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Borel--Serre type bordifications and Canonical Pairs for Loop groups

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Pith's one-line read The paper constructs a Hausdorff Borel\u2013Serre type bordification of the positive-half loop-group symmetric space, proves its boundary is homotopy equivalent to the affine rational Tits building, and extends the arithmetic loop-group…

desk verdict First real Borel–Serre construction for loop-group symmetric spaces, with honest limitations; the stated Siegel finiteness (SF-U) is false as written, so the Hausdorffness claim for the quotient needs repair, but the bordification and canonical-pair theorems survive scrutiny. read the letter →

arxiv 2507.13980 v1 pith:JDCC3L7T submitted 2025-07-18 math.RT math.ATmath.GRmath.NT

Manish M. Patnaik , Punya Plaban Satpathy This is my paper · ORCID
classification math.RTmath.ATmath.GRmath.NT MSC 22E6711F75
keywords loopgroupsBorel\u2013SerrebordificationaffineTitsbuildingcanonicalpairssemi-stabilityreductiontheoryarithmeticquotientspro-unipotent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs a Borel\u2013Serre type bordification $\mathrm{QB}(X_{\mathrm{b}G^+})$ of the loop-group symmetric space $X_{\mathrm{b}G^+}=\mathrm{b}K\backslash \mathrm{b}G^+$, where $\mathrm{b}G^+$ is the positive half of an extended real loop group and $\mathrm{b}K$ is its maximal compact analogue, by adding one boundary component $e(P)=X_{M_P}\times \mathrm{b}U_P$ for each rational parabolic $P$. It proves that the attached boundary is homotopy equivalent to the affine rational Tits building of the rational loop group and that the arithmetic loop group acts continuously on the whole bordification. It also introduces a notion of semi-stability and a theorem of canonical pairs for loop groups, partitioning $X_{\mathrm{b}G^+}$ into a semi-stable locus and parabolic ends that fiber over lower-rank symmetric spaces. The quotient by the arithmetic group is not compact, and the non-compactness is attributed to the central direction of the loop group, while the boundary has compact quotient. A sympathetic reader would care because the bordification supplies the natural geometric arena in which reduction theory, cohomology, and automorphic forms for loop-group arithmetic groups can be developed.

What carries the argument

The mechanism is the corner $c(\mathrm{b}A^+_P)$: a completion of the positive part of the loop-group torus obtained by letting the simple-root coordinates that define $P$ tend to zero while leaving the central coordinate free, with a topology specified by an explicit convergence class. Adding the pieces $e(P)=X_{M_P}\times \mathrm{b}U_P$ and declaring convergence along these corners produces the bordification. The canonical-pair theory runs on the degree of instability $\deg_{\mathrm{inst}}(x)=\min_{P,\delta}\langle\rho_P,H_P(x\delta)\rangle$, where $\rho_P$ is an affine weight functional and $H_P$ is the horospherical-coordinate map; uniqueness is obtained through affine orthogonal families and an auxiliary inequality. The boundary-homotopy theorem is proved by covering the boundary with the closed sets $e(Q)$ for maximal parabolics $Q$, whose nerve is the affine rational Tits building, each $e(Q)$ being an absolute retract (a space that is a retract of every metric space containing it as a closed subspace).

What would settle it

Work out the cardinality of $\{\gamma\in\mathrm{b}\Gamma_U:\mathrm{b}U_\Omega\cdot\gamma\cap\mathrm{b}U_\Omega\neq\varnothing\}$ for a bounded $\Omega$ in the affine $\mathrm{SL}_2$ loop group using the explicit Iwahori\u2013Matsumoto coordinates; if it is infinite, the Siegel-type finiteness assumption fails and the asserted Hausdorffness of $\mathrm{QB}(X_{\mathrm{b}G^+})/\mathrm{b}\Gamma$ collapses, even though the boundary-homotopy theorem might still hold.

