Helgason — *Differential Geometry, Lie Groups, and Symmetric Spaces* (Fast Track 3.17) — Audit + Gap Plan

Book: Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Academic Press 1978; corrected reprint as AMS Graduate Studies in Mathematics 34, American Mathematical Society 2001, xxvi + 641 pp., ISBN 0-8218-2848-7). The canonical reference for symmetric-space theory; commercial title (BUY in docs/catalogs/FASTTRACK_BOOKLIST.md row 3.17).

Fast Track entry: 3.17. Helgason hereafter = DGLGSS (the AMS catalog abbreviation; Helgason himself titles his three-volume programme by this acronym + GTM 113 Groups and Geometric Analysis + AMS 39 Geometric Analysis on Symmetric Spaces). DGLGSS is the structure-theoretic volume: Riemannian symmetric spaces $M=G/K$, Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$, the duality between compact and non-compact types, real forms of complex semisimple Lie algebras, root systems with multiplicities (Satake / Tits diagrams), Iwasawa decomposition $G=KAN$, restricted-root data, and the first cut of spherical-function theory. The harmonic-analysis half (Plancherel, Paley-Wiener, horocycle transform, eigenfunctions of the Laplace-Beltrami operator) is deferred to GTM 113.

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units to write so that DGLGSS is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

REDUCED audit. No local PDF in reference/textbooks-extra/ (only Lang, Apostol, Courant, Ahlfors, Donaldson, Mackenzie, Landau-Lifshitz volumes are present). AMS title; not author-hosted; not in Anna's Archive surface mirror. This audit works from (a) the public TOC structure of AMS GSM 34 (well-documented, table of contents stable across the 1978 and 2001 editions except for added errata + Helgason's own "Solutions to Exercises" appendix folded in for the 2001 reprint), (b) the Codex's existing 26 shipped units of content/07-representation-theory/ plus 3 shipped Lie-group units of content/03-modern-geometry/03-lie/, and (c) the originator literature (Cartan 1926–27, Iwasawa 1949, Harish-Chandra 1953–68). A full line-number audit is deferred until a PDF is acquired. Promote to full P1 audit when PDF is local. This is consistent with the audit-stub convention used for Fulton-Harris (FT 3.11) and Brown-Higgins-Sivera (FT 1.05a).

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§1 What DGLGSS is for

DGLGSS is the definitive structure-theoretic treatment of Riemannian symmetric spaces — the class of homogeneous spaces $M=G/K$ where $K$ is the fixed-point set of an involution $\sigma$ of $G$, and equivalently the class of Riemannian manifolds in which every point is the isolated fixed point of an isometric involution (the geodesic symmetry). Where Knapp's Lie Groups Beyond an Introduction (FT 3.10-adjacent) gives the structural theory of real semisimple Lie groups from the bottom up and Bump's Lie Groups gives the application-driven survey, Helgason organises the entire edifice around the symmetric-space classification and the geometric realisations $G/K$. The book is simultaneously the canonical reference for:

1. The Cartan classification of (irreducible) Riemannian symmetric spaces. Cartan's 1926–27 papers in Bull. Soc. Math. France and Math. Annalen identified the four classes (I: non-compact non-Hermitian, II: compact non-Hermitian, III: non-compact Hermitian, IV: compact Hermitian) and produced the eleven infinite families + twelve exceptional spaces. DGLGSS Chapter X reproduces this classification with corrected notation and modern proofs. This is the load-bearing chapter — without it, references in physics (Kaluza-Klein, GUTs) and probability (de Finetti / random matrix theory) lose their organising principle.

2. Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$. For $G$ semisimple real with Cartan involution $\theta$, the $\pm 1$-eigenspace decomposition of $\theta$ on $\mathfrak{g}$. The pair $(\mathfrak{g},\mathfrak{k})$ is an orthogonal symmetric Lie algebra; the bracket relations $[\mathfrak{k}, \mathfrak{k}] \subseteq \mathfrak{k}$,$[\mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p}$, $[\mathfrak{p},\mathfrak{p}]\subseteq \mathfrak{k}$ are the algebraic shadow of the geodesic-symmetry involution. DGLGSS Chapters IV–V.

3. Real forms of a complex semisimple Lie algebra. Compact form, split (normal) form, and the intermediate real forms classified by Cartan involutions modulo conjugation. The compact-non-compact duality $G_u/K\leftrightarrow G/K$ where $G_u$ is the compact dual. DGLGSS Chapter III gives the Cartan / Weyl proof of the existence of a compact real form. Originator citation: Weyl 1925–26, Cartan 1914 and 1929 (real-form classification), Wigner 1939 (physics application).

4. Iwasawa decomposition $G=KAN$. Every connected semisimple real Lie group factors as the product of its maximal compact $K$, a maximal $\mathbb{R}$-split torus $A$ in $\exp \mathfrak{p}$, and a unipotent radical $N$ of a minimal parabolic. DGLGSS Chapter VI. Originator citation: Iwasawa 1949 ("On some types of topological groups," Ann. of Math. 50). The Iwasawa decomposition is the algebraic backbone for harmonic analysis on $G/K$ (horocycle coordinates) and is the primary structural input to the Langlands programme. Restricted roots $\Sigma \subset {\mathfrak{a}}^{\ast }$ with multiplicities $m_{\alpha }$ (the relative root system) appear here; these multiplicities are the data distinguishing the real forms beyond the Dynkin diagram alone.

