Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)

Kei Yuen Chan

doi:10.1515/crelle-2021-0066

Artikel

Restriction for general linear groups: The local non-tempered Gan–Gross–Prasad conjecture (non-Archimedean case)

Kei Yuen Chan

Veröffentlicht/Copyright: 2. Dezember 2021

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2022 Heft 783

Abstract

We prove a local Gan–Gross–Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean fields. We also generalize to Bessel and Fourier–Jacobi models and study a possible generalization to Ext-branching laws.

A Some homological algebra

Let 𝒜=Alg⁢(Gl). Let ℬ=Alg⁢(Gn). Via Yoneda extension, any element in Ext𝒜1⁢(X,Y) corresponds to a short exact sequence in 𝒜, and zero element corresponds to the split sequence. Then, for an additive exact functor ℱ, ℱ sends a short exact sequence to a short exact sequence, and this defines a map from Ext𝒜1⁢(X,Y) to Extℬ1⁢(ℱ⁢(X),ℱ⁢(Y)).

Lemma A.1.

Let C be a full Serre subcategory of A=Alg⁢(Gl). Let B=Alg⁢(Gn) and let D be a Serre full subcategory of B. Let F:C→D be an exact additive functor. We also regard objects in C as objects in A via the inclusion. Assume that:

any object in 𝒞 is of finite length,
for any simple objects X,Y in the subcategory 𝒞, the induced map of ℱ, from Ext𝒜1⁢(X,Y) to Extℬ1⁢(ℱ⁢(X),ℱ⁢(Y)) is an injection,
ℱ⁢(X) is a simple object in 𝒟 if Xis simple in 𝒞,
for any simple objects X and Y in 𝒞, ℱ⁢(X)≅ℱ⁢(Y) if and only if X≅Y.

Then for any objects X,Y in C, the induced map from ExtA1⁢(X,Y) to ExtB1⁢(F⁢(X),F⁢(Y)) is also injective, and F:C→D is fully-faithful, i.e.

Homℬ⁢(ℱ⁢(X),ℱ⁢(Y))≅Hom𝒟⁢(ℱ⁢(X),ℱ⁢(Y))≅Hom𝒞⁢(X,Y)≅Hom𝒜⁢(X,Y)

for any objects X,Y in C.

Proof.

Let X and Y be objects in 𝒞. When both lengths of X and Y are 1 in 𝒞,

Hom𝒟⁢(ℱ⁢(X),ℱ⁢(Y))≅Hom𝒞⁢(X,Y),Ext𝒜1⁢(X,Y)↪Extℬ1⁢(ℱ⁢(X),ℱ⁢(Y))

are guaranteed by (2), (3) and (4). We first fix the length of X to be at most some n. We shall prove the statement for arbitrary Y by induction on the length of Y.

For an object Y in 𝒞, let Y1 be an irreducible quotient of Y. Then we have a short exact sequence

0→Y2→Y→Y1→0.

Since 𝒞 is Serre, it follows that Y1 and Y2 are in 𝒞.

Note that we have the following commutative diagram:

where the horizontal maps come from long exact sequences, in which the connecting homomorphism is the Yoneda product, and vertical maps for Ext1 are described in the beginning of this section, and the vertical map for Hom is the map induced from the functor.

We have the first vertical arrow is isomorphism and the second and forth vertical arrows are injections by induction hypothesis. Then it is direct to check that the third vertical arrow is also an injection.

Now we consider another commutative diagram:

The first and third vertical arrows are isomorphisms by induction and the last vertical arrow is an injection by induction again. Thus we have that the second vertical arrow is an isomorphism.

Now we switch the role of X and Y, and use similar argument to prove that the assertion is true for X and Y of arbitrary finite length. ∎

Remark A.2.

The above lemma is also valid for arbitrary abelian categories 𝒜 and ℬ which are Schurian k-categories, where k is a field, i.e.

Hom𝒜⁢(X,X)≅k and Homℬ⁢(Y,Y)≅k

for any simple objects X and Y in 𝒜 and ℬ, respectively.

Acknowledgements

This project grows out from discussions with Dipendra Prasad, and the author would like to thank him for helpful discussions and comments. He would also like to thank Gordan Savin for discussions on various topics and helpful comments. The author would also like to thank Max Gurevich for helpful correspondences on the preprint. The author would like to thank the referee for careful reading and useful comments.

