# Literature report 2 — Iwahori–Whittaker sheaves (Setting A) and symplectic periods / Iwahori-fixed vectors (Setting B)

SC = /private/tmp/claude-501/-Users-miltonlin-Projects-RamifiedRelativeLanglands/71142ca6-8ef2-43c4-985e-911cbbfea870/scratchpad
All new downloads are under SC/dl2/<name>/ (one directory per paper; text layers <name>.txt made with `pdftotext -layout`). Page images used for transcription are in SC/shots2/. Scripts in SC/scripts2/.

Three arXiv ids supplied in the task were wrong and were corrected via the arXiv API:
- "1806.08116" is a glass-physics paper; BGMRR is **arXiv:1806.06527**.
- "math/0601112" is Vershynin (probability); Offen–Sayag JNT is **arXiv:0806.4031**.
- "0706.1101" is Remling (Jacobi matrices); Offen–Sayag IMRN is **arXiv:0710.3492**.

Page conventions: "PDF p." = 1-based index in the file; "printed" = number printed on the page. For arXiv files PDF p. = printed p. (journal pagination differs and is given in the metadata). Scans: HR printed = PDF + 245 (PDF p.1 is the msp cover; PDF pp.35–37 are issue back matter); Casselman printed = PDF + 385; Tadić printed = PDF + 333; Laumon printed = PDF + 129; Borel printed = PDF + 231 (image-only, see SC/dl/borel/png/).

## Verified metadata (arXiv abstract pages fetched to SC/dl2/meta/abs_*.html; journal records from Crossref API, SC/dl2/meta/cr_*.json; zbMATH Open for IMRN)

| key | file | authors, title, venue | arXiv / DOI |
|---|---|---|---|
| AB | SC/dl2/ab/ab.pdf | S. Arkhipov, R. Bezrukavnikov, "Perverse sheaves on affine flags and Langlands dual group", Israel J. Math. 170 (2009), no. 1, 135–183 | arXiv:math/0201073 **v6** (11 Feb 2020; "32 pages; some corrections, epigraph added. See arXiv:1807.07614 for corrected proof of Lemma 5"); DOI 10.1007/s11856-009-0024-y |
| BGMRR | SC/dl2/bgmrr/bgmrr.pdf | R. Bezrukavnikov, D. Gaitsgory, I. Mirković, S. Riche, L. Rider, "An Iwahori–Whittaker model for the Satake category", J. Éc. polytech. Math. 6 (2019), 707–735 | arXiv:1806.06527 **v3** (17 Dec 2019; "25 pages - some references added"); DOI 10.5802/jep.104 |
| BY | SC/dl2/by/by.pdf | R. Bezrukavnikov, Z. Yun, "On Koszul duality for Kac–Moody groups", Represent. Theory 17 (2013), 1–98 | arXiv:1101.1253 **v2** (22 Jul 2014; "92 pages; with appendices by Zhiwei Yun; final version"); DOI 10.1090/s1088-4165-2013-00421-1 |
| CD | SC/dl2/cd/cd.pdf | J. Campbell, G. Dhillon, "Affine Harish-Chandra bimodules and Steinberg–Whittaker localization" | arXiv:2108.02806 **v1** (5 Aug 2021; 44 pages); no journal record found |
| LY | SC/dl2/ly/ly.pdf | G. Lusztig, Z. Yun, corrigendum + "Endoscopy for Hecke categories, character sheaves and representations" | arXiv:1904.01176 v3 (22 Aug 2021) — checked only for A3; nothing affine/Kac–Moody there (only cites BY) |
| Laumon | SC/dl2/laumon/laumon.pdf | G. Laumon, "Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil", Publ. Math. IHÉS 65 (1987), 131–210 | numdam PMIHES_1987__65__131_0; DOI 10.1007/bf02698937 |
| HR | SC/dl2/hr/hr.pdf | M. J. Heumos, S. Rallis, "Symplectic-Whittaker models for Gl_n", Pacific J. Math. 146 (1990), no. 2, 247–279 (msp.org open copy: https://msp.org/pjm/1990/146-2/pjm-v146-n2-p05-s.pdf; received 9 Dec 1988, revised 24 Oct 1989, published 1 Dec 1990) | DOI 10.2140/pjm.1990.146.247 |
| OS07 | SC/dl2/os2007/os2007.pdf | O. Offen, E. Sayag, "On unitary representations of GL_{2n} distinguished by the symplectic group", J. Number Theory 125 (2007), no. 2, 344–355 | arXiv:0806.4031 **v1** (25 Jun 2008; "Two references added and minor editing of the published version"); DOI 10.1016/j.jnt.2006.10.018 |
| OS08a | SC/dl2/os2008/os2008.pdf | O. Offen, E. Sayag, "Global mixed periods and local Klyachko models for the general linear group", Int. Math. Res. Not. IMRN 2008, Article ID rnm136, 25 pp. (zbMATH Open; Crossref lists volume "2007", online 2007) | arXiv:0710.3492 **v1** (18 Oct 2007; 22 pages); DOI 10.1093/imrn/rnm136 |
| OS08b | SC/dl2/os2008b/os2008b.pdf | O. Offen, E. Sayag, "Uniqueness and disjointness of Klyachko models", J. Funct. Anal. 254 (2008), no. 11, 2846–2865 | arXiv:0711.2884 **v1** (19 Nov 2007); DOI 10.1016/j.jfa.2008.01.004 |
| Casselman | SC/dl2/casselman/casselman.pdf | W. Casselman, "The unramified principal series of p-adic groups. I. The spherical function", Compositio Math. 40 (1980), no. 3, 387–406 | numdam CM_1980__40_3_387_0 (no Crossref DOI found) |
| Tadić | SC/dl2/tadic/tadic.pdf | M. Tadić, "Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case)", Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 335–382 | numdam ASENS_1986_4_19_3_335_0; DOI 10.24033/asens.1510 |
| Borel | SC/dl/borel/borel.pdf (already present) | 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 | GDZ scan, image-only |

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# Setting A

## A1. Arkhipov–Bezrukavnikov (AB), arXiv:math/0201073v6

Conventions (PDF p.2–3, printed 2–3; §1.1–1.2): k = F̄_p (fixed in §1.2), G split reductive over F = k((t)), I an Iwahori, Fℓ = G_F/I, W the (extended) affine Weyl group, W_f the finite Weyl group, and (PDF p.3, §1.1.1) "let W_f ⊂ W be the finite Weyl group, and ^fW ⊂ W, ^fW^f ⊂ W be the set of minimal length representatives of, respectively, left and two-sided cosets of W_f in W." Standard/costandard objects are j_{w!} = j_{w!}(Q̄_ℓ[ℓ(w)]), j_{w∗} (PDF p.6, §1.5); L_w the IC sheaves; P_I = I-equivariant perverse sheaves, D_I its derived category; ^fP_I = P_I/⟨L_w | w ∉ ^fW⟩ (Serre quotient, §1.5, PDF p.6).

### A1(a) Definition of the Iwahori–Whittaker category — §1.6, PDF p.6 (printed 6)
Verbatim (text layer): "1.6. Iwahori-Whittaker category. Let B = T·N, B− = T·N− be opposite Borel subgroups, and assume that I, I− ⊂ G_O, I ⊃ B_O, I− ⊃ B−_O for Iwahori group schemes I, I−. Let I−_u ⊂ I− be the pro-unipotent radical. Let also N−_F ⊂ G_F be the group ind-scheme, N−_F(k) = N−(F). For a simple root α let u_α : N−_F → G_a be the corresponding homomorphism. We define Ψ : N−_F → G_a by Ψ(n) = Res(dt/t Σ u_α(n)). We also define ψ_I : I−_u → G_a by ψ_I(g−g^{≥0}) = Ψ(g−) for g− ∈ I−_u ∩ N−_F, g^{≥0} ∈ I−_u ∩ B_F. Let D_IW, P_IW be, respectively, the I−_u, ψ-equivariant derived category of l-adic sheaves on Fℓ, and the subcategory of perverse sheaves therein. Since the group scheme I−_u is pro-unipotent, it follows that the forgetful functor D_IW to D(Fℓ) is a full embedding, thus D_IW ⊂ D(Fℓ), P_IW ⊂ P(Fℓ) are full subcategories."
Anchors: first "1.6. Iwahori-Whittaker category. Let B = T ·N," — last "PIW ⊂P(Fℓ) are full subcategories."
Note: the character is Res(dt/t · Σ_α u_α) over the **finite** simple roots α of N− only (nondegenerate on the finite simple root subgroups, trivial on the affine one); Lemma 1 (PDF p.7): "Lemma 1. We have a natural equivalence D_IW ≅ D^b(P_IW)." (proof: "parallel to that of Lemma 4.4.6, and Corollary 3.3.2 in [BGS]").

