:47) (cid:47) “ (cid:15) (cid:15) i p T q (cid:13) (cid:5) i p T q m (cid:15) (cid:15) i p T q η (cid:47) (cid:47) (cid:15) (cid:15) i p L q (cid:15) (cid:15) T η (cid:47) (cid:47) L . We begin with the anti-clockwise composition. When we pre-compose it by τ pp , , qq ´ , the isomorphism (cid:5) (cid:13) (cid:5) i p T q – i p T q is identified with: Y Ñ S p U z G q Ψ p g, y q ÞÑ ż Ψ ˆˆ u ˙ g ´ T , det p g q ˙ d u. Finally, by the adjunction defining i , this corresponds to the map: Y Ñ T given by: Ψ p g, y q ÞÑ ż Ψ ˆˆ u ˙ ˆ a ´ d ´ ˙ , ad ˙ d u. (2)This finishes the anti-clockwise composition.27et us now compute the clockwise composition. Consider Ψ p g, y q P Y . ByClaim B.5.9, applying the clockwise composition to it is the same as applying F ˝ µ ˝ τ p w, w, w q to the distribution:Ψ p g, y q “ Ψ p g, det p g q ´ q δ p y det g ´ q . Let f p g q “ Ψ p g, det p g q ´ q . We must apply F ˝ µ ˝ τ p w, w, w q to the distri-bution Ψ . We get: µ ˝ τ p w, w, w qp Ψ qp λ, y q ““ | λ | ż | y | ż f p w ´ gw ´ q e ˆ ´ y (cid:28) g, ˆ u ˙ ˆ λ ˙ (cid:29) ˙ δ p y det g ´ q d g d u. Simplifying this expression, we have µ ˝ τ p w, w, w qp Ψ qp λ, y q ““ ˇˇ λy ˇˇ ż f ˆˆ a bc d ˙˙ e p λy p bu ` d qq δ p y p ad ´ bc q ´ q d a d b d c d d d u ““ | y | ż f ˆˆ a c d ˙˙ e p λyd q δ p yad ´ q d a d c d d ““ ż f ˆˆ {p yd q c d ˙˙ e p λyd q d c d d | d | . Applying F , we get F ˝ µ ˝ τ p w, w, w qp Ψ qp λ, y q ““ | y | ż f ˆˆ {p yd q c d ˙˙ e p αy p d ´ λ qq d c d d | d | d α “ | λ | ´ ż f ˆˆ {p yλ q c λ ˙˙ d c “ ˇˇ λ y ˇˇ ´ ż f ˆˆ c ˙ ˆ {p yλ q λ ˙˙ d c. This corresponds to the map Y Ñ L given by Ψ p g, y q ÞÑ ˇˇ λ y ˇˇ ´ ż Ψ ˆˆ c ˙ ˆ {p yλ q λ ˙ , y ˙ d c. (3)Clearly, the map (2) matches up with the map (3) up to: η : T Ñ L f p a, d q ÞÑ ˇˇ λ y ˇˇ ´ f p λy, λ ´ q , as we wanted to show. 28inally, we end this section with the proof of Claim B.5.9. Proof of Claim B.5.9.
Because of the uniqueness result Proposition B.3.6, thecomposition F ˝ µ ˝ τ p w, w, w q ˝ τ pp , , qq factors through µ , and thus defines a map L Ñ L . We must check that this map is the identity map. Therefore, it is sufficient totest this on the indicators of sufficiently small neighborhoods of λ “ y “ e : F Ñ C ˆ is unramified. Pick some ε P O of sufficiently small absolute value. Set Ψ p g, y q to be the indicator function of the set: "ˆˆ a bc d ˙ , y ˙ ˇˇˇˇ a P ε O , b P ε O ,c P ε O , d P ` ε O , y P ` ε O * . It is clear that µ p Ψ qp λ, y q “ ` ε O p λ q ` ε O p y q . We must verify that the composition F ˝ µ ˝ τ p w, w, w q ˝ τ pp , , qqp Ψ q gives the same function.Indeed, we obtain that 1 | ε | τ pp , , qqp Ψ q is the indicator function of the set "ˆˆ a bc d ˙ , y ˙ ˇˇˇˇ a P ε ´ O , b P ε O ,c P ε ´ O , d P ` ε O , y P ` ε O * . Moreover, applying τ p w, w, w q , we get: τ p w, w, w q ˝ τ pp , , qqp Ψ q ˆˆ a bc d ˙ , y ˙ “ | ε | e p d q a P ε O , b P ε ´ O ,c P ε O , d P ε ´ O , y P ` ε O . We apply µ to obtain: µ ˝ τ p w, w, w q ˝ τ pp , , qqp Ψ qp λ, y q “ | ε | ε ´ O p λ q e p λ q ` ε O p y q . Finally, using F gives: ` ε O p λ q ` ε O p y q , as we wanted to show. 29 .6 Associativity In this section, we will prove the associativity of the multiplication on p i : Mod p T q Ñ Mod p G q constructed in Claim B.4.4.Specifically, our main theorem for this section is: Theorem B.6.1.
The functor p i : Mod p T q Ñ Mod p G q is non-unital right-laxsymmetric monoidal. As discussed in Remark B.5.5 above, Theorem B.6.1 immediately impliesthat:
Corollary B.6.2.
The restriction p i r η ´ s : Mod p T qr η ´ s Ñ Mod p G q of p i is uni-tal right-lax symmetric monoidal. We dedicate the rest of this section to proving Theorem B.6.1.Our strategy for proving Theorem B.6.1 is as follows. We are essentiallytrying to show that the two multiplication maps p i p T q (cid:13) (cid:5) p i p T q (cid:13) (cid:5) p i p T q Ñ p i p T q are the same. We will do so by showing that applying Φ ´ sends the two mapsinto the same one, which will follow immediately from the following result,Proposition B.6.3. Note that the triple-functors Φ ´ ´p i p A q (cid:13) (cid:5) p i p B q (cid:13) (cid:5) p i p C q ¯ andΦ ´ ´p i p A b T B b T C q ¯ are symmetric with respect to permutations of A, B, C . Proposition B.6.3.
Let φ be a natural map of functors: φ : Φ ´ ´p i p A q (cid:13) (cid:5) p i p B q (cid:13) (cid:5) p i p C q ¯ Ñ Φ ´ ´p i p A b T B b T C q ¯ . Then φ commutes with the above action of S . Before proving Proposition B.6.3, let us finish the details of the proof.
Proof of Theorem B.6.1.
