Relative hard Lefschetz for Soergel bimodules

We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that the structure constants of the Kazhdan-Lusztig basis are unimodal. We explain why the relative hard Lefschetz theorem implies that the tensor category associated by Lusztig to any 2-sided cell in a Coxeter group is rigid and pivotal.


Introduction
Let (W, S) denote a Coxeter system and H its Hecke algebra. It is an algebra over Z[v ±1 ] with standard basis {H x | x ∈ W } and Kazhdan-Lusztig basis {H x | x ∈ W }. The Kazhdan-Lusztig positivity conjectures are the statements: (1) ("positivity of Kazhdan-Lusztig polynomials") if we write H x = h y,x H y , then h y,x ∈ Z ≥0 [v]; (2) ("positivity of structure constants") if we write H x H y = µ z x,y H z then µ z x,y ∈ Z ≥0 [v ±1 ]. These conjectures have been known since the 1980s for Weyl groups of Kac-Moody groups [KL80,Spr82], using sophisticated geometric technology. More recently in [EW14] the authors proved these conjectures algebraically for arbitrary Coxeter systems by establishing Soergel's conjecture.
Let us briefly recall the setting of Soergel's conjecture. For a certain reflection representation h of (W, S) over the real numbers, Soergel constructed a category B of Soergel bimodules, which is a full subcategory of the category of graded R-bimodules, where R denotes the polynomial functions on h. The category of Soergel bimodules B is monoidal under tensor product of bimodules, and is closed under grading shift. Soergel .) In proving this isomorphism, Soergel constructed certain bimodules B x for each x ∈ W which give representatives for all indecomposable Soergel bimodules (up to isomorphism and grading shift). Soergel's conjecture is the statement that ch([B x ]) = H x , which immediately implies the Kazhdan-Lusztig positivity conjectures. (Property (1) follows because the coefficient of H y in ch([B]) is given by the graded dimension of a certain hom space. Property (2) follows because µ z x,y gives the graded multiplicity of B z as a summand in B x ⊗ R B y .) The geometric techniques used to understand the Kazhdan-Lusztig basis yield another remarkable property of the structure constants µ z x,y . Using duality, one can show that µ z x,y is preserved under swapping v and v −1 . The quantum numbers for m ≥ 1 give a Z-basis for those elements of Z[v ±1 ] preserved under swapping v and v −1 . A folklore conjecture states: 1 3) ("unimodality of structure constants") if we write µ z x,y = m≥1 a m [m], then a m ≥ 0 for all m.
(In other words, each µ z x,y is the character of a finite-dimensional sl 2 (C)representation .) In geometric settings unimodality follows from the relative hard Lefschetz theorem of [BBD82]. Recall that the relative hard Lefschetz theorem states that if f : X → Y is a projective morphism of complex algebraic varieties and if η is a relatively ample line bundle on X then for all i ≥ 0, η induces an isomorphism: (Here IC X denotes the intersection cohomology complex on X and p H i denotes perverse cohomology .) In this paper we prove unimodality for all Coxeter groups, by adapting the relative hard Lefschetz theorem to the context of Soergel bimodules.
Inside the category of Soergel bimodules we consider the full subcategory p B consisting of direct sums of the indecomposable self-dual bimodules B x without shifts. We call p B the subcategory of perverse Soergel bimodules. Soergel's conjecture implies that each B ∈ p B admits a canonical isotypic decomposition for certain real (degree zero) vector spaces V x . If a Soergel bimodule is not perverse, its decomposition into indecomposable summands of the form B x (i) is not canonical. However, there is a canonical filtration on any Soergel bimodule called the perverse filtration, whose i-th subquotient has indecomposable summands of the form B x (−i) for some x ∈ W . Taking the subquotients of this filtration and shifting them appropriately, one obtains for each i the perverse cohomology functor Remark 1. 1. The category B is an analogue of semi-simple complexes, p B is an analogue of the category of semi-simple perverse sheaves and H i is an analogue of the perverse cohomology functor.
