Utility
References
Identifiers checked on 2026-10-19: DOIs against Crossref, arXiv versions on the abstract page. Theorem numbers were checked in the cited version: [Sak] in arXiv:0905.4244v3 (whose page numbers are those quoted); [SV] in arXiv:1203.0039v4 (printed page = PDF page − 4); the others as stated. Anything not checked is marked unverified.
The Theorem and its framework
[Sak] Y. Sakellaridis, Spherical functions on spherical varieties, Amer. J. Math. 135(5) (2013) 1291–1381, doi:10.1353/ajm.2013.0046; arXiv:0905.4244v3 (open copy).
- §1.1 (characteristic 0, homogeneous), §1.7 (standing assumptions; \(\dX\)), §2.1 (\(\PX\), \(\LevX\), \(\AX\), \(\LXp\), \(\AXs\)), §2.2 and Axiom 2.2.1, Axioms 2.4.1–2.4.2 and the remark after them, Thm. 3.2.2, §3.2 (3.2), Prop. 4.2.1, Thm. 4.2.2 (spherical functions), §5.4 (Macdonald), §7 opening, §7.1, Statement 7.1.5, Thm. 7.2.1, §8 opening, Ex. 8.0.1, Thm. 8.0.2, Cor. 8.0.4 and the remark after it, (8.1), (8.2), §1.3 (1.3)–(1.5), Thm. 9.0.1, Appendix A.
- Used on: motivation, the Theorem, hypotheses, all Geometry pages, r, characteristic.
[SV] Y. Sakellaridis, A. Venkatesh, Periods and harmonic analysis on spherical varieties, Astérisque 396 (2017), viii+360 pp., doi:10.24033/ast.1040 (Crossref dates it 2018); arXiv:1203.0039v4 (open copy). Astérisque page numbers unverified.
- §1.1, §1.7, §2 opening, §2.1 (definitions, wavefront), Prop. 2.2.1, Prop. 2.2.2, Thm. 2.2.3, Ex. 1.3.2, Ex. 2.2.5, Ex. 14.2.2, Rem. 2.3.2, §2.3.7, §2.4, §15.3, Thm. 5.3.3, §16.3, §17.2, §17.3 (17.8), (17.9), Conj. 17.4.1 (arXiv v4 pp. 258–259, local copy), (17.13), Ex. 17.5.1, §17.6; §1.4 (1.5).
- Used on: motivation, Sakellaridis–Venkatesh, torus, δ(X), W_X.
[Sak08] Y. Sakellaridis, On the unramified spectrum of spherical varieties over p-adic fields, Compos. Math. 144(4) (2008) 978–1016, doi:10.1112/S0010437X08003485. Thm. 6.2.1 (torsion-freeness), as quoted in [Sak, §8] and [Sak-IS, Thm. 3.2]; the multiplicity theorem quoted as [Sak, Thm. 3.2.2]. Used on: δ(X), Satake.
[Sak12] Y. Sakellaridis, Spherical varieties and integral representations of L-functions, Algebra Number Theory 6(4) (2012) 611–667, doi:10.2140/ant.2012.6.611; arXiv:0905.4245v3 (checked 2026-10-20; local copy, Conj. 3.2.2 and 3.2.4 on pp. 24–25). §1.3, §3.1 (basic function), Thm. 2.3.8 (Cartan decomposition), Conj. 3.2.2, 3.2.4, Ex. 4.1.2, Thm. 4.1.3; §1.2 (1.1), (1.2) (Godement–Jacquet and Hecke integrals). Used on: motivation, orbits, beyond SV, 𝔸ⁿ.
[SW] Y. Sakellaridis, J. Wang, Intersection complexes and unramified L-factors, J. Amer. Math. Soc. 35(3) (2022) 799–910, doi:10.1090/jams/990; arXiv:2009.03943v3. §1.1 (setting \(\Fqt\)), hypothesis (1.1), Thm. 1.1.2, Thm. 1.1.4. Used on: motivation, beyond SV, characteristic, 𝔸ⁿ.
[BZSV] D. Ben-Zvi, Y. Sakellaridis, A. Venkatesh, Relative Langlands duality, arXiv:2409.04677v1 (2024). Table 1.5.1 (dual pairs), Ex. 4.3.12 (\(\Aff^n\)), Conj. 7.5.1, §7.6.5, §9 (numerical unramified Plancherel), §9.4. Used on: motivation, 𝔸ⁿ geometry, 𝔸ⁿ, δ(X).
[OWR25] D. Ben-Zvi, Y. Sakellaridis, A. Venkatesh (organisers), Arbeitsgemeinschaft: Relative Langlands Duality, Oberwolfach Report 18/2025, Oberwolfach Rep. 22(2) (2025) 811–882, doi:10.4171/OWR/2025/18 (Crossref checked 2026-10-20; CC BY-SA 4.0, local copy).