We have two compositions p i p A q (cid:13) (cid:5) p i p B q (cid:13) (cid:5) p i p C q m ˝p m (cid:13) (cid:5) id q (cid:47) (cid:47) m ˝p id (cid:13) (cid:5) m q (cid:47) (cid:47) p i p A b T B b T C q that are given by re-ordering each other’s inputs. Therefore, Proposition B.6.3implies that: Φ ´ p m ˝ p m (cid:13) (cid:5) id qq “ Φ ´ p m ˝ p id (cid:13) (cid:5) m qq . ´ is exact, the image of m ˝ p m (cid:13) (cid:5) id q ´ m ˝ p id (cid:13) (cid:5) m q is killed byΦ ´ and is a degenerate representation. Because of Remark B.2.3, it is enoughto show that the two maps p i p T q (cid:13) (cid:5) p i p T q (cid:13) (cid:5) p i p T q m ˝p m (cid:13) (cid:5) id q (cid:47) (cid:47) m ˝p id (cid:13) (cid:5) m q (cid:47) (cid:47) p i p T q are equal. However, the image of their difference is a degenerate representation,which must be 0, as p i p T q has no vector invariant to SL p F q .The proof of Proposition B.6.3 comprises the rest of this section. We willprove the proposition using a uniqueness result. That is, we will show thatall natural maps of the requisite form factor through a specific, universal one.The specific map we obtain will be manifestly symmetric with respect to the S action on Y , which will prove the result.Our first goal is to explicitly write down the map we are after. To do so, weneed an appropriate target space. Construction
B.6.4 . Consider the multiplicative convolution product: L b T L Ñ r L , where r L is the contragradient. Denote the image by L . Remark
B.6.5 . We have an isomorphism: L b T L „ ÝÑ L . Moreover, recall that L consists of functions of p λ, y q that are locally constantnear λ “
0. Then L also allows functions that grow logarithmically as ν p λ q near 0.Under the Mellin transform, the T -module L allows functions that have adouble pole at a specific location.We will now construct the desired universal mapΦ ´ ´p i p A q (cid:13) (cid:5) p i p B q (cid:13) (cid:5) p i p C q ¯ Ñ Φ ´ ´p i p A b T B b T C q ¯ . We will do so by identifying the functor Φ ´ ´p i p A q (cid:13) (cid:5) p i p B q (cid:13) (cid:5) p i p C q ¯ with ´p i p A q b p i p B q b p i p C q ¯ b G ˆ G ˆ G Y, which means that all we have to do is construct a map out of Y with theappropriate equivariance properties.Our candidate is the map: Construction
B.6.6 . We let the map of T ˆ T ˆ T -modules µ : U ˆ U ˆ U z Y pp´ , q , p´ , q , p´ , qq Ñ L p´ , q be given byΨ p g, y q ÞÑ | λ | ż Ψ ˆˆ b ˙ ˆ λ ˙ ˆ b ˙ , y ˙ d b d b .
31e now claim that
Proposition B.6.7.
The map of T -modules µ : U ˆ U ˆ U z Y pp´ , q , p´ , q , p´ , qq b T ˆ T ˆ T T Ñ L p´ , q given by forcing the three T actions on U ˆ U ˆ U z Y pp´ , q , p´ , q , p´ , qq to bethe same is an isomorphism.Proof. The proof is essentially identical to the proof of Proposition B.3.6.This immediately implies that the following construction is universal.
Construction
B.6.8 . Consider the composition F ˝ µ ˝ τ p w, w, w q : U T ˆ U T ˆ U T z Y pp , ´ q , p , ´ q , p , ´ qq Ñ L , with F : L p w qp , q „ ÝÑ L the partial Fourier transform on L , given by: F b F : L p w qp , q b T L p w qp , q „ ÝÑ L b T L . Here, τ : S ˙ G Ñ End p Y q is the action on Y and F : L p w qp , q „ ÝÑ L is as inDefinition B.5.8.Applying the functor p A b B b C q b T ˆ T ˆ T p´q to the composition F ˝ µ ˝ τ p w, w, w q gives a natural map: φ : Φ ´ ´p i p A q (cid:13) (cid:5) p i p B q (cid:13) (cid:5) p i p C q ¯ Ñ Φ ´ ´p i p A b T B b T C q ¯ . It immediately follows from Proposition B.6.7 that:
Corollary B.6.9.
Let F : Mod p T q Ñ Vect be any functor. Then any naturalmap: φ : Φ ´ ´p i p A q (cid:13) (cid:5) p i p B q (cid:13) (cid:5) p i p C q ¯ Ñ F p A b T B b T C q factors uniquely through the map φ : Φ ´ ´p i p A q (cid:13) (cid:5) p i p B q (cid:13) (cid:5) p i p C q ¯ Ñ Φ ´ ´p i p A b T B b T C q ¯ above. Finally, we can prove Proposition B.6.3:
Proof of Proposition B.6.3.
We need to show that all natural maps φ as in theclaim are symmetric under the action of S . We begin by observing that itis sufficient to show that they are all symmetric under the action of a singletransposition.Indeed, the image σ p φ q of φ under any permutation σ P S is still such anatural map. Thus, if all such maps σ p φ q are invariant under a specific trans-position, then they are invariant under all of its conjugates as well. But theconjugates of a transposition in S generate it.32herefore, by Corollary B.6.9, it is enough to show that the specific map φ is invariant under a specific transposition of our choice. We will show that φ is symmetric under the action of the transposition exchanging the left andright actions of G on Y . Note that this is the transposition of M p F q inside Y “ S p M p F q ˆ F q .However, because µ is clearly invariant under the transposition of M p F q ,the desired result is proven. B.7 Intertwining Operator
Our goal for this section is to understand the intertwining operator in terms ofthe multiplicative structure on p i . Let us begin by an informal explanation of ourapproach. In this section, we will make extensive use of the notions introducedin Subsection B.2.2. B.7.1 Intertwining as Duality
In this subsection, we will give an informal explanation of how the intertwiningoperator can arise out of the multiplicative structure of p i . We will formalize andprove this explanation in Subsection B.7.2 below.The description of our approach is cleanest using the language of the Dayconvolution product on the category of C -linear colimit preserving functorsFun L p Mod p T q , Mod p G qq . Specifically, it turns out that one can assign Fun L p Mod p T q , Mod p G qq a symmet-ric monoidal structure b Day , satisfying the universal property that a map: F b Day F Ñ F is the same as a natural morphism F p A q (cid:13) (cid:5) F p B q Ñ F p A b T B q . This symmetric monoidal structure has the property that its algebras are pre-cisely the right-lax functors Mod p T q Ñ Mod p G q .For the moment, let us ignore the various normalization factors of L . Ourconstructions so far turn p i into a right-lax symmetric monoidal functor. Thismeans that we have a unital, commutative and associative multiplication map m Day : p i b Day p i Ñ p i. However, we also have a trace map on the algebra p i , via Remark B.2.4 (recallthat we are ignoring factors of L , for the moment):Φ ´ ˝ p i p A q – A.
33t makes sense to talk about trace maps, because the categoryFun L p Mod p T q , Mod p G qq carries a trace map of its own. That is, we observe that Fun L p Mod p T q , Mod p G qq carries an action of the symmetric monoidal category Fun L p Mod p T q , Vect q ,equipped with its own Day convolution product. The trace functorΦ ´ : Mod p G q Ñ Vectnow turns Fun L p Mod p T q , Mod p G qq into a Frobenius algebra in the (2-)categoryof Fun L p Mod p T q , Vect q -linear presentable categories.Thus, the functor p i becomes a commutative unital algebra with a trace mapin the category Fun L p Mod p T q , Mod p G qq . In particular, it has a bi-linear pairinggiven by composing the multiplication map with the trace map:Φ ´ ˝ p p i b Day p i qp A q Ñ Φ ´ ˝ p i p A q – A. (4)Now, Remark B.2.9 implies that the functor p i is dualizable with respect to thepairing Φ ´ p´b Day ´q , and its dual is the functor A ÞÑ p i p A p w qq .Once again ignoring factors of L , the pairing (4) induces a map betweenthe functor A ÞÑ p i p A q and its dual A ÞÑ p i p A p w qq . This map turns out to be(a re-parametrization of) the intertwining operator for GL p q . Moreover, since A ÞÑ A p w q is a symmetric monoidal functor, the composition A ÞÑ p i p A p w qq is also an algebra with respect to b Day . Our goal in this section will be toessentially show that this map between A ÞÑ p i p A q and A ÞÑ p i p A p w qq is anisomorphism of algebras (up to L ).This will show that the trace pairing on p i is non-degenerate, and thus thatit is a Frobenius algebra. We will also show that the intertwining operatorrespects the trace map, which will automatically give the usual result that it isself-inverse up to factors of L-functions. B.7.2 Construction of the Intertwining Operator
Let us re-state things more formally. The end result is made much more cumber-some by the presence of the normalization factors L . Therefore, before statingour main theorem for this section, Theorem B.7.8, we give the following corol-lary: Corollary B.7.1.