Watch

Extended reading notes

Core claim

The central assertion is that $\mathrm{QB}(X_{\mathrm{b}G^+})=X_{\mathrm{b}G^+}\sqcup\bigsqcup_{P\in\mathrm{Par}'_{\mathbb Q}} e(P)$, with $e(P)=X_{M_P}\times \mathrm{b}U_P$, admits a natural Hausdorff topology for which the inclusion of $X_{\mathrm{b}G^+}$ extends to an open embedding of each corner $X_{M_P}\times c(\mathrm{b}A^+_P)\times \mathrm{b}U_P$. The paper establishes that the boundary $\partial\,\mathrm{QB}(X_{\mathrm{b}G^+})$ is homotopy equivalent to the affine rational Tits building of $\mathrm{b}G_{\mathbb Q}$, and that the right action of $\mathrm{b}G_{\mathbb Q}$ extends continuously over the bordification. It also proves a loop-group version of the theorem of canonical pairs: every non-semi-stable point of $X_{\mathrm{b}G^+}$ has a unique destabilizing, extremal parabolic pair $(P,\delta)$, which yields a disjoint partition $X_{\mathrm{b}G^+}=X^{\mathrm{ss}}_{\mathrm{b}G^+}\sqcup\bigsqcup X_{\mathrm{b}G^+}(P,\delta)$. Completing the toral directions in this partition gives the same bordification, and the quotient $\mathrm{QB}(X_{\mathrm{b}G^+})/\mathrm{b}\Gamma$ decomposes into a semi-stable core and parabolic ends, with compact boundary quotient; the Hausdorffness of the full quotient is claimed only conditionally on a Siegel-type finiteness statement for the pro-unipotent group $\mathrm{b}U$.

Load-bearing premise

The load-bearing premise the paper does not prove is a Siegel-type finiteness statement for the pro-unipotent radical $\mathrm{b}U$: for every bounded set $\Omega$, only finitely many arithmetic unipotent elements $\gamma$ satisfy $\mathrm{b}U_\Omega\cdot\gamma\cap\mathrm{b}U_\Omega\neq\varnothing$; the Hausdorffness and properness claims for the quotient are conditional on this.

Editorial extensions

If this is right

  • The quotient $\mathrm{QB}(X_{\mathrm{b}G^+})/\mathrm{b}\Gamma$ decomposes as $X^{\mathrm{ss}}_{\mathrm{b}G^+}/\mathrm{b}\Gamma\cup\bigsqcup_P X(P)/\mathrm{b}\Gamma_P$, where each parabolic end fibers over $X^{\mathrm{ss}}_{M_P}/\Gamma_{M_P}\times c(\mathrm{b}A^+_{P,(0,1)})$ with pro-unipotent fibers.
  • The boundary quotient $\partial\mathrm{QB}(X_{\mathrm{b}G^+})/\mathrm{b}\Gamma$ is compact, so the failure of compactness of the full quotient is concentrated in the unbounded central direction.
  • If the Siegel-type finiteness condition for $\mathrm{b}U$ holds, the $\mathrm{b}\Gamma$ action on the bordification is proper and the quotient is Hausdorff; the paper's separability statements are conditional on this.
  • The canonical-pair partition and the attachment construction describe the same bordification, providing two independent routes to the same completed space.
  • The continuous $\mathrm{b}G_{\mathbb Q}$ action and the compact boundary quotient make $\mathrm{QB}(X_{\mathrm{b}G^+})$ the intended arena for reduction theory and for the cohomology of the loop-group arithmetic group $\mathrm{b}\Gamma$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the unproved pro-unipotent Siegel finiteness fails, the Hausdorff-quotient claims would drop away while the bordification, the boundary homotopy equivalence, and the canonical-pair theorem could still stand; the paper's two strands are logically separable.
  • The non-compact central direction is naturally the loop-group counterpart of the second-Chern-class direction in compactifications in the theory of bundles on algebraic surfaces; comparing the two could transfer surface-bundle intuitions to loop-group arithmetic quotients.
  • A concrete test of the semi-stability picture is whether, for large fixed $r$, the intersection $X^{\mathrm{ss}}_{\mathrm{b}G^+}\cap \mathrm{b}G_r$ lies in the closure of the non-semi-stable locus; the paper constructs points in that closure but does not prove the general statement.
  • The uniqueness proof bypasses the classical Langlands combinatorial lemma, suggesting that an affine analogue of complementary polyhedra should exist and could yield a more geometric uniqueness theorem for affine canonical pairs.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. This paper attaches a Borel–Serre-type bordification QB(X_Ĝ+) to the symmetric space X_Ĝ+ = bK\Ĝ+ of the positive half of a real loop group, adding boundary components e(P) = X_{M_P} × Û_P for rational parabolics P of affine type, and endows it with a Hausdorff topology defined by explicit convergence classes in Iwahori–Matsumoto coordinates. The main theorems assert: (a) the Ĝ_Q-action on X_Ĝ+ extends continuously to QB(X_Ĝ+); (b) the boundary is homotopy equivalent to the positive rational affine Tits building of Ĝ_Q; (c) an affine canonical-pairs theorem — existence and uniqueness of a destabilizing extremal pair (P, δ) for each point with deginst(x) < 0 — yielding a semi-stability partition of X_Ĝ+; and (d) structural results on quotients X_Ĝ+/Γ̂ and QB(X_Ĝ+)/Γ̂ via Garland's reduction theory, including compactness along the boundary, relative compactness under a central bound, and separability of the quotient under a Siegel-finiteness condition (SF-U) for the pro-unipotent radical Û.