References

[1] A. Aizenbud, D. Gourevitch, S. Rallis and G. Schiffmann, Multiplicity one theorems, Ann. of Math. (2) 172 (2010), no. 2, 1407–1434. 10.4007/annals.2010.172.1407Suche in Google Scholar

[2] A. Aizenbud and E. Sayag, Homological multiplicities in representation theory of p-adic groups, Math. Z. 294 (2020), no. 1–2, 451–469. 10.1007/s00209-019-02262-4Suche in Google Scholar

[3] J. Arthur, Unipotent automorphic representations: conjectures, Orbites unipotentes et représentations, II, Asterisque 171–172, Société Mathématique de France, Paris (1989), 13–71. Suche in Google Scholar

[4] H. Atobe, The local theta correspondence and the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case, Math. Ann. 371 (2018), no. 1–2, 225–295. 10.1007/s00208-017-1620-5Suche in Google Scholar

[5] I. N. Bernstein, P-invariant distributions on GL⁢(N) and the classification of unitary representations of GL⁢(N) (non-Archimedean case), Lie group representations, II (College Park 1982/1983), Lecture Notes in Math. 1041, Springer, Berlin (1984), 50–102. 10.1007/BFb0073145Suche in Google Scholar

[6] I. N. Bernstein and A.V̇. Zelevinsky, Induced representations of reductive 𝔭-adic groups. I, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), no. 4, 441–472. 10.24033/asens.1333Suche in Google Scholar

[7] I. N. Bernstein and A. V. Zelevinsky, Representations of the group GL⁢(n,F) where F is a local non-Archimedean field, Russian Math. Surveys 31 (1976), no. 3, 1–68. 10.1070/RM1976v031n03ABEH001532Suche in Google Scholar

[8] R. Beuzart-Plessis, Endoscopie et conjecture locale raffinée de Gan–Gross–Prasad pour les groupes unitaires, Compos. Math. 151 (2015), no. 7, 1309–1371. 10.1112/S0010437X14007891Suche in Google Scholar

[9] R. Beuzart-Plessis, A local trace formula for the Gan–Gross–Prasad conjecture for unitary groups: The archimedean case, preprint (2015), https://arxiv.org/abs/1506.01452. 10.24033/ast.1120Suche in Google Scholar

[10] K. Y. Chan, Duality for Ext-groups and extensions of discrete series for graded Hecke algebras, Adv. Math. 294 (2016), 410–453. 10.1016/j.aim.2016.03.002Suche in Google Scholar

[11] K. Y. Chan, Some methods of computing first extensions between modules of graded Hecke algebras, Algebr. Represent. Theory 21 (2018), no. 4, 859–895. 10.1007/s10468-017-9742-8Suche in Google Scholar

[12] K. Y. Chan, Ext-multiplicity theorem for standard representations of (GLn+1,GLn), preprint (2021), https://arxiv.org/abs/2104.11528. Suche in Google Scholar

[13] K. Y. Chan, Homological branching law for (GLn+1⁢(F),GLn⁢(F)): Projectivity and indecomposability, Invent. Math. 225 (2021), no. 1, 299–345. 10.1007/s00222-021-01033-5Suche in Google Scholar

[14] K. Y. Chan and G. Savin, Iwahori component of the Gelfand–Graev representation, Math. Z. 288 (2018), no. 1–2, 125–133. 10.1007/s00209-017-1882-3Suche in Google Scholar

[15] K. Y. Chan and G. Savin, Bernstein–Zelevinsky derivatives: A Hecke algebra approach, Int. Math. Res. Not. IMRN 2019 (2019), no. 3, 731–760. 10.1093/imrn/rnx138Suche in Google Scholar

[16] K. Y. Chan and G. Savin, A vanishing Ext-branching theorem for (GLn+1⁢(F),GLn⁢(F)), Duke Math. J. 170 (2021), no. 10, 2237–2261. 10.1215/00127094-2021-0028Suche in Google Scholar

[17] F. Chen and B. Sun, Uniqueness of Rankin–Selberg periods, Int. Math. Res. Not. IMRN 2015 (2015), no. 14, 5849–5873. 10.1093/imrn/rnu110Suche in Google Scholar

[18] W. T. Gan, B. H. Gross and D. Prasad, Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups, Sur les conjectures de Gross et Prasad. I, Asterisque 346, Société Mathématique de France, Paris (2012), 1–109. Suche in Google Scholar

[19] W. T. Gan, B. H. Gross and D. Prasad, Branching laws for classical groups: the non-tempered case, Compos. Math. 156 (2020), no. 11, 2298–2367. 10.1112/S0010437X20007496Suche in Google Scholar

[20] W. T. Gan and A. Ichino, The Gross–Prasad conjecture and local theta correspondence, Invent. Math. 206 (2016), no. 3, 705–799. 10.1007/s00222-016-0662-8Suche in Google Scholar

[21] D. Gourevitch and E. Sayag, Annihilator varieties of distinguished modules of reductive lie algebras, preprint (2020), https://arxiv.org/abs/2001.11746v2. 10.1017/fms.2021.42Suche in Google Scholar

[22] B. H. Gross and D. Prasad, On the decomposition of a representation of SOn when restricted to SOn-1, Canad. J. Math. 44 (1992), no. 5, 974–1002. 10.4153/CJM-1992-060-8Suche in Google Scholar

[23] M. Gurevich, On restriction of unitarizable representations of general linear groups and the non-generic local Gan–Gross–Prasad conjecture, preprint (2020), https://arxiv.org/abs/1808.02640v3; J. Eur. Math. Soc. (JEMS), to appear. 10.4171/JEMS/1093Suche in Google Scholar