The unital commutative algebra p i r η ´ , η p w q ´ s P Fun L ` Mod p T qr η ´ , η p w q ´ s , Mod p G q ˘ along with the trace map Φ ´ ˝ p i r η ´ , η p w q ´ sp A q – A is a Frobenius algebra.Moreover, the functor p i r η ´ , η p w q ´ s is dual to its twist by A ÞÑ A p w q as aFrobenius algebra. η p w q is the image of η under the functor A ÞÑ A p w q of Definition B.2.8.Moreover, we assign an algebra structure and trace map to p i p w qr η ´ , η p w q ´ s by transporting the algebra structure and trace functional of p i r η ´ , η p w q ´ s along A ÞÑ A p w q .In order to state Theorem B.7.8, let us first more formally define the relevantFrobenius structures, and construct the intertwining operator. Definition B.7.2.
We denote the
Day convolution product on the categoriesof C -linear colimit preserving functorsFun L p Mod p T q , Mod p G qq , Fun L p Mod p T q , Vect q by b Day .We think of Fun L p Mod p T q , Mod p G qq as a Fun L p Mod p T q , Vect q -linear cate-gory. Remark
B.7.3 . For the sake of being explicit, we can also give the followingdescription. If F : Mod p T q Ñ Mod p G q , F : Mod p T q Ñ Mod p G q are given by A ÞÑ M b T A, A ÞÑ M b T A respectively, then F b Day F is given by A ÞÑ p M (cid:13) (cid:5) M q b T ˆ T T b T A. It is possible to give a similar explicit description for the Day convolution F b Day F when both of F, F lie in Fun L p Mod p T q , Vect q , or when one lies inFun L p Mod p T q , Vect q and the other lies in Fun L p Mod p T q , Mod p G qq . Remark
B.7.4 . The map m of Claim B.4.4 gives an associative multiplicationmap: m Day : p i b Day p i Ñ p i. This turns p i into a non-unital commutative algebra in Fun L p Mod p T q , Mod p G qq . Definition B.7.5.
We give the category Fun L p Mod p T q , Mod p G qq the structureof a Frobenius algebra in Fun L p Mod p T q , Vect q -linear presentable categories in-herited from the Frobenius algebra structure of Mod p G q . Remark
B.7.6 . More explicitly, we equip Fun L p Mod p T q , Mod p G qq with the tracefunctor Φ ´ ˝ ´ : Fun L p Mod p T q , Mod p G qq Ñ Fun L p Mod p T q , Vect q . b Day :Fun L p Mod p T q , Mod p G qq b Fun L p Mod p T q , Vect q Fun L p Mod p T q , Mod p G qqÑ Fun L p Mod p T q , Mod p G qq induces a self-duality on Fun L p Mod p T q , Mod p G qq relative to the base categoryFun L p Mod p T q , Vect q .We now construct the intertwining operator, keeping careful track of factorsof L . Construction
B.7.7 . The trace map:Φ ´ ˝ p i p V q – H om T p L , V q , and the multiplication m : p i p A q (cid:13) (cid:5) p i p B q Ñ p i p A b T B q induce a duality: M : p i p A q Ñ p i p L b T A p w qq using Remark B.2.9.Our main theorem for this section is: Theorem B.7.8.
The following diagrams commute.1. The intertwining operator is self-inverse: p i p A q M (cid:47) (cid:47) η b η p w q (cid:54) (cid:54) p i p L b T A p w qq M (cid:47) (cid:47) p i p L b T L p w q b T A q .
2. The intertwining operator does not affect traces: Φ ´ ˝ p i p A q (cid:47) (cid:47) Φ ´ p M q (cid:15) (cid:15) H om T p L , A q η (cid:47) (cid:47) H om T p L , L b T A q “ (cid:15) (cid:15) Φ ´ ˝ p i p L b T A p w qq (cid:47) (cid:47) A p w q .
3. The image of the unit (cid:5) is invariant to the intertwining operator: (cid:5) id (cid:15) (cid:15) (cid:47) (cid:47) p i p L q p i p L b T η p w qq (cid:47) (cid:47) p i p L b T L p w qq id (cid:15) (cid:15) (cid:5) (cid:47) (cid:47) p i p L q M (cid:47) (cid:47) p i p L b T L p w qq . . The intertwining operator is an algebra map: p i p A q (cid:13) (cid:5) p i p B q M (cid:13) (cid:5) M (cid:15) (cid:15) m (cid:47) (cid:47) p i p A b T B q M (cid:15) (cid:15) p i p L b T p A b T B qp w qq η (cid:15) (cid:15) p i p L b T A p w qq (cid:13) (cid:5) p i p L b T B p w qq m (cid:47) (cid:47) p i p L b T L b T p A b T B qp w qq . Before proving Theorem B.7.8, there are two things that we need. The firstis to have an explicit description of the map M : p i p A q Ñ p i p L b T A p w qq of Construction B.7.7. The second will be a variant of Proposition B.3.6.We begin with the description of the intertwining operator M . The readershould note that the result is slightly different from the usual normalization ofthe intertwining operator. Specifically, we claim that this map is the Fouriertransform of the usual expression: Claim B.7.9.
The map M : i p T q Ñ i p L p w qq is given explicitly by the formula: U z i p T q “ U z S p U z G q Ñ L p w q f p g q ÞÑ | y | ż f ˆˆ t ´ yt ˙ ˆ u ˙˙ e p´ yλt q | t | d t d u, Proof of Claim B.7.9.
Let us prove this. The pairing I RL p i p A qq b G i p B q Ñ L b T A b T B induced by the multiplication is explicitly given by the map U T ˆ U T z S p G qpp , ´ q , p , ´ qq Ñ L defined as: f p g ´ q ÞÑ | y | ż f p g q δ p y det g ´ q e ˆ ´ y (cid:28) g, ˆ λ ˙ (cid:29) ˙ d g. Let us simplify this expression: f p g ´ q ÞÑ | y | ż f ˆˆ a bc d ˙˙ δ p y p ad ´ bc q ´ q e p´ yλa q d a d b d c d d ““ | y | ż f ˆˆ u ˙ ˆ a ya ˙ ˆ v ˙˙ e p´ yλa q | a | d a d u d v. M : i p A q Ñ i p L p w qq is induced from the map U T z i p T qp , ´ q “ U T z S p U z G qp , ´ q Ñ L given by f p g q ÞÑ | y | ż f ˆˆ a ya ˙ ˆ v ˙˙ e p´ yλa q | a | d a d v. This gives the desired result.Finally, we need the following variant of the uniqueness result of Proposi-tion B.3.6:
Proposition B.7.10.
The map m Day : p i b Day p i Ñ p i given by the multiplication is an isomorphism.Remark B.7.11 . More explicitly, we claim that the map ´p i p T q (cid:13) (cid:5) p i p T q ¯ b T ˆ T T Ñ p i p T q is an isomorphism. Proof of Proposition B.7.10.