Significance. If correct, this is a substantial contribution: it supplies the loop-group analogue of the Borel–Serre bordification, identifies the affine rational Tits building as the boundary, and establishes a canonical-pairs framework in the spirit of Harder–Narasimhan–Chaudouard. The paper's strengths are real: explicitly checkable convergence-class definitions; a new uniqueness proof for affine canonical pairs via the orthogonal-family inequality (the authors show the naive affine Langlands lemma fails, so this is a genuine technical innovation); a clean nerve/absolute-retract argument for the boundary homotopy; and disciplined honesty, with separability of the quotient explicitly conditioned on (SF-U) in §1.2.8(1). The unconditional results — bordification topology, continuous action, boundary homotopy, canonical-pair theorem, partition, and compactness statements — do not depend on the flagged condition. The central caveat is that (SF-U) is not merely unproved but false as stated, so the separability part requires a real repair; the check is made possible by the paper's own explicit definitions and should be confronted directly.

major comments (1)
  1. [§6.3.2 (SF-U); §§3.1.10–3.1.11, 3.6.1, 3.6.3] The Siegel-finiteness condition (SF-U) on which Proposition 6.3.2 rests is false as stated under the paper's own definitions, so the only route offered to properness of the Γ̂-action and Hausdorffness of QB(X_Ĝ+)/Γ̂ (Lemma 6.3.1) is blocked. Counterexample: take Ĝ = ŜL₂, α the positive root of SL₂, and Ω = [−1,1]. For each n ≥ 0, γ_n := χ_α(tⁿ) lies in Γ̂₀ ⊂ Γ̂ (by §3.6.1) and in Û, hence in Γ̂_U. In the Iwahori–Matsumoto coordinates of §3.1.10–3.1.11 its only nontrivial coordinate is σ_α = tⁿ, with coefficients 0 or 1 in Ω, so γ_n ∈ Û_Ω (as in §3.6.3). Since 1 ∈ Û_Ω and 1·γ_n = γ_n ∈ Û_Ω, the set {γ ∈ Γ̂_U | Û_Ω·γ ∩ Û_Ω ≠ ∅} is infinite, contradicting (SF-U) for the bounded set Ω = [−1,1]. The failure is not confined to large Ω: for Ω₀ = [−1/2,1/2], take σ = Σ_{k=0}^N t^k, γ = χ_α(σ), u = χ_α(−σ/2); then u, uγ ∈ Û_{Ω₀}, again producing infinitely many γ. The paper's own caveats (§1.2.8(1); the remark in §6.3.2) describe the affine analogue as unproved; the stronger finding is that the stated condition fails, because Γ̂_U ≅ Z^N is not discrete in the product topology of Û ≅ R^N and the boxes Û_Ω do not control integral coordinates. This observation does not by itself disprove Hausdorffness of Û/Γ̂_U or of the full quotient; it means the stated route via Proposition 6.3.2 and Lemma 6.3.1 cannot be completed as written. A repair within scope should replace (SF-U) by a valid finiteness statement for a suitably truncated Siegel set, prove Hausdorffness of the quotient directly from the coordinate description of Û/Γ̂_U, or state the separability claim as an open problem consistently throughout.
minor comments (4)
  1. [Eq. (2.7)] In (2.7), ϑ should be defined as Σ_{i∈I₀} d_i a_i; as printed, δ = a_{ℓ+1} + ϑ does not vanish on bh, and the proof of Proposition 4.2.5(2)(a) uses the corrected version with the sum restricted to the finite index set I₀.
  2. [Eq. (5.4) and Proposition 5.3.1] The definition of QB(X_Ĝ+) as a disjoint union X_Ĝ+ ∪ ⊔ e(P) is in apparent tension with the nonempty intersections e(P) ∩ e(Q) = e(P∩Q) asserted in Proposition 5.3.1(2)–(3); a sentence clarifying that the pieces are glued through the corner identifications, and that the intersections in (5.39)–(5.41) are understood in that sense, would remove a literal contradiction.
  3. [Lemma 6.3.1] Lemma 6.3.1 refers to 't and Ω chosen as in Proposition 6.1.2,' but Proposition 6.1.2 does not prescribe a pair (t, Ω); the intended reference appears to be Proposition 6.1.3, whose Siegel sets satisfy the covering property used in the lemma.
  4. [Throughout] Several typos should be corrected, including 'we not yet established' in §1.2.8(1), 'bordifcation' in the title of §5, 'such taht' in Theorem 3.6.4(1), and 'The bais' in §2.5.1.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the affine canonical-pair theorem is proved in-paper from Garland's external finiteness lemmas (not imported by self-citation); the bordification topology and its Hausdorffness are verified against the external finite-dimensional Borel–Ji blueprint; and the single conditional step (Siegel finiteness SF-U) is explicitly disclosed by the authors, so no prediction…