### A1(b) Orbits and ψ-relevance — PDF p.7 (printed 7), paragraph before Lemma 2 and Lemma 2
Verbatim: "We have an injection W ֒→ Fℓ, w ↦ wI; for each one of the (ind) group schemes N−_F, I−_u, I the image of this map is a set of representatives for the orbits of the group on Fℓ. For w ∈ W let Fℓ^w (respectively, Σ^w) be the corresponding orbit of I−_u (respectively, N−_F), and i_w : Fℓ^w ֒→ Fℓ, ι_w : Σ^w ֒→ Fℓ be the embeddings. The proof of the next Lemma is left to the reader.
Lemma 2. a) For w ∈ W the following are equivalent i) w ∈ ^fW. ii) Stab_{I−_u}(wI) ⊂ Ker(ψ_I). iii) Stab_{N−_F}(wI) ⊂ Ker(Ψ). b) For w satisfying the equivalent conditions in (a) the I−-orbit Fℓ^w is contained in one N−_F-orbit."
Anchors: first "Lemma 2. a) For w ∈W the following are equivalent" — last "Fℓw is contained in one N− F-orbit." (text layer splits the sub/superscripts: "...is contained in one" / "N−" / "F-orbit.").
Proof: none ("left to the reader", same page).
Applicability: this is exactly the statement that the I−_u-orbits on Fℓ are indexed by W and that ψ-equivariant objects can live only on orbits with w ∈ ^fW (minimal length representatives of W_f\W). AB work with I−_u (pro-unipotent radical of the *opposite* Iwahori) acting on Fℓ = G_F/I, over k = F̄_p with ℓ-adic sheaves and the Artin–Schreier sheaf; transfer to C and D-modules/constructible sheaves replaces AS by the exponential sheaf and is routine but not written in AB.

### A1(c) Whittaker standard objects and the averaging functor Av_Ψ — PDF p.7 (printed 7)
Verbatim: "If w ∈ ^fW then there exist unique maps ψ_w : Fℓ^w → G_a, Ψ_w : Σ^w → G_a defined by ψ_w(g·wI) = ψ(g), Ψ_w(n·wI) = Ψ(n). Define ∆_w, ∇_w ∈ P_IW by ∆_w = i_{w!}ψ_w^∗(AS)[ℓ(w)], ∇_w = i_{w∗}ψ_w^∗(AS)[ℓ(w)] where AS in the Artin-Schreier sheaf. Define the functor Av_Ψ : D_I → D_IW, by F ↦ ∆_0 ∗ F."
Anchors: first "If w ∈fW then there exist unique maps" — last "by F 7→∆0 ∗F." (text layer: "Define the functor AvΨ : DI →DIW, by F 7→∆0 ∗F."). Note ∆_0 is written ∆_e from Lemma 4 onward (e = identity).
Module property: not a separate numbered statement; by definition Av_Ψ(F) = ∆_e ∗ F, so Av_Ψ(F ∗ G) ≅ Av_Ψ(F) ∗ G for G ∈ D_I by associativity of convolution (AB use this silently, e.g. in the proof of Lemma 4(c)).

### A1(d) Exactness and full embedding — Proposition 2, PDF p.7 (printed 7); Theorem 2
Verbatim: "Theorem 2. The functor Av_Ψ|_{P_I} induces an equivalence ^fP_I → P_IW." (anchors: "Theorem 2. The functor AvΨ|PI induces an equivalence" / "fPI g −→PIW.")
"Proposition 2. a) We have Av_Ψ : P_I ↦ P_IW; thus Av_Ψ|_{P_I} induces an exact functor P_I → P_IW. This functor factors through ^fP_I. b) The functor ^fP_I → P_IW induced by Av_Ψ is a full embedding."
Anchors: first "Proposition 2. a) We have" — last "induced by AvΨ is a full embedding."
Proof of Prop. 2: §2, PDF pp.7–10 (Lemmas 3–7; §2.0.1 "Right inverse to Av_Ψ" p.9, §2.0.2 "Proof of Proposition 2 (conclusion)" p.10; §3 starts on p.10). Proof of Theorem 2 (essential surjectivity): §4.5, PDF p.29 (printed 29), using Theorem 3 (equivalence D_{Ǧ}(Ñ) ≅ D_IW, p.7, proved §4).
Applicability: exactness of Av_ψ on perverse sheaves and the factorization through the anti-spherical quotient are exactly the properties needed; note the proof of Theorem 2 depends on the whole paper (and the arXiv comment notes a corrected proof of Lemma 5 in arXiv:1807.07614, which concerns Av_I(∆_0), not Lemma 4).

### A1(e) Av_Ψ of standard objects — Lemma 4, PDF p.8 (printed 8)
Verbatim: "Lemma 4. a) We have Av_Ψ(L_w) = 0 ⟺ w ∉ ^fW. b) We have ∆_e ≅ ∇_e. c) For w = w_f · w′, w_f ∈ W_f, w′ ∈ ^fW we have Av_Ψ(j_{w!}) ≅ ∆_{w′}, Av_Ψ(j_{w∗}) ≅ ∇_{w′}."
Anchors: first "Lemma 4. a) We have" — last "AvΨ(jw∗) ∼= ∇w′." (text layer: "AvΨ(jw!) ∼= ∆w′," / "AvΨ(jw∗) ∼= ∇w′.").
Proof: PDF p.8 (same page), ~20 lines; (a) uses that for w ∉ ^fW, L_w is I_α-equivariant for a finite simple root α ≠ α_0 so F ↦ F ∗ L_w factors through π_{α∗}, and π_{α∗}(∆_0) = 0 since ψ_I is nontrivial on stabilizers; (c) uses Lemma 3 (nonzero maps δ_e → j_{w!} with no δ_e in the cokernel) and Lemma 8(b) (invertibility of j_{w!} under convolution).
Applicability: this is the statement "Av_ψ(∆_w) is a standard object": for w ∈ ^fW, Av_Ψ(j_{w!}) ≅ ∆_w; for general w = w_f w′ one gets ∆_{w′}. In particular Av_Ψ kills all L_w with w ∉ ^fW.

## A2. Bezrukavnikov–Gaitsgory–Mirković–Riche–Rider (BGMRR), arXiv:1806.06527v3

Conventions (§2.1–2.2, PDF pp.5–6): F algebraically closed of characteristic p > 0; G connected reductive over F with B− ⊃ T, B+ the opposite Borel, U+ its unipotent radical; X∨ = X_∗(T), X∨_+ dominant and X∨_++ strictly dominant cocharacters, ∆_s simple roots (of B+); "We will assume that there exists ς ∈ X∨ such that ⟨ς, α⟩ = 1 for any α ∈ ∆_s; then we have X∨_++ = X∨_+ + ς." (PDF p.5, with footnote 3: holds e.g. for G semisimple of adjoint type). Gr = G_K/G_O, Fl = G_K/I; I+ = Iwahori associated with B+, I+_u its pro-unipotent radical (PDF p.10). Coefficients k = any of K, O, L (ℓ-modular triple).

