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Non-degeneracy of the local intersection forms detects Soergel's conjecture one step higher

Facts & Assumptions

Given: Elements $x, s$ with $x s > x$ , the assumption $S (y)$ for every $y < x s$ , the module $M = B_{x}^{R} B_{s}^{R}$ , the KL multiplication formula ${\underline{H}}_{x} {\underline{H}}_{s} = {\underline{H}}_{x s} + \sum_{y < x s} μ_{y} {\underline{H}}_{y}$ with integer coefficients $μ_{y}$ , and for every $y < x s$ the finite-dimensional real space $E_{y} = {Hom}^{0} (B_{y}^{R}, M)$ with its local intersection form $(-, -)_{y}^{x, s}$ (The local intersection form on Hom(B_y, B_x B_s), Multiplication by a generator in the Kazhdan–Lusztig basis).

[F1]

Under $S (y)$ for all $y < x s$ , the comparison lemma gives the decomposition $M = B_{x s} \oplus ⨁_{y < x s} B_{y}^{R, a_{y}}$ with nonnegative integers $a_{y}$ and no grading shifts, with the top summand $B_{x s}$ occurring exactly once; moreover $ch (B_{z}) = {\underline{H}}_{z}$ and $h_{\nabla} (B_{z}) = d (h_{Δ} (B_{z})) = {\underline{H}}_{z}$ for every $z < x s$ , while the characters of $B_{x} B_{s}$ are computed multiplicatively as $h_{Δ} (M) = h_{\nabla} (M) = {\underline{H}}_{x} {\underline{H}}_{s}$ (Indecomposable labels, characters and filtration ranks agree after real scalar extension, The type-A standard character is multiplicative).

[F2]

The graded rank of $Hom (B_{y}^{R}, M)$ is $⟨ h_{Δ} (B_{y}^{R}), h_{\nabla} (M) ⟩$ for the standard pairing with $⟨ H_{u}, H_{v} ⟩ = δ_{u v}$ on the normalized standard basis, and the trace-form values on the Kazhdan-Lusztig basis are $({\underline{H}}_{u}, {\underline{H}}_{v}) \in v Z [v]$ for $u \neq v$ and $\in 1 + v Z [v]$ for $u = v$ ; the Kazhdan-Lusztig coefficients satisfy $p_{u, w} \in v Z [v]$ for $u < w$ (The type-A Soergel Hom formula, The Hecke trace form, its values on the Kazhdan-Lusztig basis, and the standard pairing).

[F3]

The local intersection form on $E_{y}$ is symmetric and bilinear, and it is non-degenerate if and only if the degree-zero composition pairing $E_{y} \times {Hom}^{0} (M, B_{y}^{R}) \to {End}^{0} (B_{y}^{R}) = R$ is a perfect pairing; the adjoint identification between the two Hom spaces preserves the rank (The local intersection form on Hom(B_y, B_x B_s)).

[F4]

Non-degeneracy of a symmetric bilinear form on a finite-dimensional real space is equivalent to the absence of a radical, i.e. to the form having maximal rank $\dim E_{y}$ (Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR, Invariant forms on an indecomposable Soergel bimodule are unique up to scalar, non-degenerate, and agree with the restricted Bott-Samelson form).

Statement

Fix $x, s$ with $x s > x$ and assume $S (y)$ for every $y < x s$ . Put $M = B_{x}^{R} B_{s}^{R}$ and write the Kazhdan-Lusztig multiplication formula ${\underline{H}}_{x} {\underline{H}}_{s} = {\underline{H}}_{x s} + \sum_{y < x s} μ_{y} {\underline{H}}_{y}$ with integer constants $μ_{y}$ . For each $y < x s$ let $E_{y} = {Hom}^{0} (B_{y}^{R}, M)$ carry the local intersection form of The local intersection form on Hom(B_y, B_x B_s). Then:

(1) $\dim_{R} E_{y} = μ_{y}$ , while the full graded Hom module has graded rank $⟨ {\underline{H}}_{y}, {\underline{H}}_{x} {\underline{H}}_{s} ⟩$ , generally not $μ_{y}$ ;

(2) the following are equivalent: $S (x s)$ ; non-degeneracy of every local intersection form on $E_{y}$ , $y < x s$ ; perfection of every degree-zero composition pairing $E_{y} \times {Hom}^{0} (M, B_{y}^{R}) \to {End}^{0} (B_{y}^{R}) = R$ ; and equality of the actual multiplicity $a_{y}$ of $B_{y}^{R}$ in $M$ with $μ_{y}$ for every $y < x s$ .

The unconditional top summand is $B_{x s}$ with multiplicity one, and under $S (< x s)$ all remaining summands are unshifted $B_{y}^{R}$ . No positivity of Kazhdan-Lusztig polynomials or conjecture at $x s$ is assumed.