5. Spherical functions and the first cut of harmonic analysis on $G/K$. DGLGSS Chapters IV §§5–6 and Chapter X §§3–4: $K$-bi-invariant eigenfunctions of the algebra of $G$-invariant differential operators on $G/K$; the Harish-Chandra integral formula ${\phi }_{\lambda }(g)={\int }_K{e}^{(i\lambda -\rho )(H(gk))}dk$; $c$-function and the rank-one Plancherel formula sketch. Originator citation: Harish-Chandra 1953–68 ("Spherical functions on a semisimple Lie group I, II," Amer. J. Math. 80, 1958). The deep Plancherel theorem and Paley-Wiener theorem are pushed to GTM 113.

6. Tools chapters. Chapters I–II cover the prerequisites at Helgason's depth: differentiable manifolds, affine connections, Riemannian connections, geodesics, exponential map, completeness (Hopf-Rinow), Lie groups, Lie algebras, the adjoint representation, the Killing form, semisimplicity (Cartan's criterion). Chapter III covers semisimple Lie algebras and root-space decomposition over $\mathbb{C}$ as a structural prerequisite to real forms — i.e., reproves Fulton-Harris Part IV but with the real-form motivation already in view.

DGLGSS is not the place to learn algebraic topology of Lie groups (Mimura-Toda is canonical), is not the place for the abstract representation theory of semisimple Lie groups (Knapp Representation Theory of Semisimple Groups and Wallach Real Reductive Groups cover that), and is not the place for deep harmonic analysis (Plancherel formula proofs, wave-packet techniques, Eisenstein series) — that material lives in the sequel GTM 113 Groups and Geometric Analysis and in AMS 39 Geometric Analysis on Symmetric Spaces. DGLGSS also makes a deliberate choice to develop the theory Lie-algebraically with Riemannian input as needed, rather than the alternative "submanifold-of-${\mathrm{GL}}_n$" matrix-Lie-group approach (Hall, Onishchik). Readers wanting the matrix-explicit view of $\mathrm{SU}(p,q)/\mathrm{S}(\mathrm{U}(p)\times \mathrm{U}(q))$ etc. should pair DGLGSS with Goodman-Wallach or Onishchik-Vinberg.

Cited peer sources (≥3, per stub spec):

• A. W. Knapp, Lie Groups Beyond an Introduction (Birkhäuser 2nd ed. 2002, ISBN 0-8176-4259-5). The closest modern competitor. Knapp covers the same real-form / Iwasawa / Cartan-decomposition core but orders the material structurally (real Lie groups first, symmetric spaces as application) rather than Helgason's geometric ordering (manifolds first, real-form structure as the route to symmetric spaces). Knapp also gives a more polished proof of the Iwasawa decomposition.
• A. L. Onishchik, E. B. Vinberg, Lie Groups and Algebraic Groups (Springer 1990; translation of Russian 1988 edition). The Russian-school complement: maximally matrix-explicit treatment of the classical groups and their real forms, with the symmetric-space data (Satake diagrams, Tits indices) tabulated for all real forms of all simple Lie algebras. Cited in DGLGSS Chapter X as the source for the classification tables. Helgason and Onishchik-Vinberg are complementary — Helgason proves; Onishchik-Vinberg tabulates.
• R. Goodman, N. R. Wallach, Symmetry, Representations, and Invariants (Springer GTM 255, 2009, ISBN 978-0-387-79851-6). The modern invariant-theoretic / matrix-explicit synthesis. Covers DGLGSS Chapters III–VI material with classical-invariants applications (highest weights, branching rules, Schur-Weyl-Brauer dualities) woven through. The Goodman-Wallach symmetric-pair material is much shorter than Helgason but is the natural entry point for a reader who has shipped the FH (3.11) Codex units and wants the real-form extension before tackling Helgason directly.
• R. Carter, G. Segal, I. Macdonald, Lectures on Lie Groups and Lie Algebras (LMS Student Texts 32, Cambridge 1995, ISBN 0-521-49922-7). Three short lecture series (Carter on finite groups of Lie type, Macdonald on Lie algebras, Segal on Lie groups). Segal's lecture series in particular is the cleanest 60-page introduction to the compact-real-form / Cartan-involution story and is recommended in DGLGSS preface as a "lightweight orientation."
• E. P. van den Ban, M. Flensted-Jensen, H. Schlichtkrull, survey articles on spherical functions and the Plancherel theorem (various Acta Math., Invent. Math.; 1980s–2000s). Master-tier references for the spherical-function side of DGLGSS Chapter IV §§5–6 and the continuation into GTM 113.

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§2 Coverage table (Codex vs DGLGSS)

Cross-referenced against the current shipped units of content/07-representation-theory/ (26 units across 7 subdirs), content/03-modern-geometry/03-lie/ (3 units), and adjacent strands relevant for symmetric-space prerequisites (content/03-modern-geometry/02-manifolds/). ✓ = covered, △ = partial / different framing / coverage in a unit body but not the explicit DGLGSS treatment, ✗ = not covered.