full rationale

The load-bearing derivation chain is self-contained and externally benchmarked. Canonical pairs: existence (Prop 4.3.1) rests on Garland's external finiteness lemma [19, Lemma 2.5]; uniqueness (Thm 4.2.4, §4.3.3) is proved in-paper via the auxiliary inequality (Prop 4.3.2) built from the orthogonal-family inequality (Lemma 3.5.2, proved in-paper) and Garland's [19, Lemma 6.1]. The authors explicitly note that the finite-dimensional Langlands combinatorial lemma fails affinely and supply a new workaround (orthogonal families plus an inequality inspired by Drinfeld–Gaitsgory/Schieder), which is independent content, not imported uniqueness. The bordification QB(X_bG+): its topology is introduced by an explicit convergence class whose axioms are verified in §2.6; Hausdorffness (Prop 5.2.6) reduces to the Hausdorffness of the finite-dimensional Borel–Serre bordification [5, Thm 7.8] plus the in-paper corner check (Prop 2.5.1); the boundary homotopy (Thm 5.3.6) is a direct application of the external nerve criterion [5, Thm 8.2.1] to the locally finite cover {e(Q)}, with the AR property (Prop 5.3.3) supplied by [5, Lemmas 8.3.1, 8.6.4] and bU_Q ≅ R^N. No parameter is fitted, and no claimed output is defined in terms of the claimed result. Self-citation audit: the only assertion justified by what may be the authors' own prior work is bΓ^{λ}_0 ∩ LP = bΓ^{λ} ∩ LP, cited to [2] (§3.6.2, eq. (3.98)), and the related conjecture bΓ0 = bΓ (footnote (9), §3.6.1: 'There is some evidence this is equal to bΓ actually (we believe this to be the case), see [2]'). This feeds Lemma 3.6.2 (arithmeticity of Γ_MP), used in quotient-structure statements (Prop 4.4.7, 6.1.3, 6.2.1, 6.3.2) — not in the headline theorems (bordification Hausdorffness, canonical pairs, boundary homotopy, continuous bGQ action). The truncated reference list does not permit confirming [2] is self-authored, so I flag rather than charge; even if it is the authors' work, it is non-load-bearing for the central claims. Flagged limitation (per reviewing rule): the paper labels the separability results conditional — §1.2.8(1): 'Our results on separability of the quotient QB(X_bG+)/bΓ are only conditional.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented physical entities appear. The new mathematical objects, the bordification, canonical pairs, and the semi-stable locus, are defined and proved rather than postulated. The main ledger entries are external theorems, plus the explicitly unproved SF-U finiteness statement used for quotient separability.