This will follow by proving that the morphism m : ´p i p T q (cid:13) (cid:5) p i p T q ¯ b T ˆ T T Ñ p i p T q is both injective and surjective.We begin by observing that Φ ´ p m q is an isomorphism. Indeed, this followsbecause the pairing ´ I RL p p i p T qq b G p i p T q ¯ b T ˆ T T Ñ H om T p L , T q is an isomorphism, which can be seen by the computation of Claim B.7.9.From this, it follows that the kernel and co-kernel of m : ´p i p T q (cid:13) (cid:5) p i p T q ¯ b T ˆ T T Ñ p i p T q are killed by Φ ´ .Let us begin by showing that m is injective. We need to show that the U -co-invariants U T ˆt uˆ U T z Y b T ˆt uˆ T T have no SL p F q -invariant vectors. Now, we note that it is enough to show that U T ˆ U ˆ U T z Y b T ˆt uˆ T T T X SL p F q . This space can bedecomposed into orbits via the same argument as in Proposition B.3.6, whichshows that there are no such invariant vectors.It remains to prove surjectivity. Because Φ ´ p m q is surjective, it is enoughto show that m surjects onto the quotient: p i p T q { SL p F q “ D, where D “ S p F ˆ q is the G -module where G acts via the determinant. Let N be the T -module whose underlying space is also S p F ˆ q , with T -action: p a, d q ¨ f p y q “ ˇˇˇ ad ˇˇˇ f p ady q . Then we have a factorization: p i p T q (cid:13) (cid:5) p i p T q m (cid:15) (cid:15) p i p T q (cid:13) (cid:5) p i p T q η (cid:15) (cid:15) p i p T q (cid:47) (cid:47) p i p N q (cid:47) (cid:47) D. We observe that the map η is induced from: η : U T ˆ U ˆ U T z Y Ñ N Ψ p g, y q ÞÑ Ψ p , y q , and thus factors through η : D (cid:13) (cid:5) D Ñ p i p N q . However, it is easy to verify that this last map η is in fact an isomorphism. Thismeans that m projects onto p i p N q , and in particular onto D .We can now prove our main result about the intertwining operator. Proof of Theorem B.7.8.
Item (3) follows by adjunction from Item (2).Moreover, Item (4) can be tested on A “ B “ T r η ´ , η p w q ´ s . Because ofProposition B.7.10, it follows that the map M respects the multiplication if itrespects the unit map. Therefore, Item (4) follows from Item (3).Additionally, since the multiplication and trace maps induce the duality M ,we see that Item (1) follows from the combination of Items (2) and (4).Hence, it remains to show that the diagram of Item (2) commutes. This canbe checked on A “ L . Moreover, since for A “ L all maps in the diagram ofItem (2) are maps of T -modules, it will suffice to verify that the two compositionsare equal on elements of Φ ´ ˝ p i p L q “ Φ ´ ˝ i p T q – T that generate Φ ´ ˝ p i p L q “ T .Indeed, suppose without loss of generality that e is unramified. Let f n p g q P i p T q “ S p U z G q
39e given by the indicator function of the set U ¨ K n , with n ě K n “ "ˆ a bc d ˙ ˇˇˇˇ | a ´ |ď q ´ n , | b |ď q ´ n | c |ď q ´ n , | d ´ |ď q ´ n * . Going along the upper horizontal composition in Item (2), the image of f n isthe function: ż f n ˆˆ ´ ad ˙ ˆ v ˙˙ e p´ v q d v “ e p´ { d q | ad ´ | ď q ´ n , | d | “ q ´ n ż f n ˆˆ a d ˙ ˆ t ´ t ˙ ˆ u ˙ ˆ ´ ˙ ˆ v ˙˙ e p´ v ´ t q| t | d t d u d v ““ ż f n ˆˆ at avt ´ dtu dt ´ dtuv ˙˙ e p´ v ´ t q| t | d t d u d v ““ ż f n ˆˆ ˚ ˚ u w ˙˙ e ˆ ´ w ´ dtu ´ t ˙ | d ¨ u | ´ d t d u d w ““ ż f n ˆˆ ˚ ˚ d w ˙˙ e p´ w { d q | d | ´ d w ““ e p´ { d q | ad ´ | ď q ´ n , | d | “ q ´ n Part C
Global Theory
C.1 Introduction
In this part, we will take the local theory developed in Part B, and use it to studythe global category Mod aut p GL p A qq of abstractly automorphic representations.Let us give more details. From now on, we let F denote a global functionfield. Set G “ GL p A q , and let T “ T p A q Ď G be its subgroup of diagonalmatrices. Then one can define a parabolic induction functor i : Mod p T q Ñ Mod p G q , with a left adjoint r : Mod p G q Ñ Mod p T q called parabolic restriction.We wish to use the adjoint pair p r, i q to study automorphic representationsof GL p A q . We will present two problematic aspects of a na¨ıve implementa-tion of this theory, and then show how the notion of abstractly automorphicrepresentations can help rectify them. 40efore we do so, let us note the part of the theory that functions well. Theparabolic induction functor i is fairly well behaved. It sends “most” irreduciblerepresentations to irreducible representations. Moreover, the functor i respectsautomorphicity: if p χ, W q is an irreducible automorphic T -module (i.e., a char-acter χ : T p F qz T p A q Ñ C ˆ ), then i p W q is an automorphic representation of G . However, the parabolic restriction functor r is much more badly behaved.If p π, V q is an irreducible automorphic representation which is principal seriesat all places, then r p V q is an infinite dimensional representation (given by atensor product of the two-dimensional Jacquet modules of V over all places).In particular, the T -module r p V q cannot be said to be automorphic in anyreasonable sense.This picture is fairly unsatisfactory, especially in light of how useful thefunctors p r, i q are for the study of local representation theory.Moreover, there is another unsatisfactory aspect of the global theory of au-tomorphic representations. As mentioned in the beginning of Part A, there is ananalogy between the local theory of p -adic representations and the global the-ory of automorphic representations. Both theories admit notions of parabolicinduction, cuspidality, constant terms and intertwining operators.This analogy between the local and global theory is fairly imprecise. Our goalfor this part is to rectify these two gaps: the lack of good parabolic restrictionand the inability to discuss the cuspidal automorphic spectrum in quite thesame terms as the supercuspidal local spectrum. We will do so using the notionof abstract automorphicity.Indeed, we will see below that while the category Mod p G q is far too big to beof use, its full subcategory Mod aut p G q is extremely well-behaved with respect toquestions of cuspidality and parabolic restriction. Having described the issueswe intend to tackle, let us dedicate the rest of this section to informally outliningour approach.We begin by observing that objects of the category Mod p T q also have a clearnotion of being automorphic: a T -module V is abstractly automorphic if andonly if it is invariant under the rationals T p F q Ď T p A q . With this notion, itturns out that the parabolic induction functor i : Mod p T q Ñ Mod p G q indeedrespects automorphicity, and restricts to a functor: i aut : Mod aut p T q Ñ Mod aut p G q . Moreover, it turns out that i aut has a left adjoint functor r aut : Mod aut p G q Ñ Mod aut p T q , which we refer to as automorphic parabolic restriction . The functor r aut is ex-tremely well-behaved: it kills irreducible cuspidal automorphic representations,and respects finite length. Moreover, automorphic parabolic restriction actsin a way analogous to the local case. If i p W q is an irreducible representationparabolically induced from p χ, W q with χ “ χ b χ a product of automor-phic characters, then r ˝ i p W q is a two-dimensional T -module, corresponding to χ b χ and χ b χ (up to twist). 41he functor r aut , along with its interaction with the symmetric monoidalstructure (cid:13) (cid:5) on Mod aut p G q , allows us to decompose the category Mod aut p G q into components , in analogy to the local classification. The cuspidal automorphicrepresentations take the role of the supercuspidal representations, and Eisensteinseries take the role of principal series representations. The only new phenomenonoccurs at the anomalous part of the spectrum, where some objects are killed by r aut despite belonging to an Eisenstein component (see also Remark A.1.22).The representations referred to as “cuspidal” in [Vig89] are a local examplewhere a similar situation occurs.More precisely, it turns out that we have an equivalenceMod aut p G q “ ź c P C p G q Mod aut c p G q , (5)with C p G q being the disjoint union of the set C Eis p G q of unordered pairs ofcharacters A ˆ { F ˆ Ñ C ˆ up to continuous twists |¨| s b |¨| s , with the set C cusp p G q of irreducible cuspidal automorphic representations up to continuoustwist | det p¨q| s .The structure of this part is as follows. In Section C.2, we will make the nec-essary adjustments to turn the local results of Part B about the interaction of p r, i q with the symmetric monoidal structures into global results. In Section C.3,we will show that the parabolic induction functor sends automorphic represen-tations to automorphic representations. In Section C.4, we will establish thedecomposition (5), and in Section C.5 we will give some corollaries, fulfillingour promises from Section A.1. C.2 Globalization
In this section, we will discuss what needs to be done in order to lift the dataof p i being a non-unital right-lax symmetric monoidal functor at every place toglobal data.Therefore, for the rest of this paper, we change our notation. We denote by F a global function field, and for every place v we let the subscript v denote thelocal constructions of the previous part over the local field F v . This means that T v , G v , (cid:5) ,v , L v , p i v etc. all denote the corresponding local constructions.For the rest of this paper, we let T “ T p A q , G “ G p A q , and let Mod p T q and Mod p G q be the corresponding categories of smooth left modules. We let L “ S p A ˆ A ˆ q , with T “ T p A q -action given by p a, d q ¨ F p λ, y q “ ˇˇˇ ad ˇˇˇ F p d ´ λ, ady q as in Definition B.3.1. We similarly define the functor p i : Mod p T q Ñ Mod p G q as p i p A q “ i p H om T p L , A qq , i : Mod p T q Ñ Mod p G q is the usual parabolic induction functor.Our main claim for this section is Theorem C.2.1.