This main result of this paper is the following: Theorem 1.2. (Relative hard Lefschetz for Soergel bimodules) Let x, y ∈ W be arbitrary and fix ρ ∈ h * dominant regular (i.e. ρ, α ∨ s > 0 for all s ∈ S). The map induces an isomorphism (for all i ≥ 0) Remark 1. 3. A stronger version of the above theorem, involving iterated tensor products of indecomposable Soergel bimodules of arbitrary length is still open (see Conjecture 3.4). It is amusing that establishing Conjecture 3.4 for Bott-Samelson bimodules (i.e., when all x i ∈ S, in the notation of Conjecture 3.4) was the authors' original plan of attack to settle Soergel's conjecture. This remains a very interesting Hodge theoretic statement that we cannot prove! As was true in our previous work on hard Lefschetz type theorems for Soergel bimodules [EW14,Wil14], the inductive proof we use to establish our main theorem actually requires proving a stronger statement, analogous to the relative Hodge-Riemann bilinear relations [dCM05]. That is, we must calculate the signatures of certain forms on the multiplicity spaces of Relative hard Lefschetz for Soergel bimodules also has important consequences for certain tensor categories associated to cells in Coxeter groups. Recall that to any two sided cell c ⊂ W in a finite or affine Weyl group Lusztig has associated a tensor category, which categorifies the J-ring of c. These categories (for finite Weyl groups) are fundamental for the representation theory of finite reductive groups of Lie type: by results of Bezrukavnikov, Finkelberg and Ostrik [BFO12] and Lusztig [Lus15], their (Drinfeld) centers are equivalent to the braided monoidal category of unipotent character sheaves corresponding to c.
Given any two sided cell c ⊂ W in an arbitrary Coxeter group Lusztig has generalised his construction to yield a monoidal category J . (Note that J is only "locally unital" unless c contains finitely many left cells, and the existence of a unit relies on a conjecture in general, see Remark 5. 1.) In the last section of this paper we explain why Theorem 1.2 implies that J is rigid and pivotal (see Theorem 5.2). (The rigidity was conjectured by Lusztig [Lus15, § 10] when W is finite). This is an important step towards the study of "unipotent character sheaves" associated to any Coxeter system.
By a theorem of [Müg03,ENO05], rigidity of J implies that the (Drinfeld) center of J is a modular tensor category. We expect cells in noncrystallographic Coxeter groups to provide many new examples of modular tensor categories (see [Ost14,5.4 where m st denotes the order (possibly ∞) of st ∈ W . 2 We have an action of W on h given by the formula Let R be the ring of polynomial functions on h, graded so that the linear terms h * have degree 2. It comes equipped with an action of W . Define a graded R-bimodule B s = R ⊗ R s R(1) for each s ∈ S, where R s denotes the s-invariant polynomial subring. We use the standard convention for grading shifts, so that the (1) above indicates that the minimal degree element 1 ⊗ 1 lives in degree −1. Given two graded R-bimodules B, B ′ their tensor product over R is denoted BB ′ := B ⊗ R B ′ . For a sequence w = (s 1 , s 2 , . . . , s d ) with s i ∈ S, the tensor product Soergel proved in [Soe07] that, when x is a reduced expression for an element x ∈ W , there is a unique indecomposable direct summand B x ⊕ ⊂ BS(x) which is not isomorphic to a summand of a shift of any Bott-Samelson bimodule corresponding to a shorter reduced expression. Moreover, this summand does not depend on the reduced expression of x, up to non-canonical isomorphism. (Using the main theorem of [EW14] one can make this isomorphism canonical.) Note that the two notations for B s agree.
Let B denote the full subcategory of graded R-bimodules whose objects are finite direct sums of grading shifts of summands of Bott-Samelson bimodules. The objects in this category B are known as Soergel bimodules, and the bimodules {B x } x∈W give a complete list of non-isomorphic indecomposable objects up to grading shift. Because Bott-Samelson bimodules are 2 The choice of roots and coroots plays a significant role in this paper, but only up to positive rescaling; what is important (in order that we may cite certain results from [Soe07] and [EW14]) is that our representations is reflection faithful [Soe07] and that there be a well-defined notion of positive roots. If the reader prefers, they may also take the representation given by a realisation of a generalised Cartan matrix.
closed under tensor product, B is as well, and inherits its monoidal structure from R-bimodules.
If B is a Soergel bimodule we will often use the symbol B to denote the identity morphism on B. For example, if f : B ′ → B ′′ is a morphism then Bf : BB ′ → BB ′′ denotes the tensor product of the identity on B with f . Similarly, given r ∈ R of degree m, rB (resp. Br) denotes the morphism B → B(m) given by left (resp. right) multiplication by r. We say that a pairing is non-degenerate if one (or equivalently both) of these morphisms is an isomorphism. 4 A (non-degenerate) form on a Soergel bimodule is a (non-degenerate) Given a map f : B → B ′ (m) between polarized Soergel bimodules its adjoint is the unique map f * :