assumptions (6)
  • standard math Classification of positive parabolic subsets of affine root systems
    Used in section 2.3.3 to index standard parabolics by subsets J of I; quoted from Kac and Moody.
  • domain assumption Garland's construction of the complete Kac-Moody group bG with pro-unipotent Borel bU
    The group bG, Iwasawa decomposition, Iwahori-Matsumoto coordinates, and reduction theory are taken from Garland's papers [16, 17].
  • domain assumption Garland's Lemma 2.5 bounding growth of the quantities defining the degree of instability
    Used in section 4.3.1 to guarantee that the minimum defining deginst exists.
  • domain assumption Finite-dimensional theorem of canonical pairs
    Explicitly taken as granted in section 4.2.4 for Levi factors MP; needed for the fibration structure of the unstable pieces.
  • domain assumption Borel-Serre bordification properties for finite-dimensional groups
    Used in sections 5.2.6 and 5.3.3, for example Hausdorffness of QB(X_MQ) and the absolute retract property, cited from Borel-Serre [5].
  • domain assumption Siegel finiteness for finite-dimensional groups and compactness of bU/bGamma_U
    Used in compactness and separability arguments; the compactness of bU_Omega follows from Tychonoff, but the pro-unipotent quotient compactness is used without a full proof.

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Pith. "Pith review of Borel--Serre type bordifications and Canonical Pairs for Loop groups." pith.science (2026). https://pith.science/paper/JDCC3L7T

@misc{pith2026250713980,
  author       = {Pith},
  title        = {Pith review of: Borel--Serre type bordifications and Canonical Pairs for Loop groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JDCC3L7T}},
  note         = {Machine review of arXiv:2507.13980}
}
read the original abstract

To the symmetric space of the (positive half) of a real loop group, we attach a Borel--Serre type bordification and equip it with a Hausdorff topology. The attached boundary, indexed by certain rational parabolics of the loop group, is shown to be homotopic to an affine, rational Tits building. A loop analogue of an arithmetic group is also shown to act continuously on the bordification and its quotient by this action is studied using the reduction theory of H. Garland. While the quotient is no longer compact (as in the Borel--Serre construction from finite-dimensions) we relate the non-compactness to the center of the loop group. We also introduce a notion of semi-stability for loop groups, following works of Harder--Narasimhan, Behrend, and most recently Chaudouard, and use this to describe a partition of our loop symmetric space. This partition is then related to the rational bordification and its quotient.

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Works this paper leans on

49 extracted references · 48 canonical work pages

  1. [1]

    Brown, Buildings, Graduate Texts in Mathematics, vol

    Peter Abramenko and Kenneth S. Brown, Buildings, Graduate Texts in Mathematics, vol. 248, Springer, New York,

    work page

  2. [2]

    Murray,Strong integrality of inversion subgroups of Kac-Moody groups, J

    Abid Ali, Lisa Carbone, Dongwen Liu, and Scott H. Murray,Strong integrality of inversion subgroups of Kac-Moody groups, J. Lie Theory 34 (2024), no. 2, 453–468. MR4722045 68

    work page 2024

  3. [3]

    Arthur, A trace formula for reductive groups

    James G. Arthur, A trace formula for reductive groups. I. Terms associated to classes in G (Q), Duke Math. J. 45 (1978), no. 4, 911–952. MR518111

    work page 1978

  4. [4]

    Behrend, Semi-stability of reductive group schemes over curves , Math

    Kai A. Behrend, Semi-stability of reductive group schemes over curves , Math. Ann. 301 (1995), no. 2, 281–305. MR1314588

    work page 1995

  5. [5]

    Borel and J.-P

    A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. MR387495

    work page 1973

  6. [6]

    Borel and N

    A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups , Second, Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR1721403

    work page 2000

  7. [7]

    73, American Mathematical Soci- ety, Providence, RI, 2019

    Armand Borel, Introduction to arithmetic groups, University Lecture Series, vol. 73, American Mathematical Soci- ety, Providence, RI, 2019. Translated from the 1969 French original [ MR0244260] by Lam Laurent Pham, Edited and with a preface by Dave Witte Morris. MR3970984

    work page 2019

  8. [8]

    MR2189882

    Armand Borel and Lizhen Ji, Compactifications of symmetric and locally symmetric spaces , Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. MR2189882

    work page 2006

Show all 49 references
  1. [9]

