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Schubert Eisenstein Series and Poisson Summation for Schubert SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION FOR SCHUBERT VARIETIES YOUNGJU CHOIE AND JAYCE R. GETZ Abstract. Schubert Eisenstein series are defined by restricting the summation in a de- generate Eisenstein series to a particular Schubert variety. In the case of GL3 over Q they proved that these Schubert Eisenstein series have meromorphic continuations in all param- eters and conjectured the same is true in general. We revisit their conjecture and relate it to the program of Braverman, Kazhdan, Lafforgue, Ngˆo, and Sakellaridis aimed at estab- lishing generalizations of the Poisson summation formula. We prove the Poisson summation formula for certain schemes closely related to Schubert varieties and use it to refine and establish the conjecture of Bump and the first author in many cases. Contents 1. Introduction 2 1.1. Generalized Schubert Eisenstein series 2 1.2. The Poisson summation conjecture 7 1.3. Conjectures 1.3 and 1.4 8 1.4. On integral representations of L-functions 9 1.5. Outline 10 Acknowledgments 10 2. Preliminaries 11 2.1. Braverman-Kazhdan spaces 11 2.2. The YP 14 2.3. Examples 17 arXiv:2107.01874v2 [math.NT] 19 Jul 2021 2.4. Function spaces 17 2.5. Measures 18 2.6. Quasi-characters 19 2.7. The unramified setting 19 3. The Schwartz space of YP,P ′ (F ) 20 3.1. Induced representations 20 2010 Mathematics Subject Classification. Primary 11F70; Secondary 11F55, 11F85. Key words and phrases. Schubert Eisenstein series, Poisson summation conjecture, Schubert varieties. The first author was partially supported by NRF 2018R1A4A1023590 and NRF 2017R1A2B2001807. The second author is partially supported by NSF grant DMS 1901883, and received support from Y. Choie’s NRF 2017R1A2B2001807 grant. 1 2 YOUNGJU CHOIE AND JAYCE R. GETZ 3.2. The Schwartz space 20 3.3. The Fourier transform 26 3.4. The unramified setting 29 3.5. The adelic setting 31 4. The Poisson summation formula on X (F ) 32 P ∩Mβ0 5. Proofs of Theorem 1.1 and Theorem 1.2 38 6. On the poles of degenerate Eisenstein series 45 References 47 1. Introduction In this paper we prove the Poisson summation conjecture of Braverman-Kazhdan, Laf- forgue, Ngˆo, and Sakellaridis for a particular family of varieties related to Schubert varieties (see Theorem 1.2). As an application we prove (in many cases) the functional equations of Schubert Eisenstein series conjectured to exist by Bump and the first author in [BC14]. We begin by recalling the definition of the Schubert Eisenstein series and then move to a discussion of the Poisson summation formula. 1.1. Generalized Schubert Eisenstein series. Let F be a global field with ring of adeles AF , and let G be a (connected) split reductive group over F. We let T ≤ B ≤ P ≤ G be a maximal split torus, a Borel subgroup, and a parabolic subgroup of G. Moreover we let T ≤ M ≤ P be the Levi subgroup of P containing T . Let NG(T ) be the normalizer of T in G and let W (G, T ) := NG(T )(F )/T (F ) be the Weyl group of T in G. Finally we let ab k+1 ωP : M −→ Gm be the isomorphism of (2.1.11). Here M ab = M/M der is the abelianization of M, where Qder is the derived group of an algebraic group Q. × × × k+1 × Let AGm < F∞ be the usual subgroup (see (2.6.1)) and let χ :(AGm F \AF ) → C be a character. For s ∈ Ck+1 we define k si χs(a0,...,ak) := χ(a0,...,ak) |ai| . i=0 Y We form the induced representation G IP (χs) := IndP (χs ◦ ωP ), SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 3 normalized so that it is unitary when s ∈ (iR)k+1. Using the Bruhat decomposition of G one has a decomposition of the generalized flag variety P \G = P \P wB. w∈W (M,Ta)\W (G,T ) Here we have used the same symbol w for a class in W (M, T )\W (G, T ) and for a represen- tative of that class. Let Xw be the (Zariski) closure of the Schubert cell P \P wB in P \G. It is a Schubert variety. χs The Schubert Eisenstein series attached to a section Φ ∈ IP (χs), g ∈ G(AF ) and w ∈ W (G, T ) is defined as χs χs SEw(g, Φ ) := Φ (γg). γ∈Xw(F ) χs It converges absolutely provided that Re(s) lies in a suitable cone. The function SEw(g, Φ ) is no longer left G(F )-invariant. However, it is invariant under the stabilizer of the Schubert cell Xw under the natural action of G on P \G. Since this stabilizer