The functor p i : Mod p G q Ñ Mod p T q is non-unital right-laxsymmetric monoidal. This essentially boils down to establishing that the multiplication on p i re-spects the distinguished vectors, which can be checked directly.It remains to discuss the issue of unitality. Here, it turns out that: Warning C.2.2.
The local map (cid:5) ,v Ñ i v p T,v q does not respect the distin-guished vectors.In particular, there is no global map (cid:5) Ñ i p T q . However, this makes someamount of sense, as this is not the map we are after anyway: we want p i to bethe right-lax functor, not i . So, consider the localization Mod p T qr η ´ s given byinverting the maps η v : T,v ã Ñ L v at every place. Then we can uniquely complete the diagram (cid:5) ,v (cid:47) (cid:47) (cid:37) (cid:37) p i v p L v r η ´ v sq p i v p T,v r η ´ v sq . η v (cid:79) (cid:79) It is easy to verify that the resulting map (cid:5) ,v Ñ p i v p T,v r η ´ v sq respects thedistinguished vectors, and therefore we obtain: Theorem C.2.3.
The restriction p i r η ´ s : Mod p T qr η ´ s Ñ Mod p G q of p i is unitalright-lax symmetric monoidal. Warning C.2.4.
Despite the suggestive notation, there is no actual map η : T Ñ L that we are inverting for the localizationMod p T q Ñ Mod p T qr η ´ s . Indeed, the maps η v : T,v Ñ L v do not define a global map, as they do notrespect the distinguished vectors. In the classical theory, this is worked aroundvia analytic continuation, see also Remark C.3.10. However, we will not needto do this. Remark
C.2.5 . There is a similar issue with regard to the trace map on p i andthe intertwining operator. Indeed, the local trace mapΦ ´ v ˝ p i v p V q – H om T v p L v , V q
43f Section B.7 does not turn into a global map, as it does not respect thedistinguished vectors. However, after inverting η v p w q : T,v p w q Ñ L v p w q atevery place v , we can uniquely complete the diagram:Φ ´ v ˝ p i v p V q (cid:47) (cid:47) (cid:42) (cid:42) H om T v p L v , V r η v p w q ´ sq H om T v p L v b T v L v p w q , V r η v p w q ´ sq . η v (cid:79) (cid:79) It is easy to verify that the resulting map respects the distinguished vectors,giving a natural map:Φ ´ ˝ p i p A q Ñ H om T p L b T L p w q , A r η p w q ´ sq and an intertwining operator: M : p i p A q Ñ p i p A p w qr η ´ sq . C.3 Parabolic Induction and Automorphicity
Our goal in this section is to show that the parabolic induction functor p i : Mod p T q Ñ Mod p G q is well-behaved with respect to the property of being automorphic. We formalizethis using the following notation: Definition C.3.1.
Let Mod aut p T q be the category of smooth T p A q{ T p F q -modules.Note that there is a forgetful functorMod aut p T q Ñ Mod p T q , induced by restriction along the map T p A q Ñ T p A q{ T p F q . This functor is fullyfaithful, and its essential image consists of the objects of Mod p T q which areinvariant under the action of T p F q Ď T p A q . We will usually identify Mod aut p T q with its essential image in Mod p T q . We refer to objects of Mod aut p T q as ab-stractly automorphic T p A q -modules.We give I T “ S p T p F qz T p A qq the structure of a unital commutative algebrausing the convolution product over T . Remark
C.3.2 . The category Mod aut p T q is equivalent to the category Mod p I T q of I T -modules in Mod p T q .Let I G Ď S p GL p F qz GL p A qq be the space of compactly supported smoothautomorphic functions which are orthogonal to one-dimensional characters. Re-call from [Dor20a] that I G acquires a unique commutative algebra structure44rom the canonical unit map (cid:5) Ñ I G induced by taking Whittaker functions.Also recall that the category Mod p I G q of I G -modules in Mod p G q is the categoryMod aut p G q of abstractly automorphic G -modules, and that the forgetful functorMod aut p G q Ñ Mod p G q is fully-faithful.Our main theorem for this section is thus: Theorem C.3.3.
The functor p i : Mod p T q Ñ Mod p G q sends abstractly auto-morphic objects to abstractly automorphic objects. In other words, we claim that p i descends to a functor p i aut : Mod aut p T q Ñ Mod aut p G q . Remark
C.3.4 . Because p i and i are non-canonically isomorphic, Theorem C.3.3shows that the un-normalized functor i also respects automorphicity, and re-stricts to: i aut : Mod aut p T q Ñ Mod aut p G q . Remark
C.3.5 . The fully-faithful forgetful functors Mod aut p T q Ñ Mod p T q andMod aut p G q Ñ Mod p G q admit left adjoints, given by I T b T p´q and I G (cid:13) (cid:5) p´q ,respectively. In particular, the functor p i aut immediately admits a left-adjoint,given by the co-invariants under the action of T p F q ˆ : p r aut : Mod aut p G q Ñ Mod aut p T q ,V ÞÑ p r p V q { T p F q ˆ . To prove Theorem C.3.3, it is sufficient to prove the following proposition:
Proposition C.3.6.
The object p i p I T r η ´ sq lies in Mod aut p G q . Indeed, this implies the theorem:
Proof of Theorem C.3.3.
Denote by C Ď Mod aut p T q the full subcategory whoseobjects are sent by p i to Mod aut p G q . We wish to prove that C “ Mod aut p T q .Note that p i is exact and respects colimits. Therefore, since Mod aut p G q is closedunder colimits, it follows that C is closed under colimits as well.Moreover, since Mod aut p G q is closed under taking sub-objects, it follows that C has this property as well.It therefore remains to show that I T P C . However, Proposition C.3.6 impliesthat I T r η ´ s is in C . Since I T is a sub-object of I T r η ´ s , we are done.It remains to show Proposition C.3.6. Because Theorem C.2.3 implies that p i p I T r η ´ sq is a unital algebra object in Mod p G q , the following claim will finishthe proof: Claim C.3.7.