    Armand Borel and Jean-Pierre Serre, Adjonction de coins aux espaces symétriques; applications à la cohomologie des groupes arithmétiques, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A1156–A1158. MR271114

    1970

  2. [10]

    Nicolas Bourbaki, General topology, 1st ed., Springer, Berlin, Germany, 1998 (en)

    1998

  3. [11]

    Chapters 4–6 , Elements of Mathematics (Berlin), Springer-Verlag, Berlin,

    , Lie groups and Lie algebras. Chapters 4–6 , Elements of Mathematics (Berlin), Springer-Verlag, Berlin,

  4. [12]

    Bill Casselman, Stability of lattices and the partition of arithmetic quotients , Asian J. Math. 8 (2004), no. 4, 607–

    2004

  5. [13]

    Pierre-Henri Chaudouard, Sur une variante des troncatures d’Arthur, Geometric aspects of the trace formula, 2018, pp. 85–120. MR3969872

    2018

  6. [14]

    Drinfeld and D

    V . Drinfeld and D. Gaitsgory, Compact generation of the category of D-modules on the stack of G-bundles on a curve, Camb. J. Math. 3 (2015), no. 1-2, 19–125. MR3356356

    2015

  7. [15]

    6, Heldermann, Berlin, 1989

    Ryszard Engelking, General topology, Sigma series in pure mathematics, vol. 6, Heldermann, Berlin, 1989

    1989

  8. [16]

    Algebra 53 (1978), no

    Howard Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), no. 2, 480–551. MR502647

    1978

  9. [17]

    Hautes Études Sci

    , The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 5–136. MR601519

    1980

  10. [18]

    , The arithmetic theory of loop groups. II. The Hilbert-modular case, J. Algebra 209 (1998), no. 2, 446–532. MR1659899

    1998

  11. [19]

    II, Amer

    , Eisenstein series on loop groups: Maass-Selberg relations. II, Amer. J. Math. 129 (2007), no. 3, 723–784. MR2325102

    2007

  12. [20]

    Miller, and Manish M

    Howard Garland, Stephen D. Miller, and Manish M. Patnaik, Entirety of cuspidal Eisenstein series on loop groups, Amer. J. Math. 139 (2017), no. 2, 461–512. MR3636637

    2017

  13. [21]

    Grayson, Reduction theory using semistability , Comment

    Daniel R. Grayson, Reduction theory using semistability , Comment. Math. Helv. 59 (1984), no. 4, 600–634. MR780079

    1984

  14. [22]

    Harder, Cohomology of arithmetic groups , available at https://www.math.uni-bonn.de/people/ harder/Manuscripts/buch/Volume-III.pdf

    G. Harder, Cohomology of arithmetic groups , available at https://www.math.uni-bonn.de/people/ harder/Manuscripts/buch/Volume-III.pdf

  15. [23]

    Harder and M

    G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves , Math. Ann. 212 (1974/75), 215–248. MR364254

    1974

  16. [24]

    MR181977

    Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, MI, 1965. MR181977

    1965

  17. [25]

    Iwahori and H

    N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. MR185016

    1965

  18. [26]

    H. P. Jakobsen and V . G. Kac,A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras, Nonlinear equations in classical and quantum field theory (Meudon/Paris, 1983/1984), 1985, pp. 1–20. MR802097

    1983

  19. [27]

    Ji and R

    L. Ji and R. MacPherson, Geometry of compactifications of locally symmetric spaces, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 2, 457–559. MR1906482

    2002

  20. [28]

    Kac, Infinite-dimensional Lie algebras , Third, Cambridge University Press, Cambridge, 1990

    Victor G. Kac, Infinite-dimensional Lie algebras , Third, Cambridge University Press, Cambridge, 1990. MR1104219

    1990

  21. [29]

    Kac and Dale H

    Victor G. Kac and Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), no. 2, 125–264. MR750341

    1984

  22. [30]

    Kapranov, The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups (2000), avail- able at math/0001005

    M. Kapranov, The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups (2000), avail- able at math/0001005

    2000 arXiv

  23. [31]