### A2(a) Definition of the Iwahori–Whittaker category of Gr — §3.2, PDF p.10 (printed 10)
Verbatim: "3.2. Category of Iwahori–Whittaker perverse sheaves. We now denote by I+ ⊂ G_O the Iwahori subgroup associated with B+. We also denote by I+_u the pro-unipotent radical of I+, i.e. the inverse image of U+ under the map I+ → B+. We assume that there exists a primitive p-th root of unity in k, and fix one. This choice determines a character ψ of the prime subfield of F (with values in k×), and we denote by L^k_ψ the corresponding Artin–Schreier local system on G_a. (…) We also consider the "generic" character χ : U+ → G_a defined as the composition U+ ↠ U+/[U+, U+] ←(∏_α u_α, ≅)− ∏_{α∈∆_s} G_a −(+)→ G_a, and denote by χ_{I+} its composition with the projection I+_u ↠ U+. We can then define the "Iwahori–Whittaker" derived category D^b_IW(Gr, k) as the (I+_u, χ^∗_{I+}(L^k_ψ))-equivariant constructible derived category of k-sheaves on Gr (see e.g. [AR1, Appendix A] for a review of the construction of this category). This category admits a perverse t-structure, whose heart will be denoted Perv_IW(Gr, k), and moreover the "realization functor" D^b Perv_IW(Gr, k) → D^b_IW(Gr, k) is an equivalence of triangulated categories."
Anchors: first "3.2. Category of Iwahori–Whittaker perverse sheaves. We now denote by" — last (of the definition paragraph) "is an equivalence of triangulated categories."
Also on p.10: "For λ ∈ X∨ we set X_λ := I+ · L_λ. Then again we have Gr = ⊔_{λ∈X∨} X_λ."
Applicability: same construction as the page's D_{(I_u^-,ψ)} but on Gr (not Fl), with the Iwahori of B+ acting (their "I+" plays the role of the page's opposite Iwahori), character nondegenerate on the finite simple roots only.

### A2(b) Orbit/relevance lemma — Lemma 3.3, PDF p.11 (printed 11)
Verbatim: "Lemma 3.3. The orbit X_λ supports an (I+_u, χ^∗_{I+}(L^k_ψ))-equivariant local system iff λ ∈ X∨_++."
Anchors: first "Lemma 3.3. The orbit Xλ supports an" — last "equivariant local system iff λ ∈X∨ ++." (text layer: "...equivariant local system iff" / "λ ∈X∨" / "++.").
Proof: "Sketch of proof", same page, 6 lines (reduce via the affine bundle Gr^λ → G/P_λ to U+-orbits on G/P_λ; a free U+-orbit exists iff λ ∈ X∨_++, and then it is unique).
Then (p.11) the standard/costandard objects: "For λ ∈ X∨_++ we will denote by ∆^IW_λ(k), resp. ∇^IW_λ(k), the standard, resp. costandard, (I+_u, χ^∗_{I+}(L^k_ψ))-equivariant perverse sheaf on Gr associated with λ, i.e. the !-extension, resp. ∗-extension, to Gr of the free rank-1 (I+_u, χ^∗_{I+}(L^k_ψ))-equivariant perverse sheaf on X_λ." and (3.2): ∆^IW_ς(k) = ∇^IW_ς(k) = IC^IW_ς(k). Corollary 3.6 (PDF p.12): "Assume that k is a field. The category Perv_IW(Gr, k) is a highest weight category with weight poset (X∨_++, ⪯), standard objects {∆^IW_λ(k) : λ ∈ X∨_++}, and costandard objects {∇^IW_λ(k) : λ ∈ X∨_++}. Moreover, if char(k) = 0 then this category is semisimple."
Applicability: the Gr analogue of AB Lemma 2 (relevant orbits ↔ strictly dominant coweights ↔ minimal-length coset representatives in W_f\W/W_f shifted by ς).

### A2(c) The averaging functor and its exactness — §3.3, PDF p.12 (printed 12); Lemma 3.8
Verbatim: "3.3. Statement. We consider the functor Φ : D^b_{G_O}(Gr, k) → D^b_IW(Gr, k) defined by Φ(F) = ∆^IW_ς(k) ⋆_{G_O} F. In view of (3.2) (or, alternatively, arguing as in [BBM]), in this definition ∆^IW_ς(k) can be replaced by ∇^IW_ς(k) or IC^IW_ς(k); in particular this shows that the conjugate of Φ by Verdier duality is the similar functor using the character ψ^{−1} instead of ψ.
Lemma 3.8. The functor Φ is t-exact for the perverse t-structures."
Anchors: first "3.3. Statement. We consider the functor" — last "Lemma 3.8. The functor Φ is t-exact for the perverse t-structures." Proof of Lemma 3.8: PDF p.13, 12 lines (fields: from Lemma 2.3 [convolution of a perverse sheaf on Gr with a G_O-equivariant perverse sheaf is perverse]; general k via the ℓ-modular triple).
Also (3.6) on PDF p.15 identifies Φ with !-induction: "Φ(F) = ^!Ind^{(I+_u,χ_{I+})}_J ∘ F_ς ∘ For^{G_O}_{z^{−ς}Jz^{ς}}(F)[−⟨ς, 2ρ⟩]", and Remark 3.12(1), PDF p.16, gives the inverse Ψ := ^∗Ind^{G_O}_{z^{−ς}Jz^ς} ∘ F_ς^{−1} ∘ For^{(I+_u,χ_{I+})}_J[⟨ς,2ρ⟩], right adjoint to Φ.

### A2(d) Main theorem — Theorem 3.9, PDF p.13 (printed 13)
Verbatim: "We will denote by Φ^0 : Perv_{G_O}(Gr, k) → Perv_IW(Gr, k) the restriction of Φ to the hearts of the perverse t-structures, so that Φ^0 is an exact functor between abelian categories. The main result of this section is the following theorem, whose proof will be given in the next subsection.
Theorem 3.9. The functor Φ^0 is an equivalence of categories."
Anchors: first "Theorem 3.9. The functor Φ0 is an equivalence of categories." (whole statement, one line; unique on the page).
Proof: §3.4, PDF pp.13–16 (geometric Casselman–Shalika (3.4), Lemma 3.10 p.13–14, Proposition 3.11 p.14–15, "Proof of Theorem 3.9" pp.15–16: Φ^0(J_!(λ,k)) ≅ ∆^IW_{λ+ς}(k), Φ^0(J_∗(λ,k)) ≅ ∇^IW_{λ+ς}(k), then Ext-comparison via Remarks 3.2(2), 3.7(2)).
Applicability: Satake category ≅ Iwahori–Whittaker category on Gr, with standard objects matching (J_!(λ) ↦ ∆^IW_{λ+ς}); valid for any k among K, O, L (char p base field). The Fl-version (AB Theorem 2 / Lemma 4) is the Iwahori analogue.

## A3. "Fully nondegenerate" Whittaker category of the affine flag variety (character nondegenerate on ALL affine simple roots) ⇒ Vect (or Vect^Ω)

Result of the search: **not found** as an explicit numbered statement in any of the sources examined (BY, CD, LY, AB, BGMRR; FG/Raskin/Gaitsgory "Local Whittaker" not downloaded because their Whittaker conditions are w.r.t. N(K) or N(O)-type groups, not I_u^-). Closest explicit statements:

(i) Bezrukavnikov–Yun, Lemma 4.2.1, PDF p.24 (printed 24), proof on PDF pp.24–25. Setting (§2.1–2.2, PDF pp.12–13; §4.2, p.24): G a Kac–Moody group over k, B = UH its Borel, Fℓ = G/B; Σ the simple reflections, "For each subset Θ of Σ, let W_Θ ⊂ W be the subgroup generated by Θ. We say Θ is of finite type if W_Θ is finite." (p.12); for Θ of finite type, P_Θ = U^Θ L_Θ, U^−_Θ ⊂ L_Θ generated by U^−_s, s ∈ Θ; "[W_Θ\W] ⊂ W (resp. {W_Θ\W}) be the minimal (resp. maximal) length representatives of cosets in W_Θ\W" (p.12). §4.2 (p.24): "Let Θ ⊂ Σ be a subset of finite type. … Let χ : ∏_{s∈Θ} U^−_s → ∏_{s∈Θ} G_a → G_a be the sum of the isomorphisms U^−_s ≅ G_a. We can view the map χ as an additive character of U^−_Θ, or even of the pro-unipotent group U^Θ U^−_Θ. Fix a non-trivial additive character ψ : k → Q̄_ℓ^×. This determines an Artin-Schreier local system AS_ψ on G_a and hence the local system χ^∗AS_ψ on U^−_Θ or U^Θ U^−_Θ. We want to define the category D_Θ of complexes on Fℓ which are (U^Θ U^−_Θ, χ)-equivariant". Verbatim lemma: "4.2.1. Lemma. The subquotient categories D_{Θ,w} = D_{Θ,≤w}/D_{Θ,<w} admit t-exact equivalences D_{Θ,w} ≅ D^b(Fr) if w ∈ [W_Θ\W], 0 if w ∉ [W_Θ\W]". Anchors: first "4.2.1. Lemma. The subquotient categories DΘ,w = D" — last "w /∈[WΘ\W]" (text layer breaks the displayed cases; "w /∈[WΘ\W]" is the last line of p.24). Proof (p.25, first paragraph): "For w ∉ [W_Θ\W], we can find some simple reflection s ∈ Θ such that ℓ(sw) < ℓ(w). Then the stabilizer of the point wB/B under U^−_Θ contains U^−_s, on which χ is nontrivial. Therefore the stalk cohomology of F at wB/B is zero, hence F has to be zero along Fℓ^Θ_w since its cohomology sheaves are locally constant along Fℓ^Θ_w. … If w ∈ [W_Θ\W], then the action of U^−_Θ on Fℓ^Θ_w is free with quotient isomorphic to an affine space A^{ℓ(w)}." Applicability: for the affine flag variety (G = affine Kac–Moody group, Fℓ = Fl) this gives "only orbits indexed by minimal length representatives of W_Θ\W carry (U^Θ U^−_Θ, χ)-equivariant objects" for every Θ of finite type — but BY explicitly restrict to Θ of finite type, so Θ = Σ (all affine simple reflections, W_Σ = W_aff infinite) is NOT covered by the statement; the one-line vanishing argument in the proof (a simple reflection s with ℓ(sw) < ℓ(w) and χ nontrivial on U^−_s ⊂ Stab(wB)) applies verbatim to any w not of minimal length in W_aff\W̃, i.e. to any w ∉ Ω, but this is an inference, not a statement in BY.

(ii) Campbell–Dhillon (CD), §5.1.2, PDF p.23 (printed 23), finite-dimensional G only: "Let ψ be a generic additive character of N−, cf. Section 2.4.5, and consider the associated category of nondegenerate Whittaker D-modules (5.2) D(G/N−, ψ) := D(G)^{N−,ψ}. This is canonically a D(G)-module via left convolution. Note that, if we write δ_e for the delta D-module at the identity of G, (5.2) is generated as a D(G)-module by δ_ψ := Av^{N−,ψ}_∗ δ_e. Passing to B invariants, it is standard that (5.3) D(G/N−, ψ)^B ≃ Vect, with canonical generator Av^B_∗ δ_ψ." Anchors: first "5.1.2. Fix a Borel B− in general position" — last "with canonical generator AvB ∗ δψ." Context: §5.1.1 "Let G be a connected reductive group with a Borel subgroup B" (although §2.2.1, PDF p.9, lets G be either a reductive group or the loop group of an almost simple simply connected group with B an Iwahori, §5 is explicitly finite-dimensional; the affine Steinberg–Whittaker categories of §1.4/§7 use characters ψ of I̊ attached to a proper parabolic W_P ⊊ W_aff, never the fully nondegenerate one). Stated without proof ("it is standard"). CD §2.4.5 (PDF p.13) also defines the exponential D-module exp(z) on G_a as a character sheaf and "generic" = all Dynkin-coordinates nonzero.

(iii) AB Lemma 2 / Lemma 4(a) and BGMRR Lemma 3.3 (above) are the finite-simple-roots versions.

## A4. Vanishing of H_c^*(A^1, Artin–Schreier / exponential sheaf)

Laumon, Publ. IHÉS 65 (1987), (1.1.3.4), PDF p.10, printed 139, inside "Exemple (1.1.3) (Deligne, [SGA 4½] [Sommes trig.])" (starts PDF p.9, printed 138). Setting: k = F_q (§1.1 "Dans tout ce numéro … k = F_q"), J a connected commutative algebraic group of finite type over F_q, Lang isogeny 1 → J(F_q) → J −L→ J → 1 with L(x) = x^{−1}·Frob_q(x), χ : J(F_q) → Q̄_ℓ^× a character, L_χ the rank-1 Q̄_ℓ-sheaf obtained by pushing the Lang extension by χ^{−1} (p.139). Transcribed from the page image (OCR of the text layer is garbled): "De plus, L_χ a les propriétés suivantes : (1.1.3.3) pour tout entier n ≥ 1, on a t_{L_χ|J⊗F_qF_{q^n}} = χ_n := χ ∘ N_{F_{q^n}/F_q}  (1.1.3.4) si χ est non trivial, on a RΓ_c(J ⊗_{F_q} k̄, L_χ) = RΓ(J ⊗_{F_q} k̄, L_χ) = 0."
Text-layer anchors (OCR): first "(1.1.3.4) si ^ est non trivial, on a" — last "Rr,(j®^, ^) = Rr(j®^, js?,) = o." (unique on the page).
Proof: none given in Laumon (listed as a property, with the attribution to Deligne, SGA 4½ "Sommes trigonométriques"); Remarque (1.1.3.5) records the function-level counterpart Σ_{x∈J(F_{q^n})} χ_n(x) = 0.
Specialization: PDF p.11, printed 140: "Dans le cas particulier J = G_{a,F_q}, l'isogénie de Lang est aussi appelée le revêtement d'Artin-Schreier et le Q̄_ℓ-faisceau L_χ est aussi appelé le Q̄_ℓ-faisceau d'Artin-Schreier associé au caractère χ : F_q → Q̄_ℓ^×." So for J = G_a = A^1 and χ nontrivial: RΓ_c(A^1_{k̄}, L_ψ) = RΓ(A^1_{k̄}, L_ψ) = 0. §1.2.1 (PDF p.11) uses it for the Fourier–Deligne transform; Théorème (1.2.1.2) gives the function–sheaf dictionary.
D-module (exp) version over C: not found in an open-access source in this search (Katz's book and Kiehl–Weissauer are not open access); CD §2.4.5 defines exp(z) as a character sheaf but does not state the vanishing.

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# Setting B

## B1. Heumos–Rallis, Pacific J. Math. 146 (1990) — SC/dl2/hr/hr.pdf (scan with OCR text layer; anchors approximate)

Conventions (§1, PDF pp.5–6, printed 250–251): F p-adic (char 0), Gl_n = GL_n(F); J_n the symplectic form, Sp_n ⊂ Gl_{2n}; U_n upper unipotent, ψ_n nondegenerate character; for k ≤ [n/2] "define the set of models for Gl_n to be the representations (1.1) M_{n,k} = Ind^{Gl_n}_{M_k} ψ ⊗ 1" (M_k ⊃ U_{n−2k} × Sp_k); M_0 = Whittaker model; "If π is a representation, we denote by ⟨π⟩ (resp. L(π)) the unique irreducible submodule (resp. quotient module) of π, when it exists."

### B1(a) Multiplicity one for symplectic models — Theorem 2.4.2, PDF p.9, printed 254
Verbatim (image): "THEOREM 2.4.2. Let π be an irreducible, admissible representation of Gl_{2n}. Then dim Hom_{Sp_n}(π, 1) ≤ 1."
Text-layer anchors: first "THEOREM 2.4.2. Let π be an irreducible, admissible representation" — last "of Gl2n - Then dimHom S p(π, 1) < 1."
Proof: same page, 2 lines: "This a symplectic restatement of Theorem 4 and its corollary in [Ge-Ka,1]. In light of Theorem 2.4.3 [sic; the involution statement Theorem 2.4.1 and Prop. 2.3.1], their proof applies mutatis mutandis." (Gelfand–Kazhdan method; Theorem 2.4.1 on pp.253–254, Prop. 2.3.1 pp.252–253.) OS08a (PDF p.5) cite this as "[HR90, Theorem 2.4.2]" for dim Hom(π, M_{0,n}) ≤ 1.
Applicability: exactly "dim Hom_H(π, 1) ≤ 1" for G = GL_{2n}(F), H = Sp_{2n}(F), π irreducible admissible.