Proof

technique · direct; constant-term Hom formula, character comparison and the multiplicity pairing

1.1givenF1F2

The dimension of $E_{y}$ . By [F1] and [F2] the graded rank of $Hom (B_{y}^{R}, M)$ equals $⟨ h_{Δ} (B_{y}^{R}), h_{\nabla} (M) ⟩ = ⟨ {\underline{H}}_{y}, {\underline{H}}_{x} {\underline{H}}_{s} ⟩$ . Expanding ${\underline{H}}_{x} {\underline{H}}_{s} = {\underline{H}}_{x s} + \sum_{z < x s} μ_{z} {\underline{H}}_{z}$ in the Kazhdan-Lusztig basis and using the trace values of [F2], this rank equals $\sum_{z} μ_{z} ({\underline{H}}_{y}, {\underline{H}}_{z})$ up to the top term, whose contribution is $δ_{y, x s} = 0$ because $y < x s$ ; each $({\underline{H}}_{y}, {\underline{H}}_{z})$ lies in $v Z [v]$ for $z \neq y$ and in $1 + v Z [v]$ for $z = y$ . Hence the constant term of the graded rank is $μ_{y}$ , and since the graded rank lies in $Z [v]$ with nonnegative $v$ -powers, the degree-zero dimension is exactly the constant term: $\dim_{R} E_{y} = μ_{y}$ . This is the first assertion of (1); the full graded rank is the polynomial displayed and has higher-degree terms in general, so it is not in general equal to the integer $μ_{y}$ .

2.1step 1.1F1F3

The top summand and the composition pairing. By [F1], $M = B_{x s} \oplus ⨁_{y < x s} B_{y}^{R, a_{y}}$ with the top summand of multiplicity one. For $y < x s$ , consider a composite $B_{y}^{R} \to M \to B_{y}^{R}$ factoring through a summand other than a copy of $B_{y}^{R}$ ; it cannot be a nonzero scalar multiple of the identity, because rescaling such a composite would exhibit a retraction splitting $B_{y}^{R}$ off that other summand, contradicting the uniqueness of the indecomposable label. Hence only the $a_{y}$ copies of $B_{y}^{R}$ contribute to the scalar-coordinate pairing, and the pairing matrix is the standard evaluation pairing of rank $a_{y}$ ; by the adjoint identification of [F3] the local intersection form has the same rank.

3.1step 2.1F3F4

Non-degeneracy and the multiplicity equalities. By step 2.1 the local form on $E_{y}$ is non-degenerate if and only if its rank equals $\dim E_{y}$ , i.e. $a_{y} = μ_{y}$ ; by [F4] this is equivalent to non-degeneracy in the sense of trivial radical. Therefore non-degeneracy of every local form for all $y < x s$ is equivalent to the system of equalities $a_{y} = μ_{y}$ for all $y < x s$ , and the same rank comparison identifies this with perfection of the composition pairings.

4.1step 1.1step 3.1F1∎

The character equivalence and conclusion. By additivity of the character and $ch (B_{z}) = {\underline{H}}_{z}$ for $z < x s$ , the character of $B_{x s}$ is $ch (M) - \sum_{y < x s} a_{y} {\underline{H}}_{y} = {\underline{H}}_{x s} + \sum_{y < x s} (μ_{y} - a_{y}) {\underline{H}}_{y}$ . Since the Kazhdan-Lusztig basis is a basis of the Hecke algebra, $ch (B_{x s}) = {\underline{H}}_{x s}$ , i.e. $S (x s)$ , holds if and only if $μ_{y} = a_{y}$ for every $y < x s$ . Combining with step 3.1 gives the equivalence of all four conditions in (2).

Remarks

The constant-term computation is the only place where the actual Hom dimension enters; the full graded rank is deliberately kept as a polynomial, since it is generally not the integer $μ_{y}$ .
The equivalence proved here is Soergel's omnibus lemma in the form needed for the simultaneous induction; no positivity of Kazhdan-Lusztig polynomials is used, and the argument is finite and choice-free.

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Sources

Elias-Williamson, The Hodge theory of Soergel bimodules (arXiv:1212.0791v2, final version; Ann. of Math. 180 (2014) 1089-1136) (standard reference, not scraped)
Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln ueber Polynomringen (arXiv:math/0403496, published J. Inst. Math. Jussieu 6 (2007) 501-525) (standard reference, not scraped)
Riche, La theorie de Hodge des bimodules de Soergel (d'apres Soergel et Elias-Williamson), Seminaire Bourbaki no. 1139 (arXiv:1711.02464, Octobre 2017) (standard reference, not scraped)