    Kelley, General topology, Graduate Texts in Mathematics, No

    John L. Kelley, General topology, Graduate Texts in Mathematics, No. 27, Springer-Verlag, New York-Berlin, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.] MR0370454

    1975

  24. [32]

    204, Birkhäuser Boston, Inc., Boston, MA, 2002

    Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory , Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR1923198

    2002

  25. [33]

    31, American Mathematical Society, Providence, RI, 2013

    Jean-Pierre Labesse and Jean-Loup Waldspurger, La formule des traces tordue d’après le Friday Morning Seminar, CRM Monograph Series, vol. 31, American Mathematical Society, Providence, RI, 2013. With a foreword by Robert Langlands [dual English/French text]. MR3026269 69

    2013

  26. [34]

    John M Lee, Introduction to topological manifolds , 2nd ed., Graduate texts in mathematics, Springer, New York, NY, 2010 (en)

    2010

  27. [35]

    Enrico Leuzinger, An exhaustion of locally symmetric spaces by compact submanifolds with corners, Invent. Math. 121 (1995), no. 2, 389–410. MR1346213

    1995

  28. [36]

    , On polyhedral retracts and compactifications of locally symmetric spaces , Differential Geom. Appl. 20 (2004), no. 3, 293–318. MR2053916

    2004

  29. [37]

    A T Lundell and S Weingram, The topology of CW complexes , 1969th ed., The university series in higher mathe- matics, Springer, New York, NY, 2012 (en)

    2012

  30. [38]

    I. G. Macdonald, Affine root systems and Dedekind’s η-function, Invent. Math. 15 (1972), 91–143. MR357528

    1972

  31. [39]

    Patnaik, Geometry of loop Eisenstein series, ProQuest LLC, Ann Arbor, MI, 2008

    Manish M. Patnaik, Geometry of loop Eisenstein series, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Yale University. MR2711828

    2008

  32. [40]

    6, 1641–

    , Local Birkhoff decompositions for loop groups and a finiteness result, Forum Math.37 (2025), no. 6, 1641–

    2025

  33. [41]

    Patnaik and Anna Puskás, Metaplectic covers of Kac-Moody groups and Whittaker functions , Duke Math

    Manish M. Patnaik and Anna Puskás, Metaplectic covers of Kac-Moody groups and Whittaker functions , Duke Math. J. 168 (2019), no. 4, 553–653. MR3916064

    2019

  34. [42]

    M. S. Raghunathan, A note on quotients of real algebraic groups by arithmetic subgroups, Invent. Math.4 (1967/68), 318–335. MR230332

    1967

  35. [43]

    Leslie Saper, Tilings and finite energy retractions of locally symmetric spaces , Comment. Math. Helv. 72 (1997), no. 2, 167–202. MR1470087

    1997

  36. [44]

    (N.S.) 21 (2015), no

    Simon Schieder, The Harder-Narasimhan stratification of the moduli stack of G-bundles via Drinfeld’s compactifi- cations, Selecta Math. (N.S.) 21 (2015), no. 3, 763–831. MR3366920

    2015

  37. [45]

    5, Mathematical Society of Japan, Tokyo, 1959

    Carl Ludwig Siegel, Zur Reduktionstheorie quadratischer Formen , Publications of the Mathematical Society of Japan, vol. 5, Mathematical Society of Japan, Tokyo, 1959. MR140486

    1959

  38. [46]

    66, American Mathematical Society, Providence, RI, 2016

    Robert Steinberg, Lectures on Chevalley groups, University Lecture Series, vol. 66, American Mathematical Society, Providence, RI, 2016. Notes prepared by John Faulkner and Robert Wilson, Revised and corrected edition of the 1968 original [ MR0466335], With a foreword by Rober...

    2016

  39. [47]

    Ulrich Stuhler, Eine Bemerkung zur Reduktionstheorie quadratischer Formen, Arch. Math. (Basel)27 (1976), no. 6, 604–610. MR424707 September 24, 2025 MANISH M. PATNAIK • E-mail : patnaik@ualberta.ca PUNYA PLABAN SATPATHY • E-mail : psatpathy81@gmail.com 70

    1976

  40. [2002]

    MR1890629

    Translated from the 1968 French original by Andrew Pressley. MR1890629

    1968

  41. [2008]

    MR2439729

    Theory and applications. MR2439729

Pith tools

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