The unit map (cid:5) Ñ p i p I T r η ´ sq factors through the unit map of I G : (cid:5) Ñ I G . emark C.3.8 . In fact, our proof of Claim C.3.7 will show that the unique mapcompleting the diagram (cid:5) (cid:47) (cid:47) (cid:36) (cid:36) I G (cid:15) (cid:15) p i p I T r η ´ sq is the usual constant term map we are familiar with from the theory of Eisensteinseries. Remark
C.3.9 . By Corollary B.7.1, the image of the unit map (cid:5) Ñ p i p I T r η ´ sq is invariant under the intertwining operator: M : p i p I T r η ´ sq Ñ p i p I T p w qr η ´ , η p w q ´ sq – p i p I T r η ´ , η p w q ´ sq . Thus, because we interpret the map I G Ñ p i p I T r η ´ sq of Claim C.3.7 as theconstant term map, we automatically get the functional equation for Eisensteinseries. Remark
C.3.10 . It is interesting to note that the above gives a description ofthe functional equation for Eisenstein series which makes no mention of analyticcontinuation. It is worthwhile to have an explanation for how this can happen.The idea is that the role of analytic continuation is to allow a canonicaltrivialization of the normalization factor L , which represents an L-function, asin Remark B.3.2.That is, by applying the Mellin transform, one can think of elements of I T “ S p T p A q{ T p F qq as analytic functions of two complex variables p s , s q P C , parametrized by χ |¨| s b χ |¨| s . If we consider the extension I T Ď I T corresponding to functions that are analytic only on some right half plane (cid:60) s " (cid:60) s "
0, then there is a canonical isomorphism I T b I T L „ ÝÑ I T . Thus, analytic continuation trivializes L . See also Section 5 of [Dor20b].In our case, by keeping track of the normalization factors, we have success-fully avoided the need for analytic continuation. Proof of Claim C.3.7.
This is a straightforward verification. Indeed, the map (cid:5) Ñ p i p I T r η ´ sq is given via the map (cid:5) b L Ñ i p I T r η ´ sq which is adjoint to the composition: U z p (cid:5) b L q Ñ T r η ´ s Ñ I T r η ´ s . U z p (cid:5) b L q Ñ T r η ´ s above sends W p g q b F p λ, y q P U z p (cid:5) b L q (with W p g q being left θ -equivariant) to the convolution: ż A ˆ ż A ˆ ż A W ˆˆ ´ ˙ ˆ u ˙ ˆ a ´ d ´ ˙˙ ˇˇˇ ad ˇˇˇ F p d ´ λ, ady q d u d ˆ a d ˆ d. Therefore, it remains to show that the map sending W p g q P (cid:5) to ÿ α,δ P F ˆ ż A W ˆˆ ´ ˙ ˆ u ˙ ˆ αa δd ˙˙ d u (6)factors through (cid:5) Ñ I G .Indeed, we observe that the expression appearing in Equation (6) is the sameas: ÿ γ P U p F qz GL p F q ż A { F W ˆ γ ˆ u ˙ ˆ a d ˙˙ d u. Since the map (cid:5) Ñ I G is given by W p g q ÞÑ ř W p γg q , we are done. C.4 Decomposition Into Components
Our goal in this section is to decompose the category Mod aut p G q of abstractlyautomorphic representations into components, in a manner analogous to thelocal classification.We will do so as follows. We will begin by separating the category Mod aut p G q into a cuspidal part and an Eisenstein part. We will then break down the cus-pidal part into components indexed by irreducible cuspidal automorphic rep-resentations up to continuous twist. Finally, we will decompose the Eisensteinpart of the category as well.We begin by separating out the cuspidal part of the category. Let I cusp G Ď I G be the kernel of the constant term map I G Ñ p i p I T r η ´ sq constructed in Claim C.3.7. Denote the subspace of I G which is orthogonal to I cusp G with respect to the Petersson inner product by I Eis G . We claim that: Proposition C.4.1.
The subspaces I cusp G , I Eis G Ď I G are ideals, and we have adecomposition I G “ I cusp G ˆ I Eis G of algebras.Proof. Because the constant term map is an algebra map, then I cusp G is an ideal.Moreover, because the Petersson inner product respects the multiplication on47 G (see Remark 4.21 of [Dor20a]), we see that I Eis G is an ideal as well. It remainsto show that I cusp G ‘ I Eis G is all of I G .Indeed, for every r ą K Ď GL p A q , let S cusp ă r p GL p F qz GL p A q{ K q be the space of K -smooth cusp forms supported on g P G with 1 { r ă | det p g q| ă r . The desired result follows because S cusp ă r p GL p F qz GL p A q{ K q is finite dimen-sional.This shows that Mod aut p G q decomposes into a cuspidal part Mod cusp p G q “ Mod p I cusp G q and an Eisenstein part Mod Eis p G q “ Mod p I Eis G q :Mod aut p G q “ Mod cusp p G q ˆ Mod
Eis p G q . Remark
C.4.2 . As an additional consequence of Proposition C.4.1, we see that I Eis G is the image of the map I G Ñ p i p I T r η ´ sq .We also get the following corollary for free: Corollary C.4.3.
Let M P Mod cusp p G q . Then p r aut p M q “ .Proof. We claim that each p i p N q is killed by the action of I cusp G . Indeed, it sufficesto prove this for N “ I T r η ´ s . The fact that I G Ñ p i p I T r η ´ sq is an algebramap now finishes the proof. Warning C.4.4.
Note that unlike the local case, the property p r aut p M q “ anomalous , or non-isobaric , spectrum), it followsthat there are objects M P Mod
Eis p G q with p r p M q “
0. See also Warning A.1.19.
C.4.1 Cuspidal Components
We are left with the task of decomposing each of I cusp G and I Eis G separately. Webegin by decomposing I cusp G as follows.Let G Ď G be the subgroup of all g P G with | det p g q| “
1. Let p π, V q beany irreducible cuspidal automorphic representation of G , with distinguishedgeneric vector v gen P Φ ´ V. Consider the G -module given by induction Ind GG V . Spectrally, the G -moduleInd GG V contains all twists π ˆ | det p¨q| s .48 emark C.4.5 . Choose a place ν of F of degree 1, and a uniformizer π ν . Thischoice induces an isomorphism Ind GG V “ S p q Z q b V , where q is the size of thebase field of F , and S p q Z q is the space of compactly supported functions on thediscrete set q Z . The group G acts on S p q Z q b V via: g ¨ p f p y q b v q ““ f p y | det g |q b π ˜ i v ˜ π ´ log q p y q ν ¸ ¨ g ¨ i v ˜ π log q p y q` log q | det g | ν ¸¸ ¨ v, for g P G , f P S p q Z q , y P q Z and v P V , where i ν : GL p F ν q Ñ GL p A q is theembedding corresponding to the place ν ..We observe that the choice of v gen is adjoint to a surjective map (cid:5) (cid:16) Ind GG V , which turns Ind GG V into a unital commutative algebra. The fact that V is automorphic means that this is in fact an algebra over I G : I G Ñ Ind GG V. Let C cusp p G q be the set of isomorphism classes of irreducible cuspidal au-tomorphic representations p π, V q of G up to conjugation by G . Observe that C cusp p G q is the same as the set of isomorphism classes of irreducible cuspidalautomorphic representations of G up to continuous twists V ÞÑ V b | det p¨q| s .We now claim that: Proposition C.4.6.
The above construction induces an isomorphism of alge-bras: I cusp G „ ÝÑ ź p π,V qP C cusp p G q Ind GG V. Remark
C.4.7 . Note that the infinite product in Proposition C.4.6 is taken inMod p G q , and is therefore also equal to an infinite co-product. That is, thisproduct is actually a direct sum. Proof of Proposition C.4.6.