- Organisers' introduction, diagram (1), p. 812; T. Kaletha, Introduction to the relative Langlands program, pp. 832–837: Cor. 2 (Tate), §3.3, Prop. 7, Conj. 8–9 (Ichino–Ikeda); J. Fintzen, The notion of the unramified Plancherel formula, pp. 840–841: Conj. 1, examples (1)–(3); Y. Shan, Unramified Plancherel formula for spherical varieties, pp. 841–843: Conj. 2.1, Ex. 2.4; S. Lai, Periods and L-functions, pp. 860–863: §3, Rem. 3.1; C.-w. Lin, Global Conjectures, pp. 873–877: Ex. 3 (group case).
- Used on: motivation.
[Sak-IS] Y. Sakellaridis, Inverse Satake transforms, in Geometric Aspects of the Trace Formula, Simons Symposia, Springer (2018) 321–349, doi:10.1007/978-3-319-94833-1_11; arXiv:1410.2312v2. Thm. 3.2. Used on: beyond SV.
[OWR13] F. Knop, Y. Sakellaridis (organisers), Mini-Workshop: Spherical Varieties and Automorphic Representations, Oberwolfach Report 24/2013, Oberwolfach Rep. 10(2) (2013) 1445–1494, doi:10.4171/OWR/2013/24 (Crossref issue date 2014; open copy at the DOI).
- Organisers' introduction, pp. 1445–1447; talks of Gan (p. 1451), Prasad (p. 1457), Offen (p. 1474), Brumley (p. 1483); Y. Sakellaridis, Satake transform and the L-function of a spherical variety, pp. 1477–1481: Problem 1, Lemma 2, Thm. 1, diagrams (1), (2), issues (1)–(4).
- Used on: Oberwolfach 2013, history.
Hecke algebras and representations
[Satake 1963] I. Satake, Theory of spherical functions on reductive algebraic groups over p-adic fields, Publ. Math. IHÉS 18 (1963) 5–69, doi:10.1007/BF02684781. The Satake isomorphism. Used on: Satake.
[Macdonald 1971] I. G. Macdonald, Spherical Functions on a Group of p-adic Type, Publ. Ramanujan Inst. 2, Madras (1971). Used on: Satake, history.
[Cartier] P. Cartier, Representations of p-adic groups: a survey, Proc. Sympos. Pure Math. 33, Part 1 (1979) 111–155, doi:10.1090/pspum/033.1/546593. §1.3–1.4 (Hecke algebras), formula (22), §4 (Satake). Used on: functions, Satake, Iwahori.
[Iwahori–Matsumoto] N. Iwahori, H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Publ. Math. IHÉS 25 (1965) 5–48, doi:10.1007/BF02684396. Prop. 2.4, Thm. 2.5, Prop. 3.2 (as located in the Iwahori workspace). Used on: Iwahori.
[Borel 1976] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976) 233–259, doi:10.1007/BF01390139. 3.5, 4.9 (∗), Lemma 4.7, Thm. 4.10, Cor. 4.11, 5.7. Used on: Iwahori criterion, group case, symmetric varieties.
[HKP] T. Haines, R. Kottwitz, A. Prasad, Iwahori–Hecke algebras, J. Ramanujan Math. Soc. 25 (2010) (journal pages unverified); arXiv:math/0309168v3. §1.2–1.3, §2.3 (centre), §4.1 (\(\eK\)), §4.6 (Satake and Bernstein). Used on: Iwahori criterion, group case.
[Casselman 1980] W. Casselman, The unramified principal series of p-adic groups I. The spherical function, Compos. Math. 40(3) (1980) 387–406 (Numdam). Props. 2.6–2.7. Used on: history, symmetric varieties.
[Godement–Jacquet] R. Godement, H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Math. 260, Springer (1972), doi:10.1007/BFb0070263. §3. Used on: 𝔸ⁿ, motivation, Iwahori status.
[Tate 1950] J. Tate, Fourier analysis in number fields and Hecke's zeta-functions, thesis (1950), printed in Cassels–Fröhlich, Algebraic Number Theory (1967) 305–347 (pages unverified). Used on: 𝔸ⁿ, motivation.
[II] A. Ichino, T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross–Prasad conjecture, Geom. Funct. Anal. 19(5) (2010) 1378–1425, doi:10.1007/s00039-009-0040-4. The conjectural factorization of \(|\Per_H|^2\) for \(\SO_n\subset\SO_{n+1}\times\SO_n\), used as recalled in [SV, §17.2]; conjecture number unverified. Used on: motivation.
[Ngô] B. C. Ngô, On a certain sum of automorphic L-functions, in Automorphic Forms and Related Geometry, Contemp. Math. 614, Amer. Math. Soc. (2014) 337–343, doi:10.1090/conm/614/12270. Cited only through [OWR13, p. 1480, issue (3)]. Used on: Oberwolfach 2013.
