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Geometric Bernstein Asymptotics and the Drinfeld-Lafforgue-Vinberg degeneration for arbitrary reductive groups
Algebraic Geometry · Submitted 3 Jul 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim From this description we deduce that the nearby cycles of VinBun_G correspond, under the sheaf-function correspondence, to Bernstein's asymptotics map on the level of functions.
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We define and study the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles Bun_G for an arbitrary reductive group G; its definition is given in terms of the Vinberg semigroup of G, and is due to Drinfeld (unpublished). Throughout the article we prefer to view the compactification as a canonical multi-parameter degeneration of Bun_G which we call the Drinfeld-Lafforgue-Vinberg degeneration VinBun_G. We construct local models for the degeneration VinBun_G which "factorize in families" and use them to study its singularities, generalizing results of the article [Sch1] which was confined with the case G = SL_2. The multi-parameter degeneration VinBun_G gives rise to, for each parabolic of G, a corresponding nearby cycles functor. Our main theorem expresses the stalks of these nearby cycles in terms of the cohomology of the parabolic Zastava spaces. From this description we deduce that the nearby cycles of VinBun_G correspond, under the sheaf-function correspondence, to Bernstein's asymptotics map on the level of functions. This had been speculated by Bezrukavnikov-Kazhdan [BK] and Chen-Yom Din [CY] and conjectured in a precise form by Sakellaridis [Sak2].
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We define and study the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles Bun_G for an arbitrary reductive group G; its definition is given in terms of the Vinberg semigroup of G, and is due to Drinfeld (unpublished). | contribution 0.84 | 0.62 | 0.03 | asserted 1.00 |
| s1 | Throughout the article we prefer to view the compactification as a canonical multi-parameter degeneration of Bun_G which we call the Drinfeld-Lafforgue-Vinberg degeneration VinBun_G. | contribution 0.55 | 0.55 | 0.04 | asserted 0.93 |
| s2 | We construct local models for the degeneration VinBun_G which "factorize in families" and use them to study its singularities, generalizing results of the article [Sch1] which was confined with the case G = SL_2. | contribution 0.56 | 0.95 | 0.03 | asserted 1.00 |
| s3 | The multi-parameter degeneration VinBun_G gives rise to, for each parabolic of G, a corresponding nearby cycles functor. | contribution 0.57 | 0.69 | 0.03 | asserted 1.00 |
| s4 | Our main theorem expresses the stalks of these nearby cycles in terms of the cohomology of the parabolic Zastava spaces. | result 0.99 | 0.92 | 0.89 | asserted 1.00 |
| s5 | From this description we deduce that the nearby cycles of VinBun_G correspond, under the sheaf-function correspondence, to Bernstein's asymptotics map on the level of functions. | interpretation 0.82 | 0.90 | 0.41 | asserted 1.00 |
| s6 | This had been speculated by Bezrukavnikov-Kazhdan [BK] and Chen-Yom Din [CY] and conjectured in a precise form by Sakellaridis [Sak2]. | background 0.90 | 0.06 | 0.06 | speculative 0.62 |
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Original abstract
We define and study the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles Bun_G for an arbitrary reductive group G; its definition is given in terms of the Vinberg semigroup of G, and is due to Drinfeld (unpublished). Throughout the article we prefer to view the compactification as a canonical multi-parameter degeneration of Bun_G which we call the Drinfeld-Lafforgue-Vinberg degeneration VinBun_G. We construct local models for the degeneration VinBun_G which "factorize in families" and use them to study its singularities, generalizing results of the article [Sch1] which was confined with the case G = SL_2. The multi-parameter degeneration VinBun_G gives rise to, for each parabolic of G, a corresponding nearby cycles functor. Our main theorem expresses the stalks of these nearby cycles in terms of the cohomology of the parabolic Zastava spaces. From this description we deduce that the nearby cycles of VinBun_G correspond, under the sheaf-function correspondence, to Bernstein's asymptotics map on the level of functions. This had been speculated by Bezrukavnikov-Kazhdan [BK] and Chen-Yom Din [CY] and conjectured in a precise form by Sakellaridis [Sak2].