2.2.
Perverse cohomology and graded multiplicity spaces. All morphisms between indecomposable self-dual Soergel bimodules are of nonnegative degree, and those of degree zero are isomorphisms. That is: Hom(B x , B y (m)) = 0 for x, y ∈ W and m < 0, (2. 2) These fundamental Hom-vanishing statements are equivalent to Soergel's conjecture (see the paragraph following [EW14, Theorem 3.6]).
A Soergel bimodule B is perverse if it is isomorphic to a direct sum of indecomposable bimodules B x without shifts. We denote by p B the full subcategory of perverse Soergel bimodules. As a consequence of (2.2), any perverse Soergel bimodule admits a canonical decomposition for some finite dimensional real vector spaces V x . (Concretely, one has V x = Hom(B x , B).) The rest of this section is dedicated to understanding what replaces this multiplicity space V x in case the bimodule B in question is not perverse.
By the classification of indecomposable bimodules, every Soergel bimodule splits into a direct sum of shifts of perverse bimodules, but this splitting is not canonical. However, it is a consequence of (2.1) and (2.2) that B admits a unique functorial (non-canonically split) filtration, the perverse filtration, whose subquotients isomorphic to a shift of a perverse Soergel bimodule, see [EW14,§6.2]. Before discussing the details, it is worth illustrating this subtle point in examples.
is canonical up to a scalar. After all, it is easy to confirm from (2.1) and and is one-dimensional in degree −1. The same can be said about the degree −1 inclusion map, that is, the map B s (+1) → B s B s . However, the degree +1 projection map B s B s → B s (+1) (resp. the degree +1 inclusion map B s (−1) → B s B s ) is not canonical; adding to it an R-multiple of the degree −1 projection map will give another valid projection map. Said another way, B s (+1) is a canonical submodule, and B s (−1) a canonical quotient, and this filtration of B s B s splits, but not canonically.
it has a one-dimensional subspace arising as the composition of the canonical projection to B sts (−1) followed by a non-split map B sts (−1) → B st , and any morphism not in this one-dimensional subspace will serve as a projection map to B st . This example is meant to loudly proclaim that even what appears to be an "isotypic component," such as the summand B st which is the only one of its kind, is not canonically a direct summand, owing to the presence of other summands with lower degree shifts.
For any i ∈ Z, define B ≤i (resp. B >i ) to be the full additive subcategory of B consisting of bimodules which are isomorphic to direct sums of B x (m) with m ≥ −i (resp. m < −i). In formulas: Similarly we define B <i and B ≥i . We have p B = B ≥0 ∩B ≤0 . We can rephrase (2.2) as the statement: by split inclusions such that τ ≤i B ⊂ B ≤i and B/τ ≤i B ∈ B >i , see [EW14,§6.2]. This is a direct consequence of (2.4). If f : B → B ′ is a morphism then f (τ ≤i B) ⊂ τ ≤i B ′ . We have: Dually, every Soergel bimodule has a unique perverse cofiltration where every arrow is a split surjection, each τ ≥i B ∈ B ≥i and the kernel of B ։ τ ≥i B belongs to B <i . We have: The perverse cohomology of a Soergel bimodule B is for certain finite dimensional vector spaces H i z (B) . We have a non-canonical isomorphism of Soergel bimodules, and hence a degree m map grf from grB to grB ′ . For any z ∈ W this induces a map of graded vector spaces. To simplify notation, we use f to denote all these maps: f , grf , gr z f for all z ∈ W . We refer to the maps grf and gr z f as the maps induced on perverse cohomology.
The following triviality is important later: then f induces the zero map on perverse cohomology. In particular, this applies to the map given by left or right multiplication by any positive-degree polynomial in R on a Soergel bimodule B.
Proof. Only the second sentence requires proof. The perverse filtration is a filtration by R-bimodules. If r ∈ R is homogenous of degree d > 0 then multiplication by r on the left (resp. right) induces a map (see (2.5)) Therefore, the hypothesis of the lemma applies to multiplication by r.

