Multiplicities of Representations in Algebraic Families

Li Cai; Yangyu Fan

doi:10.5802/crmath.623

Article de recherche - Théorie des représentations

Multiplicities of Representations in Algebraic Families
[Multiplicités des représentations dans les familles algébriques]

Li Cai ¹ ; Yangyu Fan ²

¹ Academy for Multidisciplinary Studies, Beijing National Center for Applied Mathematics, Capital Normal University, Beijing, 100048, People’s Republic of China
² School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1097-1107.

Résumé
Abstract

Dans cette courte note, nous considérons les multiplicités de représentations dans des familles algébriques générales,en particulier la semi-continuité supérieure des multiplicités homologiques et la constance locale des nombres d’Euler–Poincaré. Ceci généralise le résultat principal d’Aizenbud–Sayag pour les familles obtenues en induisant une représentation fixée que l’on tord par des caractères non ramifiés.

In this short notes, we consider multiplicities of representations in general algebraic families, especially the upper semi-continuity of homological multiplicities and the locally constancy of Euler–Poincaré numbers. This generalizes the main result of Aizenbud–Sayag for unramified twisting families.

Reçu le : 2023-09-06
Révisé le : 2024-02-07
Accepté le : 2024-02-07
Publié le : 2024-11-05

DOI : 10.5802/crmath.623

Classification : 22E45, 20G25
Keywords: Branching laws, Homological multiplicities, Spherical varieties
Mot clés : Lois de branchement, multiplicités homologiques, variétés sphériques

Affiliations des auteurs :

Li Cai ¹ ; Yangyu Fan ²

¹ Academy for Multidisciplinary Studies, Beijing National Center for Applied Mathematics, Capital Normal University, Beijing, 100048, People’s Republic of China
² School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Multiplicities of {Representations} in {Algebraic} {Families}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1097--1107},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.623},
     language = {en},
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Li Cai; Yangyu Fan. Multiplicities of Representations in Algebraic Families. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1097-1107. doi : 10.5802/crmath.623. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.623/

Bibliographie
Cité par

[1] Avraham Aizenbud; Nir Avni; Dmitry Gourevitch Spherical pairs over close local fields, Comment. Math. Helv., Volume 87 (2012) no. 4, pp. 929-962 | DOI | MR | Zbl

[2] Avraham Aizenbud; Eitan Sayag Homological multiplicities in representation theory of $p$ -adic groups, Math. Z., Volume 294 (2020) no. 1-2, pp. 451-469 | DOI | MR | Zbl

[3] Philippe Blanc; Patrick Delorme Vecteurs distributions $H$ -invariants de représentations induites, pour un espace symétrique réductif $p$ -adique $G / H$ , Ann. Inst. Fourier, Volume 58 (2008) no. 1, pp. 213-261 | DOI | Numdam | MR | Zbl

[4] Li Cai; Yangyu Fan Top degree Ext-groups and relatively supercuspidal spectra, Adv. Math., Volume 410 (2022) no. part A, p. Paper No. 108744, 25 | DOI | MR | Zbl

[5] Li Cai; Yangyu Fan Families of canonical local periods on spherical varieties, Math. Ann., Volume 389 (2024) no. 1, pp. 209-252 | DOI | MR | Zbl

[6] R. Chen Ext-Bessel model vanishes for tempered representations (2023) (https://arxiv.org/abs/2303.12619)

[7] Kei Yuen Chan; Gordan Savin A vanishing Ext-branching theorem for $({GL}_{n + 1} (F), {GL}_{n} (F))$ , Duke Math. J., Volume 170 (2021) no. 10, pp. 2237-2261 | DOI | MR | Zbl

[8] Daniel Disegni Local Langlands correspondence, local factors, and zeta integrals in analytic families, J. Lond. Math. Soc. (2), Volume 101 (2020) no. 2, pp. 735-764 | DOI | MR | Zbl

[9] Daniel Disegni The universal $p$ -adic Gross-Zagier formula, Invent. Math., Volume 230 (2022) no. 2, pp. 509-649 | DOI | MR | Zbl

[10] Matthew Emerton; David Helm The local Langlands correspondence for ${GL}_{n}$ in families, Ann. Sci. Éc. Norm. Supér. (4), Volume 47 (2014) no. 4, pp. 655-722 | DOI | MR | Zbl

[11] Yuval Z. Flicker Bernstein’s isomorphism and good forms, $K$ -theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) (Proc. Sympos. Pure Math.), Volume 58, American Mathematical Society, Providence, RI, 1995, pp. 171-196 | DOI | MR | Zbl

[12] Brooke Feigon; Erez Lapid; Omer Offen On representations distinguished by unitary groups, Publ. Math. Inst. Hautes Études Sci., Volume 115 (2012), pp. 185-323 | DOI | Numdam | MR | Zbl

[13] Sergei I. Gelfand; Yuri I. Manin Methods of homological algebra, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003, xx+372 pages | DOI | MR | Zbl

[14] Eric Opdam; Maarten Solleveld Resolutions of tempered representations of reductive $p$ -adic groups, J. Funct. Anal., Volume 265 (2013) no. 1, pp. 108-134 | DOI | MR | Zbl

[15] Dipendra Prasad Ext-analogues of branching laws, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, World Sci. Publ., Hackensack, NJ (2018), pp. 1367-1392 | DOI | MR | Zbl

[16] Dipendra Prasad Generalizing the MVW involution, and the contragredient, Trans. Amer. Math. Soc., Volume 372 (2019) no. 1, pp. 615-633 | DOI | MR | Zbl

[17] Stack projects, 2022 (https://stacks.math.columbia.edu)

[18] Peter Schneider; Ulrich Stuhler Representation theory and sheaves on the Bruhat–Tits building, Inst. Hautes Études Sci. Publ. Math. (1997) no. 85, pp. 97-191 | DOI | Numdam | MR | Zbl

[19] Yiannis Sakellaridis; Akshay Venkatesh Periods and harmonic analysis on spherical varieties, Astérisque, 396, Société Mathématique de France, Paris, 2017, viii+360 pages | MR | Zbl

[20] Chen Wan On a multiplicity formula for spherical varieties, J. Eur. Math. Soc. (JEMS), Volume 24 (2022) no. 10, pp. 3629-3678 | DOI | MR | Zbl

[21] Amnon Yekutieli A course on derived categories (2015) (https://arxiv.org/abs/1206.6632)

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