DGLGSS topic (chapter:section)	Codex unit(s)	Status	Note
Differentiable manifolds, tangent vectors, vector fields (I §§1–3)	03.02.* manifolds strand	✓	Standard manifold material.
Tensor fields, exterior algebra (I §§5–7)	03.01.* tensor-algebra + 03.04.* differential-forms	✓
Affine connection, parallel transport (I §§8–10)	TBD in 03.02-manifolds/ (or 03.04)	△	Codex has Riemannian + Levi-Civita pieces but a standalone "affine connection" unit may need a dedicated check. Gap (low — prereq, may already be in 03.02).
Riemannian metric, Riemannian connection, geodesics (I §§11–14)	TBD	△	Likely partial; verify against 03-modern-geometry/02-manifolds/ once PDF is local.
Exponential map of a manifold, normal coordinates (I §§15)	TBD	△	Pre-Lie-group exp map; verify against shipped units.
Hopf-Rinow completeness (I §§16)	—	✗	Gap (low). Foundational Riemannian-geometry theorem; not currently a Codex unit. Pre-requisite for symmetric-space global structure.
Lie group, Lie algebra of a Lie group (II §§1–2)	03.03.01 lie-group	✓	Direct correspondent.
Exponential map of a Lie group, Baker-Campbell-Hausdorff (II §§3–5)	—	△	Gap (medium). No dedicated Lie-group-exp / BCH unit. Touched in passing in 03.03.01. Candidate: 03.03.04 lie-group-exponential-map (with BCH master-tier extension).
Adjoint representation, $\mathrm{Ad}$ and $\mathrm{ad}$ (II §§5)	—	△	Gap (low). Referenced inside 07.06.* but not a standalone unit; folds into 03.03.04 or a new 03.03.05.
Killing form, semisimplicity, Cartan's criterion (II §§6, III §§1)	— (referenced in 07.06.03 root-system master)	△	Gap (medium). Killing form has no dedicated unit despite being load-bearing for every classification result. Candidate: 07.06.10 killing-form-and-cartan-criterion.
Semisimple Lie algebras over $\mathbb{C}$, root-space decomposition (III §§4–5)	07.06.03 root-system, 07.06.04 weyl-group, 07.06.05 dynkin-diagram, 07.04.01 cartan-weyl-classification	✓	Codex covers the complex-semisimple case via the FH route.
Compact real form of a complex semisimple Lie algebra (III §§6–7)	—	✗	Gap (high). Foundational: every complex semisimple Lie algebra has a compact real form unique up to conjugation. Originator: Weyl 1925. Candidate: 07.04.02 compact-real-form.
Cartan involution $\theta$ on a real semisimple Lie algebra (III §§7, V §§1)	—	✗	Gap (very high — load-bearing). No Codex unit. Candidate: 07.04.03 cartan-involution.
Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ (V §§1)	—	✗	Gap (very high — the central algebraic object). Candidate: 07.04.04 cartan-decomposition.
Real forms of a complex semisimple Lie algebra; Cartan classification (III §§7, X §§6)	△ (mentioned in 07.04.01 as "Cartan-Weyl" but real forms not explicit)	△→✗	Gap (very high). 07.04.01 covers the complex classification; the real-form classification (compact, split, intermediate) is absent. Candidate: 07.04.05 real-forms-classification.
Orthogonal symmetric Lie algebra (IV §§1)	—	✗	Gap (high). The algebraic abstraction Helgason works with throughout. Candidate: 07.04.06 orthogonal-symmetric-lie-algebra.
Riemannian symmetric space (IV §§3)	—	✗	Gap (very high — the namesake object). Definition via geodesic symmetry $s_p$ at each point; equivalence with $G/K$ for $(\mathfrak{g},\mathfrak{k})$ orthogonal symmetric. Candidate: 03.02.0X riemannian-symmetric-space (lives in modern-geometry/02-manifolds adjacent OR in rep-theory; decision needed — likely 07-representation-theory/08-symmetric-space/ new subdir).
The four types of irreducible symmetric spaces (compact / non-compact / Hermitian / non-Hermitian) (IV §§3, X)	—	✗	Gap (high). Candidate: §Master of 07.04.05 real-forms-classification or its own unit 07.04.07 four-types-of-symmetric-spaces.
Compact / non-compact duality $G_u/K\leftrightarrow G/K$ (V §§2)	—	✗	Gap (high). Candidate: §Master extension of 07.04.04 cartan-decomposition.
Curvature of $G/K$, sectional curvature signs by type (V §§3)	—	✗	Gap (medium). Type I/III have non-positive curvature; type II/IV have non-negative curvature. Candidate: §Master extension to the riemannian-symmetric-space unit.
Totally geodesic submanifolds; Lie triple system (IV §§7)	—	✗	Gap (low — Master deepening).
Restricted roots, $\Sigma \subset {\mathfrak{a}}^{\ast }$, multiplicities $m_{\alpha }$ (VI §§3)	—	✗	Gap (high). The data refining the complex root system into real-form invariants. Candidate: 07.04.08 restricted-root-system.
Satake / Tits diagram of a real form (X §§3–4)	—	✗	Gap (medium). The combinatorial classification table. Candidate: §Master extension of 07.04.05.
Iwasawa decomposition $G=KAN$ (VI §§3)	—	✗	Gap (very high). Originator: Iwasawa 1949. Candidate: 07.04.09 iwasawa-decomposition.
$KAK$ decomposition (Cartan / polar decomposition); $A_{+}$ as a fundamental domain (IX §§1)	—	✗	Gap (medium). Candidate: §Master extension of 07.04.09.
Bruhat decomposition (IX §§1)	—	✗	Gap (medium). Candidate: §Master extension of 07.04.09 or new 07.04.10 bruhat-decomposition (decision later).
Minimal / standard parabolic subgroup (VI §§3, IX)	—	✗	Gap (medium). Parabolic-subgroup machinery; folds with Iwasawa.
Algebra of $G$-invariant differential operators $\mathbb{D}(G/K)$ (IV §§5)	—	✗	Gap (medium). Helgason's central tool for harmonic analysis. Candidate: 07.04.11 invariant-differential-operators.
Spherical function ${\phi }_{\lambda }$ (IV §§5–6)	—	✗	Gap (high). Originator: Harish-Chandra 1953–58. Candidate: 07.04.12 spherical-function.
Harish-Chandra integral formula ${\phi }_{\lambda }(g)={\int }_K{e}^{(i\lambda -\rho )(H(gk))}dk$ (IV §§5)	—	✗	Gap (medium). Folds into 07.04.12 spherical-function.
Plancherel formula on $G/K$ (sketch) (IV §§7)	—	✗	Gap (low for DGLGSS — explicitly deferred to GTM 113). Pointer-only candidate.
Symmetric spaces of compact type: classical examples (Grassmannians, Lagrangian Grassmannians, $\mathrm{SU}(n)/\mathrm{SO}(n)$, etc.) (X)	△ (Grassmannians referenced in 07.04.01)	△	Gap (medium). The 11 infinite families need a tabular pointer unit. Candidate: §Master extension or 07.04.13 classification-tables.
Hermitian symmetric spaces; Borel embedding; bounded symmetric domains (VIII)	—	✗	Gap (medium). A whole DGLGSS chapter; Hermitian type intersects complex geometry. Candidate: 07.04.14 hermitian-symmetric-space (or defer to a later DGLGSS+Helgason-GTM-113 pass).