Polarizations of Soergel bimodules.
In [EW14,§3.4, see also Corollary 3.9], the Bott-Samelson bimodule BS(w) was equipped with a nondegenerate form called the intersection form. By restriction, one obtains a form on any summand of a Bott-Samelson bimodule. By [EW14, Lemma 3.7], there is, up to an invertible scalar, a unique non-zero form on an indecomposable Soergel bimodule B x (this statement is equivalent to Soergel's conjecture), and it is non-degenerate. Thus, letting x be any reduced expression for x, the restriction of the intersection form to B x ⊕ ⊂ BS(x) is non-zero, hence is non-degenerate and hence is a polarization of B x . For all x ∈ W we fix a reduced expression x of x and an embedding B x ⊂ BS(x), and hence a polarization −, − Bx on B x . We refer to −, − as the intersection form on B x . The intersection form has the following important positivity property: If B and B ′ are two polarized Soergel bimodules, we define a form on BB ′ by the formula It is an exercise to confirm that the induced form on BB s is defined precisely in this fashion.
The induced form on BB ′ is non-degenerate, and thus is a polarization of BB ′ .
By iteration, we have an induced form on any tensor product of the form B x 1 B x 2 · · · B xm , which we continue to call the intersection form. One could also view B x 1 · · · B xm as a summand (via the tensor products of our fixed embeddings) of a Bott-Samelson bimodule BS(w), where w is a concatenation of our chosen reduced expression for each x i . The induced form agrees with the restriction of the intersection form on BS(w) to this summand. All tensor products of the form B x 1 B x 2 . . . B xm are always assumed to be polarized with respect to their intersection form.
Let (B, −, − B ) be a polarized Soergel bimodule. If B is also perverse then by considering the the isotypic decomposition (see ( We say that B is positively polarized if B = 0 or the following conditions are satified: (1) B is perverse and vanishes in even or odd degree (because B x is non-zero in degree −ℓ(x), the second condition is equivalent to the existence of q ∈ {0, 1} such that V x = 0 for all x with ℓ(x) of the same parity as q); )/2 times a positive definite form, for all y ∈ W . The canonical example of a positively polarized Soergel bimodule is given by the following lemma: , Proposition 6.12). Suppose that y ∈ W and s ∈ S with ys > y (resp. sy > y). Then B y B s (resp. B s B y ), equipped with its interesection form, is positively polarized. then we deduce from the functoriality of the perverse filtration that: and (2.9) tells us that this pairing is non-degenerate. By (2.1) the canonical decompositions . Applying (2.1) again we conclude that (2.10) is completely determined by the non-degenerate bilinear pairing on the vector spaces The left hand side is (a summand of) the pairing in (2.10) between H i (B) and H −i (B), and the right hand side is the pairing in (2.11) multiplied by the intersection form on B z . Reassembling this data, we conclude that −, − descends to a symmetric non-degenerate form −, − : grB × grB → R. and that this form is determined by the symmetric non-degenerate graded bilinear forms −, − : Here is another important triviality:

Relative hard Lefschetz and Hodge-Riemann
3. 1. Statement. We fix once and for all a dominant regular ρ ∈ h * , that is, an element such that ρ, α ∨ s ≥ 0 for all s ∈ S. Let x := (x 1 , . . . , x m ) be a sequence of elements in W , and fix scalars a := (a 1 , . . . , a m−1 ) ∈ R m−1 . Consider the operator In words, L a is the sum of the operators of multiplication by a i ρ in the gap between B x i and B x i+1 .
We have explained that to any z ∈ W we may associate a graded vector space (1) a symmetric graded non-degenerate form −, − V • obtained from the intersection form on B x 1 . . . B xm ; (2) a degree two Lefschetz operator L a : V • → V •+2 obtained by taking perverse cohomology of L a .
Remark 3. 1. The operator L a involves only internal multiplication by polynomials. One could also consider the Lefschetz operator L a + a 0 ρ · (−) + a m (−) · ρ which includes multiplication on the left and right. However, as observed in Lemma 2.4, left and right multiplication by polynomials act trivially on perverse cohomology, so this does not affect the degree 2 operator on V • .
We say that L a satisfies relative hard Lefschetz if for any d ≥ 0, L a induces an isomorphism: We say that L a satisfies relative Hodge-Riemann if L a satisfies relative hard Lefschetz and the restriction of the Lefschetz form Note that relative hard Lefschetz and relative Hodge-Riemann are both statements about H • z which are required to hold for all z ∈ W .
Remark 3. 2. The sign (−1) ε(x,z,d) might appear mysterious. The following is a useful mnemonic. Set B := B x 1 . . . B xm and consider the finite dimensional graded vector space We have a non-canonical isomomorphism Now ε(x, z, d) has the following meaning: it is half the difference between the smallest non-zero degree in on the right hand side (i.e. −ℓ(z) − d) and the smallest non-zero degree in B (i.e. − ℓ(x i ) ). In this way one may see that the above definition is compatible with the signs predicted by Hodge theory in the geometric setting (see [dCM05] and [Wil,Theorem 3.12], where the signs are made explicit).
RHR(x 1 , . . . , x m ) : L a satisfies relative Hodge-Riemann for all a := (a 1 , . . . , a m−1 ) ∈ R m−1 >0 . As always, it is implicitly assumed in these statements that all tensor products of the form B x 1 . . . B xm are equipped with their intersection form.
The main theorem of this paper is: For any x, y ∈ W , RHR(x, y) holds.
More generally, relative Hodge-Riemann should hold for any operator of the form where ρ 1 , . . . , ρ m−1 is any sequence of dominant regular elements. (Such elements span the cone of relatively ample classes in the Weyl group case. ) For the conjecture above, one sets ρ i = a i ρ.
Proof. The only nonvanishing H • z (B x ) occurs when z = x, and this multiplicity space is concentrated in degree zero. Thus RHL(x) is trivial, and RHR(x) is equivalent to the statement that the form H 0 x (B x )×H 0 x (B x ) → R is positive definite, which holds by Lemma 2.9.
Proof. Let us compare RHL(x 1 , x 2 , . . . , x m ) and RHL(x 1 , . . . , x m , id). Because B = B x 1 · · · B xm = B x 1 · · · B xm B id , the multiplicity spaces H • z (B) being studied are the same. The operator L a on B is different, because in the latter case, one is also permitted to multiply by a m ρ in the slot before the final B 1 . However, this is equal to right multiplication by a m ρ, which acts trivially on perverse cohomology. See Lemma 2.4 and Remark 3. 1

