Alphabeta MathSoergel Intersection Forms and Hodge Theory
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2 · Logical flowchart
3 · Items
Definition
The real type-A Soergel category by scalar extension
Lemma
The Hecke trace form, its values on the Kazhdan-Lusztig basis, and the standard pairing
Lemma
Indecomposable type-A Soergel bimodules are classified by their normalized top support
Lemma
Normalized type-A Soergel bimodules are self-dual and have length degree bounds
Definition
The reduced Soergel module and the dominant Lefschetz operator
Lemma
Indecomposable labels, characters and filtration ranks agree after real scalar extension
Definition
The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction
Definition
Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR
Definition
Minimal complexes and the perverse truncations in the real type-A Soergel category
Lemma
The Lefschetz operator factors through deleted Bott-Samelson subwords, with positive rescalings
Lemma
Invariant forms on an indecomposable Soergel bimodule are unique up to scalar, non-degenerate, and agree with the restricted Bott-Samelson form
Lemma
Hard Lefschetz gives the primitive decomposition, orthogonality and the signature formula
Definition
The deformed Lefschetz operator L_ζ on Bₓ Bₛ
Lemma
Radical quotients, minimal complexes and perverse filtrations in the Soergel category
Theorem
Positive Rouquier complexes are Delta-split
Lemma
Signature constancy in a Lefschetz family, the L-stable subspace lemma and the weak-Lefschetz substitute
Definition
The local intersection form on Hom(B_y, Bₓ Bₛ)
Lemma
Tensoring a direct sum of Hodge-Riemann bimodules with Bₛ preserves the signs
Lemma
Right-descent Soergel bimodules descend to simple invariants under the character hypothesis
Theorem
Minimal positive Rouquier complexes are linear under the lower Soergel induction
Lemma
Undeformed Lefschetz and Hodge-Riemann restrict to the next Soergel summand
Theorem
Non-degeneracy of the local intersection forms detects Soergel's conjecture one step higher
Lemma
Hodge-Riemann relations hold for large positive deformation, and the deformation principle
Lemma
Embedding theorem: local intersection forms are isometric to primitive Lefschetz forms
Theorem
Minimal positive Rouquier complexes carry the induced Hodge-Riemann signs
Lemma
The minimal Rouquier differential supplies the injectivity input of the weak-Lefschetz substitute
Theorem
Hard Lefschetz for the deformed operator on Bₓ Bₛ, in the three cases
Theorem
Simultaneous induction: Soergel's conjecture, hard Lefschetz and the Hodge-Riemann relations hold for all elements
Theorem
Soergel's character conjecture for type A
Corollary
Positivity of Kazhdan-Lusztig polynomials, mu-coefficients and structure constants
Theorem
Existence and uniqueness of the Kazhdan–Lusztig basis
Sources
- Elias-Williamson, The Hodge theory of Soergel bimodules (arXiv:1212.0791v2, final version; Ann. of Math. 180 (2014) 1089-1136)
- Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln ueber Polynomringen (arXiv:math/0403496, published J. Inst. Math. Jussieu 6 (2007) 501-525)
- Riche, La theorie de Hodge des bimodules de Soergel (d'apres Soergel et Elias-Williamson), Seminaire Bourbaki no. 1139 (arXiv:1711.02464, Octobre 2017)
- Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln, Satz 6.14 and its full proof
- Elias-Williamson, The Hodge theory of Soergel bimodules, Sections 3.5 and 6.1
- Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln, Satz 6.14(3) and its proof
- Elias-Williamson, The Hodge theory of Soergel bimodules, Sections 3.3-3.6, Lemma 3.8
- Gobet, Twisted filtrations of Soergel bimodules and linear Rouquier complexes (arXiv:1601.00339; J. Algebra 484 (2017) 275-309)
- Libedinsky-Williamson, Standard objects in 2-braid groups (arXiv:1205.4206; Proc. London Math. Soc. 109 (2014) 1264-1280)
- G. Lusztig, Hecke Algebras with Unequal Parameters (revised book version, arXiv:math/0208154v2) — Theorem 5.2, Proposition 5.4 and §5.6 (printed pp. 27–30): bar-invariant triangular basis, coefficient degree/parity and inverse-index symmetry; the complete argument was read. The split-parameter convention is translated to this page by inverting Lusztig's parameter.
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, arXiv:1212.0791 (45 pp.) — Corollary 1.2(1) (printed p. 5): coefficientwise positivity; §3.2 and Remark 3.2 (printed pp. 15–16): the Hecke normalization and characterization of the Kazhdan–Lusztig basis.