[24] M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich, Ann. of Math. Stud. 151, Princeton University Press, Princeton 2001. Suche in Google Scholar

[25] H. He, On the Gan–Gross–Prasad conjecture for U⁢(p,q), Invent. Math. 209 (2017), no. 3, 837–884. 10.1007/s00222-017-0720-xSuche in Google Scholar

[26] G. Henniart, Une preuve simple des conjectures de Langlands pour GL⁢(n) sur un corps p-adique, Invent. Math. 139 (2000), no. 2, 439–455. 10.1007/s002220050012Suche in Google Scholar

[27] H. Jacquet, I. I. Piatetskii-Shapiro and J. A. Shalika, Rankin–Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. 10.2307/2374264Suche in Google Scholar

[28] A. Kret and E. Lapid, Jacquet modules of ladder representations, C. R. Math. Acad. Sci. Paris 350 (2012), no. 21–22, 937–940. 10.1016/j.crma.2012.10.014Suche in Google Scholar

[29] E. Lapid and A. Mínguez, On a determinantal formula of Tadić, Amer. J. Math. 136 (2014), no. 1, 111–142. 10.1353/ajm.2014.0006Suche in Google Scholar

[30] E. Lapid and A. Mínguez, On parabolic induction on inner forms of the general linear group over a non-archimedean local field, Selecta Math. (N.S.) 22 (2016), no. 4, 2347–2400. 10.1007/s00029-016-0281-7Suche in Google Scholar

[31] G. Laumon, M. Rapoport and U. Stuhler, D-elliptic sheaves and the Langlands correspondence, Invent. Math. 113 (1993), no. 2, 217–338. 10.1007/BF01244308Suche in Google Scholar

[32] C. Mœglin and J.-L. Waldspurger, La conjecture locale de Gross–Prasad pour les groupes spéciaux orthogonaux: le cas général, Sur les conjectures de Gross et Prasad. II, Asterisque 347, Société Mathématique de France, Paris (2012), 167–216. Suche in Google Scholar

[33] E. Opdam and M. Solleveld, Extensions of tempered representations, Geom. Funct. Anal. 23 (2013), no. 2, 664–714. 10.1007/s00039-013-0219-6Suche in Google Scholar

[34] S. Orlik, On extensions of generalized Steinberg representations, J. Algebra 293 (2005), no. 2, 611–630. 10.1016/j.jalgebra.2005.03.028Suche in Google Scholar

[35] D. Prasad, On the decomposition of a representation of GL⁢(3) restricted to GL⁢(2) over a p-adic field, Duke Math. J. 69 (1993), no. 1, 167–177. 10.1215/S0012-7094-93-06908-6Suche in Google Scholar

[36] D. Prasad, Ext-analogues of branching laws, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Scientific Publishing, Hackensack (2018), 1367–1392. 10.1142/9789813272880_0101Suche in Google Scholar

[37] P. Scholze, The local Langlands correspondence for GLn over p-adic fields, Invent. Math. 192 (2013), no. 3, 663–715. 10.1007/s00222-012-0420-5Suche in Google Scholar

[38] J. A. Shalika, The multiplicity one theorem for GLn, Ann. of Math. (2) 100 (1974), 171–193. 10.2307/1971071Suche in Google Scholar

[39] B. Sun, Multiplicity one theorems for Fourier–Jacobi models, Amer. J. Math. 134 (2012), no. 6, 1655–1678. 10.1353/ajm.2012.0044Suche in Google Scholar

[40] B. Sun and C.-B. Zhu, Multiplicity one theorems: the Archimedean case, Ann. of Math. (2) 175 (2012), no. 1, 23–44. 10.4007/annals.2012.175.1.2Suche in Google Scholar

[41] M. Tadić, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. Éc. Norm. Supér. (4) 19 (1986), no. 3, 335–382. 10.24033/asens.1510Suche in Google Scholar

[42] A. Venkatesh, The Burger–Sarnak method and operations on the unitary dual of GL⁢(n), Represent. Theory 9 (2005), 268–286. 10.1090/S1088-4165-05-00226-8Suche in Google Scholar

[43] J.-L. Waldspurger, La conjecture locale de Gross–Prasad pour les représentations tempérées des groupes spéciaux orthogonaux, Sur les conjectures de Gross et Prasad. II, Asterisque 347, Société Mathématique de France, Paris (2012), 103–165. Suche in Google Scholar

[44] A. V. Zelevinsky, Induced representations of reductive p-adic groups. II. On irreducible representations of GL⁢(n), Ann. Sci. Éc. Norm. Supér. (4) 13 (1980), no. 2, 165–210. 10.24033/asens.1379Suche in Google Scholar

[45] The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu, 2020. Suche in Google Scholar

Received: 2020-07-23

Revised: 2021-10-04

Published Online: 2021-12-02

Published in Print: 2022-02-01

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.1515/crelle-2021-0066