### B1(b) Generic representations are not symplectically distinguished — Theorem 3.1 (PDF p.9, printed 254) and Theorem 3.2.2 (PDF p.11, printed 256)
Theorem 3.1 verbatim: "THEOREM 3.1. Let π be an irreducible, unitary representation of Gl_n. Let s_1, s_2 be distinct integers, 0 ≤ s_1, s_2 ≤ [n/2]. Then Hom(π, M_{s_i}) is nonzero for at most one i." Anchors: "THEOREM 3.1. Let π be an irreducible, unitary representation" / "is nonzero for at most one i." Proof pp.254–255 (orbit analysis of M_2\Gl_n under M_1, citing [Kl], Proposition 1.3 for the vanishing on each orbit).
Theorem 3.2.2 verbatim: "THEOREM 3.2.2. An irreducible, admissible representation cannot have both a Whittaker model and a symplectic model." Anchors: "THEOREM 3.2.2. An irreducible, admissible representation cannot" / "have both a Whittaker model and a symplectic model". Proof: 4 lines on p.256 ("If we have π → Ind^{Gl_{2n}}_{U_{2n}} ψ_{2n}, we obtain π′ → Ind ψ_{2n}. As in §3.1, dualizing gives ind ψ_{2n} → π (cf. [Ge-Ka,1] §3). Thus if π has a symplectic model, we obtain the composite ind^{Gl_{2n}}_{U_{2n}} ψ_{2n} → Ind^{Gl_{2n}}_{Sp_n} 1."), i.e. it reduces to the orbit argument of §3.1 (preceded by Prop. 3.2.1, p.255: contragredients preserve Whittaker/symplectic models).
**Caveat (verified in OS08a, PDF p.5, printed 5):** "Another result claimed in [HR90, Theorem 3.1] is disjointness of Klyachko models for irreducible, unitary representations. Unfortunately, the proof is based on [Kl84, Proposition 1.3] which is false. … We will obtain the disjointness and uniquness of Klyachko models over a p-adic field in an upcoming paper." That paper is OS08b (JFA), Theorem 1 (PDF p.2, printed 2): "Theorem 1. Let F be a non-archimedean local field and let π be an irreducible representation of GL_n(F) then (3) dim_C(Hom_{GL_n(F)}(π, ⊕_{k=0}^{[n/2]} M_{n−2k,2k})) ≤ 1." (anchors "Theorem 1. Let F be a non-archimedean local field and let π be an irreducible" / "Mn−2k,2k)) ≤1."; proof §§3–4, PDF pp.4–14, via invariant distributions and [GK75]/[BZ76, Thm 6.9]). This gives, for every irreducible (not nec. unitary) π of GL_{2n}(F): Whittaker ⇒ not Sp_{2n}-distinguished, and dim Hom_{Sp_{2n}}(π,1) ≤ 1. So cite OS08b Theorem 1 for "generic ⇒ not distinguished"; HR Theorem 3.2.2 is the original claim with an incomplete proof. (OS07 p.4 still cite "[HR90, Theorem 3.1]" for this fact.)

### B1(c) Tadić's classification as used — Theorem 6.1, PDF p.15, printed 260
Verbatim: "THEOREM 6.1 (Tadic). Let B = {u(δ, n), Π(u(δ, n), a) | δ ∈ D_0, 0 < a < 1/2}. (i) If σ_1, …, σ_r ∈ B, then σ_1 × ⋯ × σ_r is irreducible and unitary. (ii) If π is an irreducible unitarizable representation, then there exist τ_1, …, τ_s ∈ B, unique up to permutation, such that π = τ_1 × ⋯ × τ_s." with (p.260) "For a positive integer n and δ ∈ D_0, set u(δ, n) = L(ν^p δ, ν^{p−1} δ, …, ν^{−p} δ), where p = (n − 1)/2" and Π(σ, a) = ν^a σ × ν^{−a} σ^+.

### B1(d) GL_4: catalog and which representations have a symplectic model — §10 (PDF pp.20–21, printed 265–266), Theorem 11.1 (PDF pp.21–22, printed 266–267), Theorem 11.5 (PDF pp.22–23, printed 267–268)
§10 catalog (transcribed from the page image, printed 266; δ_n ∈ D_0(n) square integrable, χ_n unitary characters of Gl_n, α ∈ (0, 1/2)): B′_4: (i) u(δ_4, 1) = δ_4; (ii) u(δ_2, 2) = L(ν^{1/2}δ_2 × ν^{−1/2}δ_2); (iii) u(δ_1, 4) = L(ν^{3/2}χ_1 × ν^{1/2}χ_1 × ν^{−1/2}χ_1 × ν^{−3/2}χ_1) ("These representations are all characters"). Complementary series off P_{2,2}: (iv-1) ν^α δ_2 × ν^{−α} δ_2; (iv-2) ν^α χ_2 × ν^{−α} χ_2. Off P_{1,3}: (v-1) χ_1 × δ_3; (v-2) χ_1 × χ_3. Off P_{1,1,2}: (vi-1) χ′_1 × χ″_1 × δ_2; (vi-2) χ′_1 × χ″_1 × χ_2; (vi-3) δ_2 × ν^α χ_1 × ν^{−α} χ_1; (vi-4) ν^α χ_1 × ν^{−α} χ_1 × χ_2. Off the Borel: (vii-1) χ′_1 × χ″_1 × ν^α χ‴_1 × ν^{−α} χ‴_1; (vii-2) χ′_1 × χ″_1 × χ‴_1 × χ⁗_1; (vii-3) ν^{α_1}χ′_1 × ν^{−α_1}χ′_1 × ν^{α_2}χ″_1 × ν^{−α_2}χ″_1. Off P_{2,2}: (viii-1) δ′_2 × δ″_2; (viii-2) δ_2 × χ_2; (viii-3) χ′_2 × χ″_2.
Theorem 11.1 verbatim (printed 266–267): "THEOREM 11.1. Let δ be an (arbitrary) irreducible admissible representation of Gl_n. (a) The representation ν^{1/2}δ × ν^{−1/2}δ admits a nontrivial Sp_n invariant functional. (b) If δ is further assumed to be square integrable, then the functional is supported on the unique irreducible quotient L(ν^{1/2}δ × ν^{−1/2}δ)." Anchors: "THEOREM 11.1. Let δ be an {arbitrary) irreducible admissible rep-" / "L{vχl2δ x v~χl2δ)." (OCR). Proof: §11.3.1, PDF pp.25–27 (printed 270–272; 11.3.1.1 and 11.3.1.2; (b) uses that the unique irreducible subrepresentation of ν^{1/2}δ × ν^{−1/2}δ is generic and Theorem 3.2.2 — so (b) inherits the caveat of B1(b); OS07 p.4 describe this argument and say "This is a special case of [HR90, Theorem 3.1]"; OS07 Theorem 1 reproves U(δ,2) distinguished by a global method and OS08b Theorem 1 repairs the disjointness).
Theorem 11.5 verbatim (printed 267–268): "THEOREM 11.5. If π is an irreducible unitary representation of Gl_4, then π can be realized in a unique way as a submodule of exactly one of the following representations: the Whittaker M_0, the mixed model M_1 or the symplectic model M_2." Anchors: "THEOREM 11.5. If π is an irreducible unitary representation of GI4," / "or the symplectic model Jti." (OCR). Proof: §11.1–11.3, PDF pp.23–32 (printed 268–277): §11.1 (p.268): Whittaker models for cases (i), (iv-1), (v-1), (vi-1), (vi-3), (vii-1), (vii-2), (vii-3), (viii-1); "Case (iii), the unitary characters, obviously have symplectic models."; §11.2 (pp.268–270): mixed models for (v-2), (vi-2), (vi-4), (viii-2); §11.3.1 (pp.270–272): case (ii) u(δ_2, 2) has a symplectic model (Theorem 11.1(b)); §11.3.3 "Proofs of Proposition 11.4 and Theorem 11.5; cases (iv-2), (viii-3)" (PDF pp.29–32, printed 274–277): symplectic models for ν^α χ_2 × ν^{−α} χ_2 and χ′_2 × χ″_2 via Bernstein's continuation of the integral (11.3.3.2) over (Sl_2 × Sl_2)\Sp_2.
So HR's list of Sp_4-distinguished irreducible unitary representations of GL_4: (ii) u(δ_2, 2) [δ_2 ∈ D_0(2), e.g. St_2 or supercuspidal], (iii) characters u(χ_1, 4), (iv-2) ν^α χ_2 × ν^{−α} χ_2, (viii-3) χ′_2 × χ″_2. In particular **U(St_2, 2) = L(ν^{1/2}St_2 × ν^{−1/2}St_2) is case (ii) and is Sp_4-distinguished (Theorem 11.1(b))**, with the model unique (Theorem 2.4.2) and disjoint from the Whittaker model. (Uniqueness of the "exactly one" part relies on Theorem 3.1 — see the caveat; the corrected statement is OS08b Corollary 1.)