contains B, it is a parabolic subgroup, and it is often larger than B. In [BC14] Bump and the first author posed the following questions: (a) Does the Schubert Eisenstein series admit a meromorphic continuation to all s? (b) Do a subset of the functional equations for the full Eisenstein series continue to hold for the Schubert Eisenstein series? (c) Is it possible to find a linear combination of Schubert Eisenstein series which is entire? We generalize the definition of Schubert Eisenstein series in higher rank in a manner that can be specialized to the definition in the G = GL3 case treated in [BC14]. We then answer all of these questions in great generality. We refer to Theorem 1.1 and the subsequent remarks for details. Before proceeding, let us examine the setting considered in [BC14] and isolate the change in viewpoint that is the germ of our work. Let G = GL3 and let B < GL3 be the Borel subgroup of upper triangular matrices. From the point of view of this paper the most interesting Schubert variety in this setting is that attached to the Bruhat cell Bs1s2B where 1 1 s1 = 1 , s2 = 1 . 1 1 In this case using standard facts on the Bruhat ordering one has (1.1.1) Bs1s2B = Bs1s2B ∐ Bs1B ∐ Bs2B ∐ B. Let R be an F -algebra. Then ab c (1.1.2) Bs1s2B(R)= def ∈ GL3(R) . g h n o Indeed, the set on the right is the R-points of an irreducible closed subscheme of GL3 that contains the irreducible closed subscheme Bs1s2B, hence we deduce equality by considering 4 YOUNGJU CHOIE AND JAYCE R. GETZ dimensions. If we let ab c ab c P2,1(R) := def ∈ GL3(R) , P1,2(R) := e f ∈ GL3(R) h g h n o n o then (1.1.3) Bs1s2B = P2,1s1s2P1,2. Indeed, it is easy to see that each Bruhat cell on the right of (1.1.1) is contained in P2,1s1s2P1,2, and it is clear on the other hand from (1.1.2) that P2,1s1s2P1,2 ⊆ Bs1s2B. The natural map P2,1 × P1,2 −→ Bs1s2B is a lift of the Bott-Samelson resolution of the image Xs1s2 of Bs1s2B in B\SL3. This was the point of departure for the arguments in [BC14]. Instead of pursuing Schubert varieties and the Bott-Samelson resolution to study higher rank analogues of the left hand side of (1.1.3), we generalize and study the right hand side directly. Let us step back to consider the situation for a general reductive group G. Consider ′ the closure P wB in G. It is the union of Bruhat cells ∪w′≤wPw B, where ≤ denotes the Bruhat order. The group G acts on itself on the left, and we let Q be the stabilizer of the closed subscheme P wB ⊂ G (i.e. the algebraic subgroup of G sending P wB to itself). This is a closed algebraic subgroup of G [Mil17, Corollary 1.81]. The group Q contains P, and hence is a parabolic subgroup. Thus rather than studying P wB, for any parabolic subgroup P ′ with P ≤ P ′ ≤ Q we can study generalized Schubert Eisenstein series built from sums over P ′wB. In fact, there is nothing special about B, and it is more interesting from an automorphic perspective to replace B by the largest possible group that preserves P wB under right multiplication. Thus we could work with (1.1.4) P ′γH where P ≤ P ′ ≤ G are a pair of parabolic subgroups, γ ∈ G(F ) and H is an arbitrary algebraic subgroup of G. From the automorphic point of view this may be the most important situation. However, not all Schubert cells are the image in P \G of a set of the form (1.1.4). Indeed, Schubert cells are often nonsmooth, whereas the image of any set of the form P ′γH in P \G is a smooth subscheme (this follows from lemmas 2.4 and 2.5). In order to treat Eisenstein series indexed by sets of the form Y and P wB simultaneously we work with an arbitrary (locally closed) subscheme Y ⊆ G that is stable under left ′ ◦ der multiplication by P . Let XP := P \G be the Braverman-Kazhdan space associated to P and G. Let ◦ (1.1.5) YP = Im(Y −→ XP ). SCHUBERT EISENSTEIN SERIES AND POISSON SUMMATION 5 ◦ To be more precise, the set theoretic image of Y → XP is locally closed by Lemma 2.4. This ◦ set is the underlying topological space of a subscheme YP of XP . For some examples when ◦ G = SLn we refer to §2.3. The subscheme YP ⊆ XP is quasi-affine. Let XP be the affine ◦ closure of XP and let (1.1.6) YP,P ′ ⊆ XP be the partial closure of YP in XP constructed in (2.2.6) below. We observe that there is a ab natural action of M on YP,P ′ that preserves YP (see (2.1.5)).
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