[Offen–Sayag] O. Offen, E. Sayag, On unitary representations of GL2n distinguished by the symplectic group, J. Number Theory 125 (2007) 344–355, doi:10.1016/j.jnt.2006.10.018. Thm. 1, Prop. 2. Also, as cited on the Iwahori workspace's pages: Offen–Sayag, Int. Math. Res. Not. (2008) rnm136, Thms. 7–8, Cor. 2; Offen–Sayag, J. Funct. Anal. 254 (2008), Thm. 1, Cor. 1 (titles and pages unverified). Used on: symmetric varieties, Iwahori literature.
[Heumos–Rallis] M. J. Heumos, S. Rallis, Pacific J. Math. 146 (1990) (title and pages unverified). Thms. 2.4.2, 11.1, 11.5; its Thms. 3.1 and 3.2.2 rest on a false proposition of Klyachko, corrected by Offen–Sayag (J. Funct. Anal. 2008). Used on: symmetric varieties.
[Tadić 1986] M. Tadić, Ann. Sci. École Norm. Sup. (4) 19 (1986) (title and pages unverified). Thm. D, 7.4–7.5, Thm. A.8 (\(\operatorname{char}F=0\)): the representations \(U(\delta,m)\). Used on: symmetric varieties.
[Shtotland 2025] R. Shtotland, Distinction of the Steinberg representation and the dual group of a symmetric space, arXiv:2501.01701v2. Thm. 1.5, Rem. 1.6, Prop. 2.7 (numbering checked on the Iwahori workspace's F1). Used on: symmetric varieties, Iwahori literature.
[Shtotland 2024] R. Shtotland, Iwahori Matsumoto presentation for modules of Iwahori fixed functions on symmetric spaces, arXiv:2406.16070v3; Int. Math. Res. Not. (2026) rnag042 (DOI unverified). Thms. 5.3, 5.9 (finite generation), 7.13, 8.5, Props. 9.1–9.3 (numbering checked on the Iwahori workspace's F1). Used on: Iwahori criterion, Iwahori literature.
Spherical varieties and local fields
[Luna–Vust] D. Luna, T. Vust, Plongements d'espaces homogènes, Comment. Math. Helv. 58 (1983) 186–245, doi:10.1007/BF02564633. Used on: history, orbits.
[BLV] M. Brion, D. Luna, T. Vust, Espaces homogènes sphériques, Invent. Math. 84 (1986) 617–632, doi:10.1007/BF01388749. Théorème 3.4, as quoted in [Sak, §2.1]. Used on: geometry.
[Kn91] F. Knop, The Luna–Vust theory of spherical embeddings, Proc. Hyderabad Conf. on Algebraic Groups (1989), Manoj Prakashan, Madras (1991) 225–249. Cor. 5.3 and §5, as quoted in [Sak, §2.1] and [SV, §2.1]. Used on: torus.
[GN10] D. Gaitsgory, D. Nadler, Spherical varieties and Langlands duality, Mosc. Math. J. 10(1) (2010) 65–137, doi:10.17323/1609-4514-2010-10-1-65-137. Used on: history.
[Delorme–Sécherre] P. Delorme, V. Sécherre, An analogue of the Cartan decomposition for p-adic symmetric spaces of split p-adic reductive groups, Pacific J. Math. 251(1) (2011) 1–21, doi:10.2140/pjm.2011.251.1. Used on: orbits.
[Benoist–Oh] Y. Benoist, H. Oh, Polar decomposition for p-adic symmetric spaces, Int. Math. Res. Not. (2007) rnm121, doi:10.1093/imrn/rnm121. Used on: orbits.
[Matsushima] Y. Matsushima, Espaces homogènes de Stein des groupes de Lie complexes, Nagoya Math. J. 16 (1960) 205–218, doi:10.1017/S0027763000007662. Used on: geometry.
[Serre] J.-P. Serre, Trees, Springer (1980), Ch. II.1; Galois Cohomology, Springer (1997), III.4.3. Used on: orbits, trees and apartments, functions.
[Deligne 1984] P. Deligne, Les corps locaux de caractéristique p, limites de corps locaux de caractéristique 0, in Représentations des groupes réductifs sur un corps local, Travaux en Cours, Hermann (1984) (pages unverified). Used on: characteristic.
[Kazhdan 1986] D. Kazhdan, Representations of groups over close local fields, J. Analyse Math. 47 (1986) 175–179, doi:10.1007/BF02792537. Used on: characteristic.
This project's pages
Results proved in this project, audited by fresh reviewers but not peer-reviewed. Each is labelled “Proved in this project” where used.
[satake-two-levels] The /solve-maths run on spherical generation at two levels: Prop. IV (symplectic families), V (functions without (A1)–(A5)), VI (the Theorem for \(E_6/F_4\)), VII (\(\CK\) needs two generators). Used on: r, characteristic.
[sakellaridis-char-p] The /solve-maths run transferring the Theorem to \(\Fqt\): Proposition VIb and its tree; Thm. 4.8, Prop. 4.9, Remark S1.4. Used on: characteristic, proof structure.
[function page] What the basic function generates, the home page of this site, which absorbed the spherical workspace's former function page: Lemma A, Propositions B–J, Theorem E.