Aggregate coverage estimate (REDUCED audit basis).

• Chapter I–II (manifolds, Lie groups foundations): ~60% covered. Codex has manifolds, Lie group definition, Lie algebra. Gaps are Hopf-Rinow, BCH formula, adjoint-action standalone unit, and the Riemannian-geometry prerequisites at Helgason's depth.
• Chapter III (complex semisimple Lie algebras, root systems, compact real form): ~70% covered for the complex side via existing 07.06.03–07.06.05 and 07.04.01; the compact-real-form half is a total gap (0%).
• Chapter IV–V (orthogonal symmetric Lie algebras, Riemannian symmetric spaces, Cartan decomposition, duality): ~0% covered. This is the book's load-bearing core and is a complete gap in the Codex.
• Chapter VI (Iwasawa, restricted roots): ~0% covered.
• Chapter VIII (Hermitian symmetric spaces, bounded domains): ~0% covered.
• Chapter IX (decompositions and integration): ~0% covered.
• Chapter X (classification tables): ~0% covered. (07.04.01 states the complex classification; real-form classification is the missing half.)
• Spherical functions (IV §§5–6, X §§3–4): ~0% covered.

Overall: ~15–20% of DGLGSS covered by existing Codex units, almost all of that being the Chapter III complex-Lie-algebra prerequisites that Codex already ships via the FH (3.11) route. The symmetric-space half of DGLGSS (chapters IV–X, the namesake material) is a complete gap. This is unsurprising — the Codex has shipped the FH "complex classification" units and the Hall-/Knapp-flavoured Lie-group-as-manifold units, but has not yet entered the real-form / symmetric-space sector at all. The gap closure is new-unit-heavy, not deepening-heavy, the inverse pattern from the FH audit.

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§3 Gap punch-list (priority-ordered)

Priority 0 — prerequisites that should ship from other plans before the priority-1 DGLGSS units begin:

• 03.02.0X Hopf-Rinow theorem (Riemannian completeness; lives in manifolds strand). Probably 1 hour.
• 03.03.04 Lie-group exponential map + BCH (master tier). Probably 3 hours. May already be covered as a §Master extension of 03.03.01 lie-group; verify.
• 07.06.10 Killing form and Cartan's semisimplicity criterion. Likely a 1500-word unit. Three-tier. ~2 hours.

These three are shared prereqs that should be coordinated with the Knapp audit (when written), the Kobayashi-Nomizu Vol 1 audit (already exists as kobayashi-nomizu-foundations-vol1.md), and the Woit audit.

Priority 1 — high-leverage, captures DGLGSS's central content (the symmetric-space classification + Iwasawa decomposition):

1. 07.04.02 Compact real form of a complex semisimple Lie algebra. Statement of the theorem (every complex semisimple Lie algebra has a compact real form, unique up to conjugation), Weyl's "unitary trick," construction via the Chevalley involution. Three-tier; master tier includes Weyl 1925 originator prose. ~1500 words. Foundational — the gateway to all real-form theory. ~3 hours.

2. 07.04.03 Cartan involution. Definition $\theta \colon \mathfrak{g} \to \mathfrak{g}$,$\theta^2 = \mathrm{id}$,$-B_\theta(X, Y) = -B(X, \theta Y)$ positive-definite (where $B$ is the Killing form). Existence and uniqueness up to inner automorphism. Three-tier; ~1500 words. High — the algebraic object that bridges the complex and real-form theories. ~3 hours.

3. 07.04.04 Cartan decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$. $\pm 1$-eigenspaces of $\theta$; bracket relations $[\mathfrak{k},\mathfrak{k}]\subseteq \mathfrak{k}$, $[\mathfrak{k},\mathfrak{p}]\subseteq \mathfrak{p}$, $[\mathfrak{p},\mathfrak{p}]\subseteq \mathfrak{k}$. Group-level counterpart $G=K\cdot \exp \mathfrak{p}$ (Cartan decomposition of the group). Three-tier; ~2000 words. Master tier sketches the compact-non-compact duality. Very high — the central algebraic object of DGLGSS. ~4 hours.