3.4.
Structure of the proof. Let us outline the major steps in the proof of Theorem 3.3, which will be carried out in the rest of this paper. The proof is by induction on ℓ(x) + ℓ(y) and then on ℓ(y). More precisely, for integers M and N , consider the statements: RHR(x ′ , s, y ′ ) holds, for all s ∈ S, whenever either  RHR(< x, s,ẏ) + RHR(x, s, <ẏ) ⇒ RHL(x, s,ẏ) (3. 2) We now distinguish two cases. If xs > x then an easy limit argument (Proposition 4.11) gives: If xs < x then a more complicated limit argument (Proposition 4.13) allows us to reach essentially the same conclusion: RHR(x,ẏ) + RHL(x, s,ẏ) ⇒ RHR(x, s,ẏ).  Finally, if x, y ∈ W and t ∈ S is such that ℓ(x) + ℓ(y) + 1 = M and ℓ(y) = N then as in (3. 2) we deduce: RHR(< x, t, y) + RHR(x, t, < y) ⇒ RHL(x, t, y). Putting these two steps together we deduce: We conclude by induction that X M,M , Y M,M hold for all M . This reduces the proof of the theorem to the propositions listed above.

The proof
4. 1. Hodge-Riemann implies hard Lefschetz. In [EW14] it was observed that homological algebra in the homotopy category of Soergel bimodules can be used to imitate the weak Lefschetz theorem. This is the key step to deduce the hard Lefschetz theorem by induction. In this section we show that the same idea is useful for studying relative hard Lefschetz.
Recall that B denotes the category of Soergel bimodules. Let denote its bounded homotopy category. As in [EW14, §6.1] we denote the cohomological degree of an object by an upper left index, so as not to get confused with the grading. Thus, an object in K is a complex with each i F ∈ B. We denote by (K ≤0 , K ≥0 ) the perverse t-structure on K (see [EW14,§6.3

]).
Lemma 4. 1 ) be a complex supported in non-negative homological degrees, and suppose that F ∈ K ≥0 . Then the induced map Proof. Because F ∈ K ≥0 then by definition we can find an isomorphism of complexes Only the summand F c contributes to H i ( 0 F ) for i < 0, but the first differential in a contractible complex is a split injection.
Given any x ∈ W we denote by a fixed choice of minimal complex for the Rouquier complex (unique up to isomorphism), see [EW14,§6.4]. The following lemma shows that tensor product with F x is left t-exact.

Proof.
Because F x is a tensor product of various F s , s ∈ S, it is enough to prove the lemma for x = s. That (−)⊗F s preserves K ≥0 is proven in [EW14, Lemma 6.6]; the proof deduces the general statement from [EW14, Lemma 6.5], which states that B x F s ∈ K ≥0 for all x ∈ W and s ∈ S. The same proof shows that F s B x ∈ K ≥0 , and consequently that F s ⊗ (−) preserves K ≥0 .
The following proposition is fundamental for what follows. (In rough form it appears first in [EW14] as Theorem 6.9, Lemma 6.15 and Theorem 6.21.)