For every cuspidal p π, V q P C cusp p G q , we have asection Ind GG V Ñ I cusp G given in the notation of Remark C.4.5 by φ p g q b f p y q ÞÑ φ ˜ g ¨ i ν ˜ π log q | det g | ν ¸¸ f p| det g |q , where φ p g q is an automorphic form in the space of V .Together with the finite dimensionality of the space S cusp ă r p GL p F qz GL p A q{ K q from the proof of Proposition C.4.1 and the orthogonality of the different cus-pidal representations, the claim follows.49 .4.2 Eisenstein Components It remains to decompose the Eisenstein part of the category Mod aut p G q . Ourstrategy for doing so is as follows. It is immediate that p i p I T r η ´ sq decomposesinto a direct product over components, because I T does so as well. Because wehave a map of algebras I G Ñ p i p I T r η ´ sq , we will be able to define sub-algebrasof I G corresponding to each component separately. Thus, it will remain to showthat every element φ P I G is the direct sum of its projections corresponding toeach component. This is a property of I G as a G -module, which can be verifiedbecause these projections can be done via elements of the center of Mod p G q .Let us formalize this. Let C Eis p G q be the set of automorphic characters χ : T p F qz T p A q Ñ C ˆ , up to continuous twists |¨| s b |¨| s and twists by theaction of w , as in Definition B.2.8. We can decompose the category Mod aut p T q into components: Mod aut p T q “ ź c P C Eis p G q Mod aut c p T q . In particular, we have a decomposition of algebras: p i p I T r η ´ sq “ ź c P C Eis p G q p i p I cT r η ´ sq as well. Note that as in Remark C.4.7, this product is actually a direct sum.Recall from Remark C.4.2 that I Eis G is isomorphic to the image of the algebramap I G Ñ p i p I T r η ´ sq . Define the algebras I cG for c P C Eis p G q via the pullback square: I cG (cid:47) (cid:47) (cid:15) (cid:15) I Eis G (cid:127) (cid:95) (cid:15) (cid:15) p i p I cT r η ´ sq (cid:31) (cid:127) (cid:47) (cid:47) p i p I T r η ´ sq . We claim that:
Proposition C.4.8.
The map: ź c P C Eis p G q I cG Ñ I Eis G is an isomorphism of algebras.Proof. Consider the composition I Eis G Ñ p i p I T r η ´ sq “ ź c P C Eis p G q p i p I cT r η ´ sq . It is sufficient to show that for each c P C Eis p G q , the projection: I Eis G ã Ñ p i p I T r η ´ sq “ ź c P C Eis p G q p i p I c T r η ´ sq (cid:16) p i p I cT r η ´ sq I cG ã Ñ I Eis G . We will do this byshowing that the image of I Eis G in p i p I T r η ´ sq is closed under the idempotent: e c : p i p I T r η ´ sq Ñ p i p I cT r η ´ sq Ñ p i p I T r η ´ sq . Indeed, we claim that e c is given by the action of an element of the center ofthe category Mod p G q on p i p I T r η ´ sq .This can be shown as follows. Consider the commutative diagram: t w v “ u T (cid:31) (cid:127) (cid:47) (cid:47) (cid:15) (cid:15) T (cid:15) (cid:15) I w “ T (cid:31) (cid:127) (cid:47) (cid:47) I T , Here, t w v “ u T is the sub-algebra of T “ S p T q consisting of functions which areseparately invariant under the Weyl action of Definition B.2.8 at every place.Moreover, I w “ T is the sub-algebra of I T consisting of functions that are invariantunder the simultaneous Weyl action at all places. Recall that t w v “ u T is a sub-algebra of the center of Mod p G q .Since the idempotent e c is given by averaging against elements of I w “ T , itis sufficient to show that the morphism t w v “ u T (cid:47) (cid:47) I w “ T is onto. This is thestatement of Lemma C.4.9 below. Lemma C.4.9.
The map t w v “ u T (cid:47) (cid:47) I w “ T from the proof of Proposition C.4.8 is onto.Remark C.4.10 . To be as explicit as possible, let us give the following equivalentformulation of Lemma C.4.9.Let f : T p F qz T p A q Ñ C be a smooth and compactly supported function suchthat it is invariant under the Weyl action: f p a, d q “ ˇˇˇ ad ˇˇˇ f p d, a q . Then there is a smooth and compactly supported function f : T p A q Ñ C suchthat:1. The function f lifts f : f p a, d q “ ÿ α P F ˆ ÿ δ P F ˆ f p αa, δd q .
2. The function f is locally invariant under the Weyl action at every place v separately: f p a, d q “ ˇˇˇˇ a v d v ˇˇˇˇ v f p ad v { a v , da v { d v q , where a v and d v denote the components of a and d at v .51 emark C.4.11 . The point of Lemma C.4.9 is to show that one cannot “re-mix” the local components of two automorphic characters into a different pairof automorphic characters.Indeed, by going to the spectral picture, Lemma C.4.9 says the following.Consider a pair of automorphic characters χ, χ : T p F qz T p A q Ñ C ˆ , and supposethat by applying the Weyl twist locally at some of the places of F , one can turn χ into χ . Then either χ “ χ or χ “ χ p w q . Proof of Lemma C.4.9.
Let us prove the explicit form of Lemma C.4.9 given byRemark C.4.10.Consider a compact subgroup K Ď A ˆ . Let x, y P F ˆ z A ˆ { K be any twoelements. It is enough to show the claim for f supported on the pair of cosets p x, y q : f p a, d q “ p xKF ˆ ,yKF ˆ q p a, d q ` ˇˇˇˇ xy ˇˇˇˇ p yKF ˆ ,xKF ˆ q p a, d q . By the Chebotarev Density Theorem applied to the ray class group F ˆ z A ˆ { K, there is a place v of F such that π v and xy ´ are equivalent in the ray classgroup, where π v P F ˆ v is a uniformizer. We choose: f p a, d q “ p π v yK,yK q p a, d q ` ˇˇˇˇ xy ˇˇˇˇ p yK,π v yK q p a, d q . Let C p G q “ C cusp p G q š C Eis p G q . We summarize our results as follows. Theorem C.4.12.
The category
Mod aut p G q decomposes into a direct product: Mod aut p G q “ ź c P C p G q Mod aut c p G q . C.5 Corollaries
In this section, we will give two structural results about the category Mod aut p G q ,which follow as corollaries of the heavy Theorem C.4.12.The first result, Theorem C.5.1, will be that irreducible objects in Mod aut p G q are indeed automorphic in the classical sense. We will prove this in Sub-section C.5.1. The second result, Theorem C.5.4, will be that automorphicparabolic restriction p r aut respects finite-length. We will prove this in Subsec-tion C.5.2. 52 .5.1 Irreducible Abstractly Automorphic Objects In this subsection, we will show that all irreducible abstractly automorphic rep-resentations are in fact irreducible automorphic representations in the classicalsense. This means that they are irreducible sub-quotients of the space of smoothfunctions on the automorphic quotient GL p F qz GL p A q . Theorem C.5.1.
Every irreducible V P Mod aut p G q is an irreducible automor-phic representation in the classical sense.Proof. Let V be as above. We must prove that V is a constituent of the contra-gradient of the space S p GL p F qz GL p A qq . Note that because V is irreducible,it lies in some component c P C p G q .If c is Eisenstein, then the action of the center of the category Mod aut p G q on V shows that it is a constituent of a representation of the form p i aut p χ q , for χ P Mod aut p T q irreducible. However, by the main result of [Lan79], this meansthat V itself is automorphic in the classical sense.If c is cuspidal, then we note that the action of the center of G shows that V is a module of an irreducible cuspidal automorphic representation W of G withrespect to (cid:13) (cid:5) . Working place-by-place, the claim follows from Lemma C.5.2.Recall that in a symmetric monoidal Abelian category C , where the monoidalstructure is right-exact, a surjection C (cid:16) A determines a unique algebra struc-ture on A . Lemma C.5.2.