4. 07.04.05 Real forms of a complex semisimple Lie algebra (Cartan's classification). Statement of the classification: real forms of ${\mathfrak{g}}_{\mathbb{C}}$ are in bijection with conjugacy classes of Cartan involutions, which in turn correspond to certain involutions of the Dynkin diagram (the Satake / Tits diagram data). Compact form, split (normal) form, and intermediate forms. Three-tier; ~2500 words. Master tier reproduces Cartan's classification tables for $A_n, B_n, C_n, D_n, G_2, F_4, E_6, E_7, E_8$. Very high. ~4–5 hours.

5. 07.04.06 Orthogonal symmetric Lie algebra. Definition: a pair $(\mathfrak{g},s)$ where $s$ is an involutive automorphism and the bilinear form $B+B\circ s$ is non-degenerate. The four classes (Euclidean, compact, non-compact, "of $\mathbb{R}$-type"). Bridge from the algebraic side to the geometric side. Three-tier; ~1500 words. High — gateway to symmetric spaces. ~3 hours.

6. 07.04.07 Riemannian symmetric space. Geometric definition: a Riemannian manifold $M$ such that at every $p\in M$ there is an isometric involution $s_p$ with $p$ as an isolated fixed point. Equivalence with $G/K$ for $(\mathfrak{g},\mathfrak{k})$ an orthogonal symmetric Lie algebra of compact, non-compact, or Euclidean type. Curvature signs by type. Three-tier; ~2500 words. Master tier includes worked examples: spheres $S^n = \mathrm{SO}(n+1)/\mathrm{SO}(n)$,hyperbolicspace$H^n = \mathrm{SO}_0(n,1)/\mathrm{SO}(n)$, complex projective space ${\mathbb{CP}}^n=\mathrm{SU}(n+1)/\mathrm{S}(\mathrm{U}(n)\times \mathrm{U}(1))$, Grassmannian $\mathrm{Gr}_k(\mathbb{R}^n) = \mathrm{SO}(n)/(\mathrm{SO}(k)\times\mathrm{SO}(n-k))$. Very high — the namesake unit. ~5 hours.

7. 07.04.09 Iwasawa decomposition $G=KAN$. Statement, proof sketch via Iwasawa's 1949 argument (or modern proof via Lie-algebra Iwasawa $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$ and exponentiation), uniqueness of decomposition, global diffeomorphism $K\times A\times N\to G$. Three-tier; ~2000 words. Master tier includes the restricted-root data $\Sigma \subset {\mathfrak{a}}^{\ast }$ with multiplicities. Very high — the structural cornerstone for everything past Chapter VI. ~4 hours.

Priority 2 — fills medium-priority DGLGSS content with new units:

8. 07.04.08 Restricted root system. $\Sigma = {\alpha \in \mathfrak{a}^* \setminus {0} : \mathfrak{g}\alpha \neq 0}$ where $\mathfrak{g}\alpha$isthe$\alpha$-restricted-root space under $\mathrm{ad}(\mathfrak{a})$. Multiplicities $m_\alpha = \dim \mathfrak{g}_\alpha$.Non-reducedrootsystems($BC_n$) appear here even when the complex root system is reduced. Three-tier; ~1500 words. Medium-high. ~3 hours.

9. 07.04.10 Bruhat decomposition. $G={\bigsqcup }_{w\in W}BwB$ for $B=MAN$ a minimal parabolic, $W=W(\mathfrak{g},\mathfrak{a})$ the restricted Weyl group. The big-cell open dense Bruhat cell. Three-tier; ~1500 words. Medium. ~3 hours.

10. 07.04.11 Algebra of $G$-invariant differential operators $\mathbb{D}(G/K)$. Definition, commutativity (the Helgason-Harish-Chandra theorem), characterisation as polynomials on ${\mathfrak{a}}^{\ast }$ invariant under the restricted Weyl group $W$. Intermediate + Master. ~1500 words. Medium. ~3 hours.

11. 07.04.12 Spherical function ${\phi }_{\lambda }$ on $G/K$. Definition as the unique $K$-bi-invariant joint eigenfunction of $\mathbb{D}(G/K)$ with eigenvalue determined by $\lambda \in \mathfrak{a}\mathbb{C}^*$,normalisedby$\varphi\lambda(e) = 1$. Harish-Chandra integral formula. Statement of the rank-one Plancherel sketch; deeper analysis deferred to GTM 113. Three-tier; ~2500 words. Master tier carries Harish-Chandra 1953–58 originator-prose. High. ~4 hours.

12. 07.04.13 Classification tables for irreducible Riemannian symmetric spaces. Reference / lookup unit. The four classes (compact non-Hermitian, non-compact non-Hermitian, compact Hermitian, non-compact Hermitian) and the explicit tables: the 11 infinite families ($\mathrm{AI},\mathrm{AII},\mathrm{AIII}$; $\mathrm{BDI},\mathrm{DIII}$; $\mathrm{CI},\mathrm{CII}$; etc.)

    • the 12 exceptional spaces. Cartan's original labelling. Intermediate + Master. ~1500 words; predominantly tabular. Medium — high reference value, low explanatory value. ~3 hours.

Priority 3 — Master-tier deepenings and pointer units:

13. §Master extension of 07.04.04: compact / non-compact duality $G_u/K\leftrightarrow G/K$ and worked examples. ~500 words added.

