Every source cited on these pages, with the exact results used and the pages that use them. Entries were checked on 2026-10-11 as stated in each entry.
Primary sources Proved (cited)
[IM65] Nagayoshi Iwahori and Hideya Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of $\mathfrak p$-adic Chevalley groups. Publ. Math. IHÉS 25 (1965), pp. 5–48. DOI 10.1007/BF02684396 (Crossref record checked 2026-10-11); open copy at Numdam, whose scan carries the running page numbers 237–279 that this site cites.
Results used (read in the Numdam scan on 2026-10-11): §1.1–1.4 (pp. 239–243), Lemma 1.5, Cor 1.6, Lemma 1.7, Cor 1.8 (pp. 244–245), Prop 1.10, Cor 1.11 (p. 245), Prop 1.15, Cor 1.16, §1.7 (p. 247), Prop 1.18, Cor 1.19 (p. 248), Prop 1.28–1.29 (p. 255); §2.1–2.2 (pp. 259–261), Prop 2.1 (p. 261), Thm 2.5, Prop 2.6 (p. 263), Cor 2.7 (p. 264), Prop 2.8, Cor 2.9 (p. 265), Prop 2.10 (p. 266), Prop 2.13, Lemma 2.14 (p. 267), Lemma 2.15, Thm 2.16, Cor 2.17–2.19 (p. 268), Prop 2.20 (p. 269); §3.1, Prop 3.1 (p. 275), the definition of the Hecke ring and of $\mathrm{ind}$, Prop 3.2, Thm 3.3 (p. 276), Prop 3.4, Thm 3.5 (p. 277), Cor 3.6 (p. 278), Prop 3.7, Prop 3.8 and §3.2 (p. 279).
Used on: the Concept, its source page, Groups, Hecke algebras, SL₂ I, II, III, History.
[HKP10] Thomas J. Haines, Robert E. Kottwitz and Amritanshu Prasad, Iwahori–Hecke algebras. J. Ramanujan Math. Soc. 25 (2010), no. 2, pp. 113–145; arXiv:math/0309168 [v3, 16 Apr 2009, the version read]. The journal reference is the one given on the arXiv abstract page (checked 2026-10-11); the journal's own record was not consulted, and the numbering cited is that of arXiv v3.
Results used (arXiv v3 pages): p. 1 (notation: $K$, $I$, $\pi^\mu$), §1.1 (p. 1), §1.2 and footnote 1 (p. 2), §1.3 (p. 2), Lemma 1.6.1 and footnote 2 (p. 3), §1.7, Lemma 1.7.1, Rmk 1.7.2, §1.8 (p. 4), §1.12 (pp. 7–8), Lemma 1.13.1, (1.14.1) (p. 8), (1.15.2) (p. 9), Lemma 2.1.1 (p. 9), Lemma 2.3.1 (p. 10), §4.1–4.3 (pp. 11–12), (4.6.1) (p. 12), proof of Thm 5.6.1 (p. 15), §7.1 and Lemma 7.2.1 (p. 19), the display of the Iwahori–Matsumoto relations in §7.2 (p. 20).
Used on: its source page, the Concept, Groups, Hecke algebras, History.
Sources
Sources for history and context Context
[Iwa64] Nagayoshi Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field. J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), no. 2, pp. 215–236. Metadata as recorded for the sibling site explainations/invariants-counit on 2026-10-07 (JaLC DOI 10.15083/00039900, Zbl 0135.07101); not re-resolved today.
Result used: Thm 4.1 (the defining relations of the finite Hecke ring), unverified cited through [IM65, p. 239], which refers to “[10, Th. 4.1]” for the finite case; the original was not opened. Used on: History.
[Bor76] Armand Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup. Invent. Math. 35 (1976), no. 1, pp. 233–259. DOI 10.1007/BF01390139 (Crossref record checked 2026-10-11).
Results used: Thm 4.10 (p. 249) and Cor 4.11 (p. 250), the equivalence between representations generated by Iwahori-fixed vectors and modules over the Iwahori–Hecke algebra; numbers as read for the sibling site on 2026-10-07 in the GDZ copy, not re-opened today. Used only as context. Used on: History, the Concept §6, SL₂ II §5.
[Lus89] George Lusztig, Affine Hecke algebras and their graded version. J. Amer. Math. Soc. 2 (1989), no. 3, pp. 599–635. DOI 10.2307/1990945 (Crossref record checked 2026-10-11).
Use: cited as the printed source of Bernstein's presentation, without a result number; the statements used on this site are taken from [HKP10], which refers to this paper for the elements $\Theta_\lambda$ (its Remark 1.7.2). Used on: History.
[BT72] François Bruhat and Jacques Tits, Groupes réductifs sur un corps local: I. Données radicielles valuées. Publ. Math. IHÉS 41 (1972), pp. 5–251. DOI 10.1007/BF02715544 (Crossref record checked 2026-10-11).
Use: cited for the existence of buildings and Iwahori subgroups of general reductive groups, without a result number; no statement on this site depends on it. Used on: History, the Concept §8.
[Ser80] Jean-Pierre Serre, Trees, translated by John Stillwell. Springer, 1980. DOI 10.1007/978-3-642-61856-7 (Crossref record checked 2026-10-11: title, author, publisher, year).
Result used: Ch. II, §1.1 “The tree” and its Theorem 1 (the graph of lattice classes is a tree); location as read for the sibling site on 2026-10-07, not re-opened today. The local count of $q+1$ neighbours is reproved on this site. Used on: SL₂ I §2, History.