## B2. Offen–Sayag

### B2(a) OS07 (JNT 125), arXiv:0806.4031v1 — sufficiency
Conventions (PDF p.1, printed 1; PDF p.3, printed 3): F p-adic, G = GL_{2n}(F), H = H_{ε_{2n}} = Sp_{2n}(F) for ε_{2n} = ((0, w_n),(−w_n, 0)); "Definition 1. A smooth representation π of G is called H-distinguished if Hom_H(π, C) ≠ 0." (PDF p.1). §2 (PDF p.3): ν = |det|; π_1 × ⋯ × π_t normalized induction; π(τ, α) = ν^α τ × ν^{−α} τ; Λ_m = ((m−1)/2, (m−3)/2, …, (1−m)/2); "For a square integrable representation δ of GL_r(F), the representation I(δ^{⊗m}, Λ_m) has a unique irreducible quotient which we denote by U(δ, m)." (so U(δ,m) = Langlands quotient of ν^{(m−1)/2}δ × ⋯ × ν^{(1−m)/2}δ); "B_u = {U(δ, m), π(U(δ, m), α) : δ-square integrable, m ∈ N, |α| < 1/2}. A representation of the form σ_1 × ⋯ × σ_t where σ_i ∈ B_u, is irreducible and unitary. Any irreducible, unitary representation of GL_q(F) for some q has this form and is uniquely determined by the multi-set of σ_i's up to reordering. This is the classification of Tadic."
Theorem 1, PDF p.3 (printed 3), verbatim: "Theorem 1. Let π = σ_1 × ⋯ × σ_t × τ_{t+1} × ⋯ × τ_s be a unitary representation of G, such that σ_i = U(δ_i, 2m_i) ∈ B_u and τ_i = π(U(δ_i, 2m_i), α_i) ∈ B_u. Then π is H-distinguished."
Anchors: first "Theorem 1. Let π = σ1×· · ·×σt×τt+1×· · ·×τs be a unitary representa-" — last "Bu. Then π is H-distinguished." (last line of the page).
Proof: Theorem 1 is a special case of Proposition 2, PDF p.6 (printed 6): "Proposition 2. The representation (3) J = ν^{α_1}U(δ_1, 2m_1) × ⋯ × ν^{α_t}U(δ_t, 2m_t) is distinguished by H." (hypotheses, same page: δ_i irreducible square integrable of GL_{r_i}(F), α_i ∈ R, m_i positive integers — no unitarity needed). Proof of Prop. 2: §4, PDF pp.6–12 ("The rest of this work is devoted to the proof of Proposition 2"; building block U(δ,2m) via Proposition 1, PDF p.5, by a global argument; induction via Bernstein's meromorphic continuation, Lemma 2 p.8 ff.). Page 4 (printed 4) explains that the HR argument covers only m = 1.
Applicability: for G = GL_4, H = Sp_4: U(St_2, 2) (δ_1 = St_2, m_1 = 1, t = 1, s = 1) is H-distinguished by Theorem 1 (and already by HR Theorem 11.1(b)).

### B2(b) OS08a (IMRN), arXiv:0710.3492v1 — existence of Klyachko models and the explicit model
Conventions (PDF pp.3–4): G = GL_n, H_{r,2k} = {(u, X; 0, h)} with u ∈ U_r, h ∈ Sp_{2k} (n = r + 2k), ψ_r the nondegenerate character of U_r inflated to H_{r,2k}; "Definition 2. We say that a representation π of G(F) is (H, χ)−distinguished if Hom_{H(F)}(π, χ) ≠ 0." (PDF p.3); Klyachko model M_{r,2k} = Ind^{G(F)}_{H_{r,2k}(F)}(ψ_r), "M_{n,0} is the Whittaker model while when n is even M_{0,n} is a purely symplectic model" (PDF p.4).
Theorem 2, PDF p.4 (printed 4): "Theorem 2. Let π be an irreducible, unitary representation of G(F). There exists an integer k, 0 ≤ k ≤ [n/2] such that π is (H_{n−2k,2k}, ψ_{n−2k})−distinguished." Anchors: "Theorem 2. Let π be an irreducible, unitary representation of G(F )." / "distinguished." (3-line statement).
Tadić's classification as used, §3.4, PDF p.18 (printed 18): "For δ ∈ D^u and a positive integer t the representation δ[(1−t)/2] × δ[(3−t)/2] × ⋯ × δ[(t−1)/2] has a unique irreducible subrepresentation which we denote by U(δ, t). It will be convenient to allow the notation U(δ, 0) for the trivial representation of G_0 = {e}. Let B be the collection of all representations of the form U(δ, t) or U(δ, t)[α] × U(δ, t)[−α] where δ ∈ D^u, t is a positive integer and 0 < α < 1/2. Theorem 7 (Tadic). For every σ_1, …, σ_t ∈ B the representation σ_1 × ⋯ × σ_t is irreducible and unitary. Any irreducible, unitary representation of G is of this form uniquely determined (up to reordering) by the multiset {σ_1, …, σ_t}." (Here δ[α] := ν^α δ; U(δ,t) as unique irreducible subrepresentation of the increasing-exponent product = unique irreducible quotient of the decreasing one; Zelevinsky form of U(δ,t) given on pp.18–19 citing [Tad86, Theorem A.10(iii)].)
Main local theorem, Theorem 8, PDF p.20 (printed 20), §3.6: "Theorem 8. Let δ_1, …, δ_q, δ′_1, …, δ′_{q′} ∈ D^u, let m_1, …, m_q, m′_1, …, m′_{q′} be non negative integers and let α_1, …, α_q, α′_1, …, α′_{q′} ∈ R. Assume that δ_i is a representation of G_{r_i}, that δ′_i is a representation of G_{r′_i} and that n = Σ_{i=1}^q (2m_i + 1)r_i + Σ_{i=1}^{q′} 2m′_i r′_i. Let n = r + 2k where r = r_1 + ⋯ + r_q and k = m_1r_1 + ⋯ + m_qr_q + m′_1r′_1 + ⋯ + m′_{q′}r′_{q′}. The representation (13) U(δ′_1, 2m′_1)[α′_1] × ⋯ × U(δ′_{q′}, 2m′_{q′})[α′_{q′}] × U(δ_1, 2m_1 + 1)[α_1] × ⋯ × U(δ_q, 2m_q + 1)[α_q] is (H_{r,2k}, ψ_r)-distinguished."
Anchors: first "Theorem 8. Let δ1 , . . . , δq , δ1′ , . . . , δq′ ′ ∈ D u" — last "is (Hr,2k , ψr )-distinguished." Proof: same page (p.20), ~15 lines, via highest derivatives (Lemmas 3, 4) and [OS07, Proposition 2].
Then (PDF p.21, printed 21): "Using the notation of the statement of Theorem 8 for a representation π of the form (13) we define κ(π) = k. Note that by Theorem 7 every irreducible, unitary representation π of G is of the form (13) with |α_i|, |α′_i| < 1/2. In particular κ(π) is defined. … Corollary 2. Let π be an irreducible, unitary representation of G then π is (H_{n−2κ(π),2κ(π)}, ψ_{n−2κ(π)})-distinguished. This proves in particular Theorem 2."
Applicability: κ(π) = 0 ⟺ all factors have odd t (Whittaker model); κ(π) = n/2 (purely symplectic model M_{0,n}) ⟺ r = 0 ⟺ every factor is U(δ′_i, 2m′_i)[α′_i] with 2m′_i even, i.e. π = ∏ U(δ_i, 2m_i) × ∏ π(U(δ_j, 2m_j), α_j) — the OS07 family. For U(St_2, 2) of GL_4: q′ = 1, δ′_1 = St_2 (r′_1 = 2), m′_1 = 1, α′ = 0 ⇒ r = 0, k = 2, κ = 2: it is (H_{0,4}, 1) = Sp_4-distinguished, consistent with OS07 Theorem 1.