14. §Master extension of 07.04.07: curvature of $G/K$, sectional curvature signs by type (non-positive for non-compact, non-negative for compact). ~500 words added. Useful cross-link to physics strand (Kaluza-Klein compactifications, sigma models).

15. §Master extension of 07.04.09: $KAK$ decomposition / polar decomposition, $A_{+}$ as a fundamental domain for $K$-bi-invariant functions. ~400 words added.

16. 07.04.14 Hermitian symmetric space (pointer unit). Brief definition: a Riemannian symmetric space with a $G$-invariant complex structure compatible with the metric. Borel embedding into its compact dual; bounded-symmetric-domain realisation (Harish-Chandra 1956). Pointer to deeper material in Knapp / Wolf Spaces of Constant Curvature. ~1200 words; master tier only. Medium-low — important for physics-bridge (string theory moduli spaces, Calabi-Yau, etc.) but not strictly required for FT equivalence on DGLGSS. ~2 hours.

17. §Master extension of 07.04.07: Lie triple system and totally geodesic submanifolds. ~400 words added.

Priority 4 — survey / exercise pack follow-ups (optional):

18. Exercise pack: explicit Iwasawa decomposition of ${\mathrm{SL}}_2(\mathbb{R})$ ($K=\mathrm{SO}(2)$, $A$ = positive diagonals, $N$ = upper unitriangulars), of ${\mathrm{SL}}_n(\mathbb{R})$ (full Gram-Schmidt), and of $\mathrm{SU}(n,1)$. Fold into 07.04.09 exercise set.

19. Exercise pack: compute restricted root systems for $\mathrm{SU}(p,q)$, ${\mathrm{SO}}_0(p,q)$, $\mathrm{Sp}(p,q)$, ${\mathrm{Sp}}_n(\mathbb{R})$. Fold into 07.04.08.

20. Pointer to physics strand: Cartan's classification of symmetric spaces underlies Altland-Zirnbauer's tenfold classification of topological insulators / random matrix universality classes. ~300 words; Master section in 07.04.13 or new unit in physics strand.

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§4 Implementation sketch (P3 → P4)

Realistic production estimate (mirroring earlier audits — Brown-Higgins-Sivera, Lawson-Michelsohn, Bott-Tu, Fulton-Harris):

• Priority 0 prereqs (3 prereq units / extensions): ~6 hours.
• Priority 1 (7 new units): ~3.5–5 hours each = ~26–28 hours. 07.04.02 compact-real-form, 07.04.03 cartan-involution, 07.04.04 cartan-decomposition, 07.04.05 real-forms-classification, 07.04.06 orthogonal-symmetric-lie-algebra, 07.04.07 riemannian-symmetric-space, 07.04.09 iwasawa-decomposition.
• Priority 2 (5 new units): ~3.5 hours each = ~17 hours.
• Priority 3 (4 master deepenings + 1 pointer unit): ~1 hour each for deepenings + 2 hours for pointer = ~6 hours.
• Priority 4 (exercise packs + physics pointer): ~3 hours.

Total: ~58–60 hours of focused production for full FT-equivalence coverage of DGLGSS. Fits an 8–10 day window. Priority 1 alone (~26 hours, ~3.5 days) raises effective coverage from ~15% to ~65% and closes the namesake-material gap.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, originator-prose treatment with primary-source citations should appear in:

• Cartan 1926–27 — Élie Cartan, "Sur une classe remarquable d'espaces de Riemann," Bull. Soc. Math. France 54 (1926) 214–264 and 55 (1927) 114–134; and "La géométrie des groupes de transformations," J. Math. Pures Appl. 6 (1927) 1–119. Foundational classification of Riemannian symmetric spaces. Cite in 07.04.05, 07.04.07, 07.04.13.
• Cartan 1914 and 1929 — Élie Cartan, "Les groupes réels simples, finis et continus," Ann. Sci. École Norm. Sup. 31 (1914) 263–355; "Groupes simples clos et ouverts et géométrie riemannienne," J. Math. Pures Appl. 8 (1929) 1–33. Real-form classification. Cite in 07.04.02, 07.04.05.
• Weyl 1925–26 — Hermann Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen," Math. Z. 23–24 (1925–26). Compact-real-form existence and the "unitary trick." Cite in 07.04.02. (Also cited by the FH audit in 07.06.07, 07.07.04 — coordinate to avoid duplication.)
• Wigner 1939 — E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group," Ann. of Math. 40, 149–204. Physics application of real-form classification (the Poincaré group's representation theory uses the same Cartan-involution machinery). Cite as application in 07.04.05 Master section.
• Iwasawa 1949 — Kenkichi Iwasawa, "On some types of topological groups," Ann. of Math. 50 (1949) 507–558. Originates the $G=KAN$ decomposition. Cite in 07.04.09.
• Harish-Chandra 1953–68 — Harish-Chandra, "Representations of semisimple Lie groups I–VI," Trans. Amer. Math. Soc. 75 (1953), 76 (1954), 76 (1954); Amer. J. Math. 77, 78 (1955–56); plus "Spherical functions on a semisimple Lie group I, II," Amer. J. Math. 80 (1958) 241–310, 553–613. Origin of spherical-function theory and the foundations for the Plancherel programme. Cite in 07.04.11, 07.04.12.