Let v be a place of F . Let W be a generic irreducible repre-sentation of GL p F v q and fix a surjection (cid:5) (cid:16) W . Let V P Mod p GL p F v qq bean irreducible W -module with the algebra structure on W induced by (cid:5) (cid:16) W .Then V is non-canonically isomorphic to W .Proof. The claim is easy to see by the action of the center when W is irreducibleprinciple series or supercuspidal. It remains to resolve the case where W isSteinberg. Assume without loss of generality that W “ St is the Steinbergrepresentation with trivial central character, and denote by triv G the trivial G -module.Observe that by Example 3.54 of [Dor20a], the unit map (cid:5) Ñ St inducesan isomorphism: St – St (cid:13) (cid:5) St . It follows that any M P GL p F v q is an M -module if and only if it satisfies M – St (cid:13) (cid:5) M. Moreover, one can observe that:St (cid:13) (cid:5) ˆ triv G St ˙ – (cid:13) (cid:5) ˆ Sttriv G ˙ – St , ` AB ˘ denotes the unique non-trivial extension with quotient A and sub-object B . This implies that the category of St-modules is projectively generatedby the single object St, and thus that every St-module is of the form:St À I for some set I . Remark
C.5.3 . Na¨ıvely, one might have guessed that the category Mod aut p G q is simply given by the subcategory of Mod p G q where the center of the categoryacts in a certain way. That is, that an irreducible object V P Mod p G q canbe tested for abstract automorphicity by only considering the action of thecenter of Mod p G q on it. However, note that there are cuspidal automorphicrepresentations which are locally Steinberg in some place v . In these cases,replacing the local Steinberg representation at v with the corresponding one-dimensional representation keeps the action of the center the same, but therepresentation is no longer automorphic. This is the difficulty that Lemma C.5.2overcomes. C.5.2 Finite-Length of Automorphic Parabolic Restric-tion
In this subsection, we will show that the functor p r aut : Mod aut p G q Ñ Mod aut p T q sends finite-length representations to finite-length representations. This willfulfill a debt from Section A.1, where we asserted without proof that p r aut iswell-behaved. Theorem C.5.4.
The functor p r aut : Mod aut p G q Ñ Mod aut p T q sends finite-length representations to finite-length representations.Proof. Because p r aut is right-exact, it is sufficient to show that p r aut p V q has finite-length for every irreducible V .If V is cuspidal, then p r aut p V q is 0 by Corollary C.4.3. It remains to checkthe case where V belongs to an Eisenstein component. We know that V is asubquotient of some p i aut p χ q , with χ : T p F qz T p A q Ñ C ˆ . We want to show thatthe space of co-invariants: p r aut p V q “ p r p V q { T p F q is finite-dimensional.To do this, we need a description of the space p r p V q . It is given by therestricted product â v p r v p V v q . Each p r v p V v q is either one of the characters χ v or χ v p w v q , or it decomposes as:0 (cid:47) (cid:47) χ v p w v q (cid:47) (cid:47) p r v p V v q (cid:47) (cid:47) χ v (cid:47) (cid:47) , χ v p w v q being the Weyl twist of χ v , as in Definition B.2.8.Note that there is some additional data needed to understand the restrictedproduct giving p r p V q , which is the distinguished vector of each local represen-tation p r v p V v q . All we need to know is that at the places where p r v p V v q is two-dimensional, it is generated as a T p F v q -module by its distinguished vector.Armed with this description, we can state an informal summary of the ideaof the proof. We want to show that p r p V q { T p F q is finite-dimensional. Essentially,we want to decompose it into a sum of characters of T p F qz T p A q . If we were todecompose each object in the restricted tensor product: â v p r v p V v q , then we would see that p r p V q is composed only of characters that are either χ v or χ v p w v q at every place. However, we want to show that only a finitenumber of these combinations are automorphic , and therefore descend to thequotient T p F qz T p A q . Indeed, as discussed in Remark C.4.11, this is the pointof Lemma C.4.9.Going back to the formal discussion, we observe that our description of p r p V q can be summarised by stating that there is a surjection of T p A q -modules: T b t wv “ u T ´ χ ˇˇˇ t wv “ u T ¯ (cid:16) p r p V q where ´ χ ˇˇˇ t wv “ u T ¯ is the restriction of χ : T Ñ C along t w v “ u T Ñ T . Here,as in Lemma C.4.9, the space t w v “ u T is the space of smooth and compactlysupported functions on T that are invariant under the Weyl action w v at everyplace v of F . This means that: p r p V q { T p F q is covered by: I T b t wv “ u T ´ χ ˇˇˇ t wv “ u T ¯ – I T b I w “ T ´ χ ˇˇˇ I w “ T ¯ , where the isomorphism follows via Lemma C.4.9. However, I T is free of rank 2over I w “ T , meaning that p r p V q { T p F q is at most two-dimensional. Remark
C.5.5 . In the course of the proof of Theorem C.5.4, we have in factshown that p r aut p V q is covered by an extension ˆ χχ p w q ˙ of some character χ : T p F qz T p A q Ñ C ˆ by its twist χ p w q .55 eferences [BZ76] I. N. Bernshtein and A. V. Zelevinskii. Representations of thegroup GL p n, F q where F is a non-Archimedean local field. Rus-sian Math. Surveys , 31(3):1–68, 1976. URL: , doi : 10 . 1070 /RM1976v031n03ABEH001532 . 14[Dor20a] Gal Dor. Exotic Monoidal Structures and Abstractly AutomorphicRepresentations for GL p q . arXiv e-prints , page arXiv:2011.03313,November 2020. arXiv:2011.03313 . 1, 3, 4, 5, 6, 14, 15, 16, 25, 26,44, 48, 53[Dor20b] Gal Dor. Modules of Zeta Integrals for GL p q . arXiv e-prints , pagearXiv:2012.03068, December 2020. arXiv:2012.03068 . 1, 8, 20, 23,25, 46[GGPS69] I. Gelfand, M. Graev, and I. Pyatetskii-Shapiro. Representation the-ory and automorphic functions. In Generalized Functions , volume 6.W. B. Saunders Company, 1969. Translated from Russian by K. A.Hirsch. 8[KS15] Robert Kurinczuk and Shaun Stevens. Cuspidal (cid:96) -modular representa-tions of p -adic classical groups. arXiv e-prints , page arXiv:1509.02212,September 2015. arXiv:1509.02212 . 9[Lan79] Robert P. Langlands. On the notion of an automorphic representa-tion. In Automorphic Forms, Representations and L-functions, Proc.Sympos. Pure Math. 33, part 1 , pages 203–207, 1979. 53[nLa21] nLab authors. monoidal functor. http://ncatlab.org/nlab/show/monoidal+functor , January 2021. Revision 43. 7[Vig89] Marie-France Vign´eras. Repr´esentations modulaires de gl p , f q en car-act´eristique l , f corps p -adique, p ‰ l . Compositio Mathematica ,72(1):33–66, 1989. URL: . 9, 10, 42[Vig01] Marie-France Vign´eras. La conjecture de langlands locale pour gl p n, f q modulo (cid:96) quand (cid:96) ‰ p , (cid:96) ą n . Annales scientifiques de l’ ´Ecole NormaleSup´erieure , 4e s´erie, 34(6):789–816, 2001. URL: , doi:10.1016/s0012-9593(01)01077-1doi:10.1016/s0012-9593(01)01077-1