### B2(c) OS08b (JFA), arXiv:0711.2884v1 — uniqueness and disjointness, hence the classification
PDF p.2 (printed 2): Theorem 1 (quoted in B1(b)); "Corollary 1. Let F be a p-adic field and let π be an irreducible, unitary representation of GL_n(F) then dim_C(Hom_{GL_n(F)}(π, ⊕_{k=0}^{[n/2]} M_{n−2k,2k})) = 1." (uses OS08a for ≥ 1). "It follows that for an irreducible, unitary representation π of GL_n(F) there is a unique integer 0 ≤ κ(π) ≤ [n/2] such that Hom_{H_{n−2κ(π),2κ(π)}(F)}(π, ψ_{n−2κ(π),2κ(π)}) ≠ 0 … and that the space of such functionals is one dimensional. Remark 1. In [OS07, Theorem 1], when F is a p-adic field and n is even, we exhibited a family of irreducible, unitary representations of GL_n(F) that are Sp_n(F)-distinguished. We promised in [OS07] that in [OS] we will show that this family exhausts all irreducible, unitary representations that are Sp_n(F)-distinguished. Eventually, we postponed the delivery of this statement to the current paper. It is immediate from Corollary 1."
Anchors: "Corollary 1. Let F be a p-adic field and let π be an irreducible, unitary represen-" / "Mn−2k,2k)) = 1." ; "Remark 1. In [OS07, Theorem 1], when F is a p-adic field and n is even, we exhib-" / "immediate from Corollary 1."
Combined classification (OS07 Thm 1 + OS08a Thm 8/Cor. 2 + OS08b Cor. 1): an irreducible unitary π = σ_1 × ⋯ × σ_t (σ_i ∈ B_u, Tadić) of GL_{2n}(F) is Sp_{2n}(F)-distinguished iff every σ_i is of the form U(δ_i, 2m_i) or π(U(δ_i, 2m_i), α_i) (all Speh "lengths" even); then dim Hom_{Sp_{2n}}(π, 1) = 1. There is no single displayed theorem with this wording in the three papers; the wording above is assembled from the quoted statements. Note the hypothesis "F p-adic" (characteristic 0) in OS07/OS08a/Cor. 1 (global methods), while OS08b Theorem 1 holds for any non-archimedean local F.

## B3. Iwahori-fixed vectors and unramified principal series

### Casselman, Compositio 40 (1980) — Propositions 2.6 and 2.7, PDF p.11, printed 396
Conventions (§1, PDF pp.2–4, printed 387–389): G reductive p-adic (split or more generally with Iwahori B fixing a chamber), P = MN minimal parabolic, M_0 ⊂ M maximal compact, B the Iwahori subgroup (text layer: "Let B be the Iwahori subgroup fixing the chamber C"), I(χ) = Ind(χ|P, G) the unramified principal series (χ an unramified character of M), V_N the Jacquet module, H(G, B) the Iwahori Hecke algebra.
Verbatim (page image): "2.6. PROPOSITION: If (π, V) is any irreducible admissible representation of G with V^B ≠ 0, then there exists a G-embedding of V into some unramified principal series. Conversely, if V is any non-trivial G-stable subspace of an unramified principal series, then V^B ≠ 0." Proof: same page, 8 lines (Frobenius reciprocity Hom_G(V, Ind(χ|P,G)) ≅ Hom_M(V_N, χδ^{1/2}) [7: 3.2.4] and Proposition 2.4 [PDF p.10, printed 395: "If (π, V) is any admissible representation of G, then the canonical projection from V^B to V_N^{M_0} is a linear isomorphism." — Casselman attributes the injectivity proof to Borel, "see Lemma 4.7 of [1]"]).
"2.7. PROPOSITION: The G-module I(χ) is generated by I(χ)^B." Proof: same page, 6 lines (if U = I(χ)/⟨G·I(χ)^B⟩ then U^B = 0, Ũ^B = 0, Ũ ⊂ I(χ^{−1}), so Ũ = 0 by 2.6).
Text-layer anchors (OCR): "2.6. PROPOSITION: If (03C0, V) is any irreducible admissible" / "VB~ 0." ; 2.7 header is missing from the OCR layer; its proof begins "PROOF: If U is the quotient of I(X) by the G-space generated by".
Applicability: with G split over C-coefficients this is exactly "irreducible with I-fixed vectors ⇒ sub of an unramified principal series; nonzero subrepresentations of I(χ) have I-fixed vectors". The subquotient version follows from 2.7 + exactness of (−)^B (B compact open) or from Borel (*) below.

### Borel, Invent. Math. 35 (1976) — Lemma 4.7, Lemma 4.8, (*)/4.9, Theorem 4.10, Corollaries 4.11–4.12 (image-only PDF; transcribed from SC/dl/borel/png/p017–p019.png)
Conventions (Part II, §3.1, printed 242–243; §4.6, printed 247): k non-archimedean local, 𝒢 connected semi-simple, G = 𝒢(k), B an Iwahori subgroup, H(G,B) the Hecke algebra; 𝒫, 𝒫^− opposite minimal parabolics, M = 𝒫 ∩ 𝒫^−, N, N^− unipotent radicals, B = N^−_0·M_0·N_0 (Iwahori factorization, 4.6(1)), M_0 the greatest compact subgroup of G_0 ∩ M; V_N the Jacquet module.
- Lemma 4.7, PDF p.17, printed 248: "4.7. Lemma. Let (π, V) be an admissible G-module. Then the canonical projection V → V_N induces an isomorphism of V^B onto (V_N)^{M_0}." Preceded by "The following lemma is a slight strengthening of Theorem 3.3.3 in [7], valid for Iwahori subgroups:". Proof: same page, 8 lines (uses [7: 4.1.4] and invertibility of π(e_a) on V^B, (3.6)).
- Lemma 4.8 (Casselman), PDF p.17–18, printed 248–249: "4.8. Lemma (Casselman). Let E be an admissible G-module. Assume that E is generated by E^B as a G-module and that, if E′ is a non-zero G-submodule of E, then E′^B ≠ 0. Then (i) Any exact sequence of admissible G-modules 0 → E → F → F′ → 0 where F′^B = 0, splits. (ii) if E′ is a G-submodule of E, then E′ is generated by E′^B as a G-module." Proof pp.248–249; it invokes, printed 249 (PDF p.18): "(∗) Let σ be a finite dimensional representation of M trivial on M_0, and L a non-zero decomposition factor of Ind(σ|P, G). Then L^B ≠ 0." and "4.9. The assertion (∗) above is essentially proved in [7]. A similar one is also announced in [14: p. 19]. We indicate briefly how to derive it from [7]. … By [7: Thm. 3.3.3], it suffices to prove that L_N ≠ 0. Since the functor σ ↦ Ind σ is exact, … we may assume it [σ] to be irreducible. The G-module L has finite length [7: 6.3.8], so we may assume it to be irreducible. But then L embeds in an induced module (Ind(τ|P, G), where τ is a finite dimensional irreducible representation of M trivial on M_0 [7: 6.3.9] and L_N ≠ 0 by [7: 3.2.5]. Here, B need not be an Iwahori subgroup. It suffices that it be a compact open subgroup of G admitting an Iwahori factorization (4.6(1))."
  (∗) is precisely "every irreducible subquotient (decomposition factor) of an unramified principal series Ind(σ|P,G), σ unramified (trivial on M_0), has nonzero Iwahori-fixed vectors." [7] = Casselman's unpublished notes "Introduction to the theory of admissible representations of p-adic reductive groups".
- Theorem 4.10, PDF p.18, printed 249 (already located earlier; restated): "4.10. Theorem. Let (r, E) be a finite dimensional H(G, B)-module. Then the canonical morphism j: I_{B,G}(E) → P_{B,G}(E) is an isomorphism. The G-module I_{B,G}(E) is irreducible if and only if E is an irreducible H(G, B)-module. The assignment E ↦ I(E) (resp. E ↦ P(E)) is an exact functor from finite dimensional H-modules to admissible G-modules. The H-module C_c(G/B) is flat." Proof pp.249–250.
- Corollary 4.11, PDF p.19, printed 250: "Let V be an admissible G-module and assume that V is generated by G·V^B. Then the canonical morphisms I(V^B) →^μ V →^ν P(V^B) are isomorphisms. Every G-submodule of V is generated as a G-module by its B-fixed vectors." Corollary 4.12: "Let V be an admissible G-module. Let M be the G-submodule generated by V^B. Then V is the direct sum of M and of a G-submodule N such that N^B = 0."
- The sentence "irreducible admissible representations with nonzero B-fixed vectors are exactly the subquotients of unramified principal series" does NOT appear verbatim in Borel; it is the conjunction of (∗) (printed 249) with Casselman Prop. 2.6 (printed 396). Borel's introduction (printed 233–234) only states the H(G,B)-module results and credits Casselman for Lemma 4.8.