Notation crosswalk. DGLGSS uses $G$ for a (connected) semisimple real Lie group, $K$ for its maximal compact subgroup (= fixed-point set of the Cartan involution lifted to $G$), $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$fortheCartandecomposition(note:$\mathfrak{p}$ not $\mathfrak{m}$ — Helgason's $\mathfrak{p}$ collides with the "Iwasawa $\mathfrak{p}=\mathfrak{m}+\mathfrak{a}+\mathfrak{n}$" parabolic-subalgebra convention used by Knapp; Codex should adopt Helgason's $\mathfrak{p}$ for the Cartan-decomposition piece and call out the collision in 07.04.04 §Notation). DGLGSS writes $\mathfrak{a}$ for a maximal $\mathbb{R}$-split abelian subalgebra of $\mathfrak{p}$ (consistent with Knapp), $\Sigma \subset {\mathfrak{a}}^{\ast }$ for the restricted-root system, $m_{\alpha }$ for the multiplicity $\dim {\mathfrak{g}}_{\alpha }$, $W(\mathfrak{g},\mathfrak{a})$ for the restricted Weyl group (distinct from the complex Weyl group $W$ familiar from 07.06.04). Codex should adopt all of these.

For symmetric spaces themselves: Helgason writes $M=G/K$ for a non-compact-type symmetric space and $M^{\ast }=U/K$ (or sometimes $M_u$) for its compact dual. Helgason's classification labels ($\mathrm{AI}, \mathrm{AII}, \ldots, \mathrm{EVII}, \mathrm{EVIII}, \mathrm{EIX}$) should be reproduced verbatim in 07.04.13.

Cross-strand weaving (Pass-W). New units should link laterally to:

• 07.06.* Lie-algebraic strand — 07.04.02 compact-real-form and 07.04.05 real-forms-classification are direct deepenings of 07.04.01 cartan-weyl-classification.
• 03.02 manifolds and 03.04 differential-forms — 07.04.07 riemannian-symmetric-space is a Riemannian-geometry application.
• 06-riemann-surfaces — Hermitian symmetric spaces of non-compact type in rank one are the unit ball ${\mathbb{B}}^n\subset {\mathbb{C}}^n$, connecting to complex geometry.
• physics/quantum-mechanics/ — Wigner's classification of particle representations uses real forms of the Poincaré algebra; Altland-Zirnbauer tenfold way for topological matter uses Cartan's classification of symmetric spaces.
• Future Knapp audit (if commissioned) — Knapp covers the same real-form material with a different ordering; coordinate prereq / prerequisite chains to avoid duplication.
• Future Kobayashi-Nomizu Vol 1 audit (already exists as kobayashi-nomizu-foundations-vol1.md) — KN Vol 1 covers Riemannian connections and curvature; coordinate the Hopf-Rinow prereq.

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§5 What this plan does NOT cover

• A line-number-level inventory of every named theorem in DGLGSS (full P1 audit; deferred until a local PDF is acquired). Coverage estimates in §2 are based on DGLGSS's well-known chapter structure (AMS GSM 34 catalog page + standard references in Knapp, Goodman- Wallach, Onishchik-Vinberg).
• Helgason's sequel volumes. Groups and Geometric Analysis (GTM 113, AMS reprint as GSM 83) covers the harmonic-analysis half: invariant differential operators on $G/K$ in depth, spherical Plancherel theorem, Paley-Wiener theorem on $G/K$, horocycle transform, eigenfunctions of the Laplace-Beltrami operator on $G/K$. Geometric Analysis on Symmetric Spaces (GSM 39) is the third volume, covering the Helgason-Fourier transform and conjectures on eigenfunctions. These two volumes should each receive their own audit plan when commissioned. Cross-references to GTM 113 are acceptable in the Codex 07.04.12 spherical-function master tier, but the deep harmonic-analysis content stays in the GTM 113 plan.
• Knapp's Lie Groups Beyond an Introduction. Substantial overlap with DGLGSS Chapters II–VI; deferred to its own audit. Note that Knapp orders the material structurally (real Lie groups → root systems → symmetric spaces), while Helgason orders it geometrically (manifolds → Lie groups → orthogonal symmetric Lie algebras → symmetric spaces). The Codex Lie-strand units 07.04.* should read cleanly under either ordering.
• Goodman-Wallach Symmetry, Representations, and Invariants. Major overlap with DGLGSS Chapter III; deferred.
• Onishchik-Vinberg Lie Groups and Algebraic Groups. Tabulation reference; consult for 07.04.13 classification tables but no separate audit needed (Onishchik-Vinberg is reference-style).
• Wolf Spaces of Constant Curvature. Covers the space-form classification side; deferred.
• The infinite-dimensional / Kac-Moody / loop-group analogues of symmetric-space theory. Live in 03-modern-geometry/11-infinite-dim-lie/ and have their own canonical references (Pressley-Segal Loop Groups, Kac Infinite Dimensional Lie Algebras).
• Modular / positive-characteristic representation theory of real reductive groups. Out of scope for DGLGSS by design.
• Number-theoretic / Langlands-programme applications. DGLGSS Chapter VI material (Iwasawa decomposition, parabolic subgroups) is the structural input but the arithmetic side is deferred to Bump Automorphic Forms, Cogdell L-Functions, Borel Automorphic Forms on Reductive Groups.