between positively polarized Soergel bimodules such that
(1) all summands of F are isomorphic to B z with z < x; (2) d x is isomorphic to the first differential on a Rouquier complex; Except for part (2) this proposition is [Wil14, Proposition 7.14]. However the reader may easily check that the inductive proof of [Wil14, Proposition 7.14] goes through if one adds the inductive assumption "d x is isomorphic to the first differential on a Rouquier complex". (Indeed, the proof mimics tensoring with a complex isomorphic to the Rouquier complex F s to carry out the induction .) Exchanging left and right actions gives: (1) all summands of G are isomorphic to B z with z < y; (2) d y is isomorphic to the first differential on a Rouquier complex; (3) if d * y : G → B y (1) denotes the adjoint of d, then d * y • d y = ρB y − B y (y −1 ρ) Putting these three statements together gives: Proposition 4. 5. Consider the map Here, d x and F are as in Proposition 4.3, and d y and G are as in Proposition 4. 4. Then (1) the induced map Proof. The first claim follows by noticing that f is isomorphic to the first differential on a Rouquier complex representing Because F x F y ∈ K ≥0 the first claim in the lemma follows from Lemma 4. 1.
The adjoint of f is given by the matrix which is the second claim in the lemma.

Similarly we have:
Proposition 4.6. Fix a, b > 0 and consider the map

Then
(1) the induced map The argument for (2) is the same as for the previous proposition.
It remains to show part (1). Note that g a,b is the first differential on a complex representing and so F x B s F y ∈ K ≥0 by Lemma 4. 2. Now (1) follows from Lemma 4. 1.
The following two propositions explain the title of this section.
Remark 4.8. This proposition is an instance of the philosophy that HR in dimension ≤ n − 1 implies HL in dimension n.
Proof. Let us keep the notation in the statement of Proposition 4.5. We assume that B x B y is standardly polarized and E is polarized with the induced form. Fix z ∈ W and consider the graded vector spaces Also, the maps f, f * of Proposition 4.5 induce maps (again by taking perverse cohomology) These maps are morphisms of R[L]-modules. We have: (1) f is injective in degrees < 0, by Proposition 4.5(1).
( Proposition 4. 9. Fix x, y ∈ W and s ∈ S and suppose HR(x ′ , s, y) and HR(x, s, y ′ ) hold for all x ′ < x, y ′ < y. Then HL(x, y) holds. Proof. The proof is the same as that of the previous proposition, replacing Proposition 4.5 with Proposition 4.6.

4.2.
Signs via limit arguments. In this section we will repeatedly appeal to the principle of conservation of signs, which states that a continuous family of non-degenerate symmetric forms on a real vector space has constant signature. The following lemma, which was one of the key techniques used by de Cataldo and Migliorini in their proof of the Hodge-Riemann bilinear relations in geometry [dCM02], is an immediate consequence. To spell out this general argument in slightly more detail: one is given a finite-dimensional polarized graded vector space V • . A degree 2 Lefschetz operator induces a symmetric form on each V −i , i ∈ Z ≥0 , which collectively are non-degenerate if and only if L satisfies hard Lefschetz. If L does satisfy hard Lefschetz, then L satisfies the Hodge-Riemann bilinear relations if and only if the signature of the Lefschetz form on each V −i agrees with a certain formula, which depends only on the graded dimension of V . From this, one deduces the lemma above. The applications will become clear immediately.
Proof. For a, b ∈ R, consider the Lefschetz operator Recall that HR(x, s, y) means that L a,b induces an operator on H • z (B x B s B y ) which satisfies hard Lefschetz and Hodge-Riemann, for any a > 0, b > 0.
However B x B s is perverse, and by RHR(x, s) (see Lemma 2.8 above) the restriction of the intersection form on B x B s to each summand B z ⊕ ⊂ B x B s is a multiple of the intersection form on B z with sign (−1) (ℓ(x)+1−ℓ(z))/2 . By RHR(≤ xs, y), L 0,b satisfies relative Hodge-Riemann on B x B s B y for any b > 0 (it is an exercise to confirm that the signs are correct). Thus L a,b satisfies relative hard Lefschetz for all a ≥ 0 and b > 0 and satisfies relative Hodge-Riemann for a = 0, b > 0. We can now appeal to the principle of conservation of signs to conclude that relative Hodge-Riemann is satisfied for all a ≥ 0, b > 0. Thus RHR(x, s, y) holds.
The previous proof uses the case special case a = 0, b > 0 to deduce the general case a > 0, b > 0. Here we go the other way: Proposition 4. 12. Suppose x, y ∈ W , s ∈ S and that sy > y. Assume RHR(x, s, y) and RHL(x, ≤ sy). Then RHR(x, sy) holds.
Proof. Let L a,b denote the Lefschetz operator considered in the previous proof. By our assumptions, L a,b satisfies Hodge-Riemann for a > 0, b > 0 and hard Lefschetz for a > 0, b = 0. By the principle of conservation of signs, Hodge-Riemann is also satisfied for a > 0, b = 0. Now B x B sy is a summand of B x B s B y and the intersection form on B x B s B y restricts to a positive multiple of the intersection form on B x B sy . We conclude 6 that L a,0 satisfies Hodge-Riemann on B x B sy , which is what we wanted.
Proposition 4. 13. Let x, y ∈ W and s ∈ S be such that xs < x. Assume HL(x, s, y), HR(x, y). Then HR(x, s, y) holds.
The proof of Proposition 4.13 is more complicated than that of Proposition 4.11, and will occupy the rest of this section. Here is a sketch of our approach. We fix a decomposition B x B s = B x (1) ⊕ B x (−1) and explicitly calculate the Lefschetz operator and forms in the decomposition in terms of the corresponding operators on B x B y . Appealing to RHR(x, y) we will see that the signs are correct for b ≫ a > 0. By the principle of conservation of signs (which is applicable by our RHL(x, s, y) assumption) we deduce that RHR(x, s, y) holds, which is what we wanted to show.
Lemma 4.14. The map r → (∂ s (−rs(ρ)), ρ∂ s (r)) gives an isomorphism Proof. R is free as an R s -module with basis {1, γ} where γ ∈ R 2 is any degree two element which is not s-invariant. In particular we can take γ = ρ. and so our map sends a basis to a basis, and the lemma follows.