## B4. Tadić, Ann. Sci. ENS 19 (1986) — Speh representations u(δ, n)

- Definition and Theorem D, PDF pp.5–6, printed 338–339 (introduction). Transcribed from the page image: "Let D^u be the set of all square integrable classes in Irr^u. Let n be a positive integer, and let δ ∈ D^u. The representation ν^{(n−1)/2}δ × ν^{((n−1)/2)−1}δ × ⋯ × ν^{−(n−1)/2}δ has a unique irreducible quotient which is denoted by u(δ, n). THEOREM D. — Let B be the set of all u(δ, n), ν^α u(δ, n) × ν^{−α} u(δ, n) where n is a positive integer, δ ∈ D^u and 0 < α < 1/2. (i) If π_1, …, π_r ∈ B, then π_1 × ⋯ × π_r ∈ Irr^u. (ii) If σ ∈ Irr^u, then there exist π_1, …, π_r ∈ B so that σ = π_1 × ⋯ × π_r. The elements π_1, …, π_r are unique up to a permutation." (Irr^u = unitarizable classes in Irr = ⊔_n GL(n,F)^~, §1.2–1.3.) Text-layer (OCR) anchors: "has a unique irreducible quotient which is denoted by u(8, n)." / "THEOREM D. — Let B be the set of all"; the (i)–(ii) continue on printed 339 ("(i) Ifn^ . . .,7^eB, then n^x . . . x^eirr"." / "ar^ unique up to a permutation.").
- Restatement in the body: 7.4 and Theorem 7.5, PDF p.37, printed 370 (image-transcribed): "7.4. For δ ∈ D^u and n ∈ N set u(δ, n) = L(ν^{−(n−1)/2}δ, ν^{1−((n−1)/2)}δ, …, ν^{(n−1)/2}δ). We can characterize u(δ, n) as the unique irreducible quotient of ν^{(n−1)/2}δ × ν^{((n−1)/2)−1}δ × ⋯ × ν^{−(n−1)/2}δ. Clearly L(a(n, d)^{(ρ)}) = u(L(∆[d]^{(ρ)}), n) and this implies {L(a(n, d)^{(ρ)}); n, d ∈ N, ρ ∈ C^u} = {u(δ, n); δ ∈ D^u, n ∈ N}. 7.5. THEOREM. — Set B = {u(δ, n), ν^α u(δ, n) × ν^{−α} u(δ, n); δ ∈ D^u; n ∈ N, 0 < α < 1/2}. (i) If π_1, …, π_r ∈ B, then π_1 × ⋯ × π_r ∈ Irr^u. (ii) Let σ ∈ Irr^u. Then there exist π_1, …, π_r ∈ B so that σ = π_1 × ⋯ × π_r. The multiset (π_1, …, π_r) is uniquely determined by σ." Anchors (OCR): "7.4. For 8eD" and ne^J set" / "7.5. THEOREM. — Set" / "(7i i, . . ., TI,.) is uniquely determined by a." Proof: Theorem 7.5 is the Langlands-classification form (via 7.3–7.4) of Theorem 7.1 (PDF p.35–36, printed 368–369, proof by induction on m using Lemma 6.2 and [28]); §7 assumes "(Irr^u)^t ⊂ Irr^u" (printed 368: "In this section and the following one we assume (Irr^u)^t ⊂ Irr^u and thus we do not make difference between t and r").
- Irreducibility of u(δ,n): by definition (unique irreducible quotient). Unitarizability: Theorem A.8, PDF p.42, printed 375 (appendix): "In the rest of this section we suppose that the characteristic of the field F is zero. Let δ ∈ GL(m, F)^~ be a square-integrable representation and n ∈ N. Then the induced representation (ν^{(n−1)/2}δ) × (ν^{((n−1)/2)−1}δ) × ⋯ × (ν^{−(n−1)/2}δ) has a unique irreducible quotient. This quotient was denoted by u(δ, n). A.8. THEOREM. — Suppose that char F = 0. Let δ ∈ Irr be a square-integrable representation and let n ∈ N. Then u(δ, n) is a unitarizable representation." Anchors (OCR): "This quotient was denoted by u (8, n)." / "A. 8. THEOREM. — Suppose that charF=0." / "is a unitarizable representation." Proof: PDF pp.42–44, printed 375–377 (global: δ ↦ cuspidal automorphic σ of a division-algebra form via [22], residual representation of GL(mn, A) via [16]; ends "Since T is a subrepresentation of L^2(ω, GL(mn, A)), T_w is unitarizable and therefore u(δ, n) is unitarizable."). A.9 Theorem (printed 377): "Representations L(a(n, d)^{(ρ)}), n, d ∈ N, ρ ∈ C^u are unitarizable." Theorem D(i) with r = 1 then gives u(δ, n) ∈ Irr^u.
Applicability: U(St_2, 2) of GL_4 in the page's notation = Tadić's u(St_2, 2) (δ = St_2 ∈ D^u, n = 2) = L(ν^{−1/2}St_2, ν^{1/2}St_2) = unique irreducible quotient of ν^{1/2}St_2 × ν^{−1/2}St_2; irreducible unitary by Theorem D / A.8 (char 0). Note HR's u(δ,n) (B1(c)) and OS's U(δ,m) are the same objects.

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## Caveats / not found
- A3: no source found stating the Vect (or Vect^Ω) description of the fully nondegenerate (all affine simple roots) Whittaker category of Fl; BY Lemma 4.2.1 is restricted to Θ of finite type, CD (5.3) is finite-dimensional and unproved ("it is standard").
- A4: only the ℓ-adic Artin–Schreier statement (Laumon (1.1.3.4), attributed to Deligne, no proof given) was found open-access; no D-module/exp statement located.
- B1: HR Theorems 3.1 and 3.2.2 (and hence the "exactly one" part of 11.5 and the quotient-support part 11.1(b) as proved there) rely on [Kl84, Prop. 1.3], which OS08a (p.5) state is false; the corrected disjointness/uniqueness is OS08b Theorem 1 / Corollary 1. HR Theorem 2.4.2 (multiplicity one) is independent of that.
- HR, Casselman, Tadić, Laumon are scans; anchors quoted from their OCR layers are approximate (verified against page images in SC/shots2/).
- IMRN citation details (2008, Article ID rnm136, 25 pp.) come from zbMATH Open (OUP page is Cloudflare-blocked); Crossref records the DOI with volume "2007".