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A topological approach to Soergel theory

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Roman Bezrukavnikov, Simon Riche

Representation Theory · Submitted 19 Jul 2018

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra.

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arXiv ID1807.07614 (version 2)
Pith IDLHW73FP2, Pith's permanent record of this paper and its review
CitationRoman Bezrukavnikov, Simon Riche. "A topological approach to Soergel theory." arXiv:1807.07614 (2018). https://pith.science/paper/LHW73FP2
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,345 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.08 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; .

We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra. We also obtain a description of the derived category of $T$-monodromic $\Bbbk$-sheaves on $G/U$ (where $U$, $T\subset B$ are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in $T^\vee_\Bbbk \times_{(T^\vee_\Bbbk)^W} T^\vee_\Bbbk$, where $T^\vee_\Bbbk$ is the $\Bbbk$-torus dual to $T$.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. contribution 1.00 0.96 0.28 asserted 1.00
s1 Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra. contribution 0.64 0.96 0.47 asserted 1.00
s2 We also obtain a description of the derived category of $T$-monodromic $\Bbbk$-sheaves on $G/U$ (where $U$, $T\subset B$ are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in $T^\vee_\Bbbk \times_{(T^\vee_\Bbbk)^W} T^\vee_\Bbbk$, where $T^\vee_\Bbbk$ is the $\Bbbk$-torus dual to $T$. result 0.73 0.94 0.17 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

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The paper
Original abstract

We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a "multiplicative" coinvariant algebra, and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of "Soergel modules" over this algebra. We also obtain a description of the derived category of $T$-monodromic $\Bbbk$-sheaves on $G/U$ (where $U$, $T\subset B$ are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in $T^\vee_\Bbbk \times_{(T^\vee_\Bbbk)^W} T^\vee_\Bbbk$, where $T^\vee_\Bbbk$ is the $\Bbbk$-torus dual to $T$.

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10 arXiv papers cite this in 77 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 62 sentences without a confident label

Builds on it 1 paper
  • The case $t=1$ was the main point of interest of [ref] ; in this paper, we begin to explore the categories given by other parameters $t$.

    Perverse Monodromic Sheaves · Motivation · 0.97

Uses its method or tool 2 papers
  • The $G$-action on the base point of $_G$ induces an isomorphism between $_G,w_0=L^+G/$ and the flag variety $G/B$, from which we obtain equivalences of categories [equation] Using these equivalences, we can apply [ref] ,

    Modular affine Hecke category and regular unipotent centralizer · Proof of Theorem · 0.96

  • Using the formalism of the free-monodromic completion from [ref] one can construct a canonical triangulated functor [equation] such that the diagram [equation] commutes.

    A geometric model for blocks of Frobenius kernels · Baby co-Verma modules are finitely generated · 0.94

Cites it as support 4 papers
Cites it as background 2 papers
  • (A Kostant section is an “additive” variant of the Steinberg section $Σ$; in particular the coadjoint quotient provides an isomorphism $^* ^*/$.) The same considerations as in [ref] lead to the definition of the universal centralizer group scheme $_^*$ over $^*$, see [ref] and [ref] for details.

    Modular affine Hecke category and regular centralizer · Additive Hecke categories and representations of the (additive) universal centralizer · 0.85

  • Recall that in [ref] a completed unipotently monodromic Hecke category $= D̂^b(G\((G/U)^2T^2))$ is defined in cases [ref], [ref], and in other cases in [ref] .

    Equivariant derived category of a reductive group as a categorical center · Monodromic Hecke categories. · 0.96

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