) yields a decomposition
Lemma 4. 15. With respect to the decomposition (4.3) the degree 2 endomorphism B x ρB s is given by the matrix: Proof. We identify B x with B s x ⊗ R s R, and write an element of it as Consider an element of the form b ⊗ 1 ∈ B x . We calculate the action of B x ρB s on the summand B x (1): BxρBs (4. 2) Similarly we calculate the action on the summand B x (−1): The lemma follows. Here and in the following proof, an invariant form on an (R, R s )-bimodule means a graded bilinear form −, − : Our fixed decomposition (4.3) gives the basic identification: The following is immediate from the definitions: Lemma 4. 17. Under (4.5) the invariant form is given by: We now put the above calculations together. Until the end of the section let us in addition fix z ∈ W and set . Then V • is equipped with a symmetric form −, − V • and a Lefschetz operator L : V • → V •+2 . This data satisfies Hodge-Riemann, by our assumption HR(x, y). Our identification (4.5) fixes an isomorphism 18. Under the identification (4.6): (1) The invariant form is given by (2)) is an immediate consequence of Lemma 4.17 (resp. Lemma 4.15).
To this end let us fix bases: Because L satisfies hard Lefschetz on V we deduce that Lx 1 , . . . , Lx m , p 1 , . . . , p n is a basis for V −d+1 .

Thus a basis for (V
. Thus if we define matrices then we can write the Gram matrix of the Lefschetz form (v, w) → v, L d a,b w as a block matrix with entries: Thus Proposition 4.13 holds (see the remarks immediately after the statement of the proposition).