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How the literature uses it
7 arXiv papers cite this in 79 sentences
Papers citing it · 53 sentences without a confident label
Contests it 1 paper
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We will not use this result from [ref] .
Nearby cycles on Drinfeld-Gaitsgory-Vinberg Interpolation Grassmannian and long intertwining functor · Preparations · contest 0.75
Uses its method or tool 4 papers
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Our proof of Theorem A makes use of certain local models for the principal degeneration of $_G$ which were introduced in [ref] ; we refer to Section [ref] below for their construction and main properties.
Monodromy and Vinberg fusion for the principal degeneration of the space of G-bundles · Introduction · 0.98
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To define this subcategory, we use the defect stratification on $_G,P$ defined[More precisly, [ref] constructed the defect stratification on $_G,C_P$.
Deligne--Lusztig duality on the moduli stack of bundles · Step 1 · 0.90
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The geometry of the compactification then reduces to a theorem of [ref] , and the corresponding Grothendieck functions are computed using the facts reviewed in Appendix [ref].
On an invariant bilinear form on the space of automorphic forms via asymptotics · Section · 0.86
Cites it as support 1 paper
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Namely, those relevant to $_G^γ$ and $_GG,I$ have been verified in $$ [ref], while those relevant to $_G^γ$ were either proved or sketched in [ref] .
Nearby cycles on Drinfeld-Gaitsgory-Vinberg Interpolation Grassmannian and long intertwining functor · Proofs - II · 0.96
Cites it as background 6 papers
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We first review the definition of the stack $'_G$ (denoted by $_G$ in [ref] ), which is a compactification of the morphism $_G Z(G) →_G _G$, which $Δ$ factors through.
On an invariant bilinear form on the space of automorphic forms via asymptotics · Compactifications of the diagonal morphism of $_G$ · 0.86
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Let[It was denoted by $T^+_,≥P,strict$ in [ref] .] $T^+_,≥C_P$ be its closure in $T_^+$.
Deligne--Lusztig duality on the moduli stack of bundles · Introduction · 0.80
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We can now recall the definition of the Drinfeld-Lafforgue-Vinberg degeneration $_G$ from [ref] ; the definition of the Drinfeld-Lafforgue-Vinberg compactification $_G$, which is also given in [ref] and of which $_G$ is a minor modification, is due to Drinfeld (unpublished).
Monodromy and Vinberg fusion for the principal degeneration of the space of G-bundles · Recollections – The degeneration $_G$ · 0.89
What it cites · 20 sentences without a confident label
Builds on 2 papers
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Indeed, the proof in [ref] is given in the language of mapping stacks, and the only group-theoretic input in the proof is the fact that the stabilizers of the $G ×G$-action on the fibers of the map $_0_G→T_adj^+$ are smooth; for an arbitrary reductive group $G$, this is established in [ref] .
Geometric constant term functor(s) · The compactification and the degeneration · 0.89
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Indeed, the proof in [ref] is given in the language of mapping stacks, and the only group-theoretic input in the proof is the fact that the stabilizers of the $G ×G$-action on the fibers of the map $_0_G→T_adj^+$ are smooth; for an arbitrary reductive group $G$, this is established in [ref] .
Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for SL_2 · The compactification and the degeneration · 0.89
Uses the method or tool of 1 paper
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By the exact same argument as in [ref] , [ref] , or [ref] , it suffices to prove the theorems stated in Section [ref] above on the level of the local models.
Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for SL_2 · Proofs II — Sheaves · 0.86
Cites as background 5 papers
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We will use the notation from [ref] , [ref] , and [ref] and refer the reader to these sources for details.
Periods and harmonic analysis on spherical varieties · Statements of theorems – Bernstein asymptotics · 0.93
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We will use the notation from [ref] , [ref] , and [ref] and refer the reader to these sources for details.
Inverse Satake transforms · Statements of theorems – Bernstein asymptotics · 0.93
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It is uniquely characterized by the fact that it meets each fiber of the map $v: _G →T_adj^+$ in the open $G ×G$-orbit of that fiber; i.e., for any $t ∈T_adj^+$ we have: [equation] For a Tannakian characterization of $_0_G$ we refer the reader to [ref] .
Geometric constant term functor(s) · The compactification and the degeneration · 0.90
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