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§6 Acceptance criteria for FT equivalence (DGLGSS)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, DGLGSS is at equivalence-coverage when:

• The 3 Priority-0 prereqs have shipped (Hopf-Rinow, Lie-group exp/BCH, Killing form + Cartan criterion).
• The 7 Priority-1 new units have shipped (07.04.02 through 07.04.07, 07.04.09). These close the namesake-material gap.
• ≥95% of DGLGSS's named theorems in Chapters I–X map to a Codex unit (currently ~15–20%; after Priority-1 this rises to ~65%; after Priority-1+2 to ~88%; full ≥95% requires Priority-3 master deepenings + the Hermitian-symmetric-space pointer unit).
• ≥90% of DGLGSS's worked examples (sphere, hyperbolic space, complex projective space, real and complex Grassmannians, the bounded symmetric domains in low rank, ${\mathrm{SL}}_2(\mathbb{R})$ Iwasawa) have a direct unit or are cross-referenced from a unit that covers them.
• Notation crosswalk recorded (see §4).
• Pass-W weaving connects the new units to 07.06.*, 03.02 manifolds, 03.04 differential-forms, and the physics strand (Wigner classification + Altland-Zirnbauer cross-link).
• Originator-prose sections (Cartan 1926–27 + 1914 + 1929, Weyl 1925, Iwasawa 1949, Harish-Chandra 1953–68, Wigner 1939) appear in the relevant Master tiers.

Equivalence verification protocol (per FASTTRACK_EQUIVALENCE_PLAN.md §4):

• 4.1 Book-as-input self-check. Sample 5 random theorem statements from DGLGSS (likely from Chapters IV, V, VI); for each, identify the Codex unit that proves it.
• 4.2 Exercise reproducibility. Sample 5 random exercises from DGLGSS (Helgason's exercises are detailed and his Solutions to Exercises appendix in the 2001 reprint is itself a 70-page document; bias the sampling toward exercises with solutions in the appendix); for each, identify the Codex unit(s) sufficient to solve it.
• 4.3 Notation comprehension. Sample 5 fragments of Helgason prose; paraphrase each into the Codex notation system (per §4 crosswalk). The $\mathfrak{p}$ / parabolic-$\mathfrak{p}$ collision is a known stumbling block — flag any tested paraphrase that mishandles it.

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§7 Sourcing

• Local PDF status. Not present in reference/textbooks-extra/ (verified — present titles are Lang Basic Mathematics, Apostol Calculus Vols 1 + 2, Ahlfors Complex Analysis + Riemann Surfaces, Courant Differential and Integral Calculus Vols 1 + 2, Donaldson Riemann Surfaces, Mackenzie Lie Groupoids and Algebroids Vols, Landau-Lifshitz Vols 1, 2, 6, 8, and an unrelated ODE epdf). No Helgason title anywhere in the Codex reference/ tree.
• Commercial source. AMS GSM 34 (2001 corrected reprint). Available at https://bookstore.ams.org/gsm-34/ (BUY). ISBN 978-0-8218-2848-9. Also available as the original Academic Press 1978 edition (ISBN 0-12-338460-5) — content essentially identical, but the 2001 reprint adds the Solutions to Exercises appendix and errata corrections. Target the 2001 reprint for the local copy.
• Library-mirror sources. Acquire via institutional access (AMS member subscription or university library) or interlibrary loan. For the local copy, target placement is reference/textbooks-extra/Helgason-Differential-Geometry-Lie-Groups-Symmetric-Spaces.pdf.
• Companion / supplementary materials.
  • S. Helgason, Groups and Geometric Analysis (AMS GSM 83, 2000 reprint of Academic Press 1984) — the harmonic-analysis sequel.
  • S. Helgason, Geometric Analysis on Symmetric Spaces (AMS Math. Surveys 39, 2nd ed. 2008) — the Helgason-Fourier-transform volume.
  • A. W. Knapp, Lie Groups Beyond an Introduction (Birkhäuser 2nd ed. 2002) — alternative coverage of the same structure theory; recommended companion for the Iwasawa / restricted-root material.
  • A. L. Onishchik, E. B. Vinberg, Lie Groups and Algebraic Groups (Springer 1990) — recommended companion for the classification tables.
  • R. Goodman, N. R. Wallach, Symmetry, Representations, and Invariants (Springer GTM 255, 2009) — matrix-explicit treatment of the same material.
• Open-access alternatives covering ~30% of DGLGSS:
  • G. Segal, "Lie Groups," in Carter-Macdonald-Segal Lectures on Lie Groups and Lie Algebras (LMS Student Texts 32, Cambridge
    1995. — 60-page introduction to compact-real-form theory. Substantial preview on Google Books / Cambridge Core.
  • Various lecture notes: A. Kirillov Jr., Introduction to Lie Groups and Lie Algebras (free, online at Stony Brook); covers DGLGSS Chapters I–III material at gentler pace, real forms only briefly.
  • D. Bump, Lie Groups (Springer GTM 225, 2nd ed. 2013) — partial Google Books preview; covers the compact-real-form material and spherical functions in the rank-one case.
• Reduced-audit flag. This plan was produced without a local Helgason PDF. Coverage estimates in §2 should be re-verified once a PDF is acquired, particularly for Chapters VIII (Hermitian symmetric spaces) and X (classification tables) where the load-bearing content is the detailed tabular data and the precise definitions of the Cartan labels ($\mathrm{AI}, \mathrm{AII}, \mathrm{AIII}, \mathrm{BDI}, \mathrm{DIII}, \mathrm{CI}, \mathrm{CII}, \mathrm{EI}, \mathrm{EII}, \ldots, \mathrm{EIX}$). Promote to full P1 audit when PDF is local.