Ridigity
Let c ⊂ W be a two-sided cell, a its a-value 7 and J = x∈c Zj x the J- [Lus15, § 10], we define a semi-simple monoidal category J (J is denoted C c in [Lus14, § 18.5] ).
We first consider the subcategory B <c ⊂ B, consisting of all direct sums of shifts of B z with z < LR c (< LR denotes the two-sided preorder ). Let I c denote the ideal in B consisting of all morphisms which factor through objects in B <c . Because B <c is closed under tensor products with arbitrary objects of B, I c is a tensor ideal in B, and we can form the quotient of additive categories B ′ c := B/I c . Then B ′ c is a graded additive monoidal category and we set B c to be the full graded additive subcategory generated by B x with x ∈ c. We denote the image of B x in B ′ c by B c x . The objects B c x (m) with x ∈ W and x < c (resp. x ∈ c) give representatives for the isomorphism classes of the indecomposable objects in B ′ c (resp. B c ). Moreover B c is a graded additive monoidal category (without unit unless c = {id} ).
The (obvious analogues of the) crucial vanishing statements (2.1) and (2.2) still hold in B ′ c and B c , and hence the perverse filtration and perverse cohomology functors descend to B ′ c and B c . We denote them by the same symbols. It is immediate from the definition of the a-function that, for all x, y ∈ c, We now come to the definition of J . It is a full subcategory of B c , although with a different monoidal structure. The objects of J are given by direct sums (without shifts) of B c x with x ∈ c, and thus by (2.1) the category is semi-simple. The monoidal product is given by Remark 5. 1. The reader is warned that in general J is a "monoidal category without unit", i.e. it has an associator but no unit. In general, Lusztig conjectures [Lus14, §13.4] that the a-function is bounded (i.e. a(z) ≤ N for all z ∈ W and some fixed constant N , which he describes explicitly). This boundedness is known to hold for finite and affine Weyl groups. Under the assumption of this conjecture, it turns out that J has a unit if and only if c contains finitely many left cells (as is always the case in finite and affine type ). In this case Lusztig proves [Lus14, §18.5] that the object x∈D∩c B c x is a unit for J (here D ⊂ W denotes the set of distinguished involutions). Even when c contains infinitely many left cells J is "locally unital" (and still under the boundedness assumption). For any given object B ∈ J , only finitely many B c x with x ∈ D ∩ c satisfy B c x * B = 0. The formal direct sum x∈D∩c B c x , while not an object in J when D ∩ c is infinite, acts on any object, and it will act as a monoidal identity would.
Our aim in this chapter is to show that the relative hard Lefschetz theorem for Soergel bimodules implies Theorem 5. 2. J is a rigid, pivotal monoidal category.
Remark 5.3. For finite and affine Weyl groups the rigidity of J has been proved by Bezrukavnikov, Finkelberg and Ostrik [BFO09,§4.3] (using the geometric Satake equivalence). Lusztig has also proven rigidity for Weyl groups (see [Lus15,§9.3] and [Lus14, §18.19] ). His techniques probably extend to crystallographic Coxeter groups. Lusztig also conjectured the rigidity to hold for any finite Coxeter group [Lus15,§10], in which case he expects the Drinfeld center Z(J ) to be related to the "unipotent characters" of W . Ostrik has informed us that for the interesting case of the two-sided cell in H 4 with a-value 6, he has been able to verify the rigidity of J by other means.
Remark 5.4. As we will see, the pivotal structure on J will depend on our fixed choice of regular dominant element ρ ∈ h * . We do not know if the structure varies in an interesting way with ρ. It is possible that the Hodge-Riemann relations might allow one to show that J is unitary, and hope to address this question in future work.
Because J does not have a unit in general the standard definition of rigidity does not make sense. We will prove the following (which is equivalent to the usual notion of rigidity if J has a unit, see Remark 5.6 below): preserves rigidity). Let us denote by B → B ∨ the duality on B. It is easy to see that B is even pivotal (i.e. we have a canonical isomorphism B ∼ → (B ∨ ) ∨ ). As quotients of a rigid, pivotal monoidal category, the monoidal categories B c and B ′ c are rigid and pivotal. We abuse notation and also denote the duality on B c by B → B ∨ .
Proof. We first establish (1). We will construct the isomorphism φ X,Y , the proof for χ X,Y is similar. Let X, Y, B ∈ J . We have canonical identifications (by definition and the analogue for B c of (2. The composition of these isomorphisms defines our isomorphism φ X,Y . It is immediate to check that this isomorphism is natural in X and Y . We now turn to (2). As before we only establish the commutativity of (5. 2), with (5.3) being similar. Choose f ∈ Hom J (X, B * Y ) and let f N E (resp. f SW ) denote the image of f in Hom J (B ∨ * X * Z, Y * Z) obtained by passing through the north-east (resp. south-west) corner of (5. 2 (Here f ′ (resp. g ′ , h ′ ) are obtained from f (resp. g, h) using the dual pair (B, B ∨ ) in B c , and h is uniquely determined by H 0 (h) = H −a (g) .) Consider the diagram given in Figure 1. The maps which have not been defined above are given as follows: (1) All maps labelled ∼ are relative hard Lefschetz isomorphisms (given by our fixed choice of ρ ∈ h * ). At the top and bottom of the middle square we use the canonical identifications: (2) We set l := H −a (ϕZ) and r := H a (h ′ ). It is straightforward but tedious to check that all squares and triangles in Figure 1 commute. If q denotes the relative hard Lefschetz isomorphism q : H −a (Y Z) → H a (Y Z) we deduce from the commutativity of the diagram that q •f N E = q •f SW , and hence that f N E = f SW , which is what we wanted to show.
Remark 5. 6. Suppose that c contains finitely many left cells. Then J has a unit (see Remark 5. 1), which we denote by 1. Applying the isomorphisms of Proposition 5.5 to the identity maps in Hom J (B, B) and Hom J (B ∨ , B ∨ ), we obtain morphisms ε : 1 → B * B ∨ and µ : B ∨ * B → 1. Using the naturality of Proposition 5.5(1) and the commutativity of Proposition 5.5(2) one may check that for f : X → B * Y , φ X,Y (f ) is given by the composition 8 Similarly, the inverse of φ X,Y sends g : B ∨ * X → Y to From this one easily deduces that B ∨ (and ε, µ) is left dual to B. Similarly, one deduces that B ∨ is right dual to B. Hence J is rigid in the usual sense.