THE NÉRON–SEVERI LIE ALGEBRA OF A SOERGEL MODULE

We introduce the Néron–Severi Lie algebra of a Soergel module and we determine it for a large class of Schubert varieties. This is achieved by investigating which Soergel modules admit a tensor decomposition. We also use the Néron–Severi Lie algebra to provide an easy proof of the well-known fact that a Schubert variety is rationally smooth if and only if its Betti numbers satisfy Poincaré duality.


Introduction
Let X be a smooth complex projective variety of dimension n and ρ ∈ H 2 (X, R) be the Chern class of an ample line bundle on X. The Hard Lefschetz Theorem states that for any k ∈ N cupping with ρ k yields an isomorphism ρ k : H n−k (X, R) → H n+k (X, R). This assures the existence of an adjoint operator f ρ ∈ gl(H * (X, R)) of degree −2 which together with ρ generates a Lie algebra g ρ isomorphic to sl 2 (R). In [LL] Looijenga and Lunts defined the Néron-Severi Lie algebra g N S (X) of X to be the Lie algebra generated by all the g ρ with ρ an ample class.
The decomposition of H(X) := H * (X, R) into irreducible g ρ -modules is called the primitive decomposition. The primitive part (i.e., the lowest weight spaces for the g ρ -action) inherits a Hodge structure from the Hodge structure of H(X) and the Hodge structure of the primitive part determines completely the Hodge structure on H(X). However, this decomposition depends on the choice of the ample class ρ. Looijenga and Lunts' initial motivation was to find a "universal" primitive decomposition of H(X), not depending on any choice: this is achieved by considering the decomposition of H(X) into irreducible g N S (X)-modules, which always exists as one can prove that g N S (X) is semisimple. One can easily generalize this construction to any complex variety, possibly singular, by replacing the cohomology H(X) with the intersection cohomology IH(X).
The category of Soergel modules of a Coxeter group W is a full subcategory of the category of graded R-modules, where R is a polynomial ring. Over a field of characteristic 0 the category of Soergel modules is a Krull-Schmidt category whose indecomposable objects (up to shifts) are denoted by {B w } w∈W . When W is a Weyl group (of a reductive group G) then B w −1 ∼ = IH(X w ), where X w is the Schubert variety corresponding to w inside the flag variety of G.
For any real Soergel module B w one can still define its Néron-Severi Lie algebra g N S (w). If W is finite, since g N S (w) is semisimple and B w is indecomposable as R-module (hence as g N S (w)-module), it follows that B w is an irreducible g N S (w)module. From this we deduce, in §2, an easy proof of the Carrell-Peterson criterion [Ca]: a Schubert variety X w is rationally smooth if and only if the Poincaré polynomial of H(X w ) is symmetric. In the Appendix we explain how to extend this proof in the setting of a general finite Coxeter group.
Looijenga and Lunts went on to compute g N S (X) for a flag variety X = G/B. They prove that it is "as big as possible," meaning that it is the complete Lie algebra of endomorphisms of H(X) preserving a non-degenerate (either symmetric or antisymmetric depending on the parity of dim X) bilinear form on H(X).
In §3 we explore the case of the Néron-Severi Lie algebra g N S (X w ) of the intersection cohomology of an arbitrary Schubert variety, a question also posed in [LL]. In Proposition 19 we show, using a result of Dynkin, that g N S (X w ) is maximal if and only if it is a simple Lie algebra. If g N S (X w ) is not simple then IH(X w ) admits a tensor decomposition IH(X w ) = A 1 ⊗ A 2 , where A 1 (resp. A 2 ) is a R 1 (resp. R 2 ) module and R 1 , R 2 are polynomial algebras with R = R 1 ⊗ R 2 .
Finally in §4 we try to characterize for which w ∈ W there is such a decomposition. To an element w ∈ W we associate a graph I w whose vertices are the simple reflections S, and in which there is an arrow s → t whenever ts ≤ w and ts = st. We prove that if the graph I w is connected and without sinks then a tensor decomposition of IH(X w ) cannot exist, hence we deduce that in this case g N S (X w ) is maximal. Thus for the vast majority of Schubert varieties the Néron-Severi Lie algebra is "as big as possible." Acknowledgements. I wish to warmly thank my PhD supervisor Geordie Williamson for introducing me to this problem, and for many useful comments and discussion. I am also grateful to him for explaining to me the content of §A.1. I would also like to thank the referee for a careful reading and many useful comments.
Some of this work was completed during a research stay at the RIMS in Kyoto. I was supported by the Max Planck Institute in Mathematics.

Notation
All cohomology and intersection cohomology groups in this paper are considered with coefficients in the real numbers, unless otherwise stated. Given a graded vector space or module M = i∈Z M i we denote by M [n], for n ∈ Z, the shifted module with M [n] i = M n+i .

Lefschetz modules
In this section we recall from [LL] the definition and the main properties of the Néron-Severi Lie algebra.
Let M = k∈Z M k be a Z-graded finite-dimensional R-vector space. We denote by h : M → M the map which is multiplication by k on M k . Let e : M → M be a linear map of degree 2 (i.e., e(M k ) ⊆ M k+2 for any k ∈ Z). We say that e has the Lefschetz property if for any positive integer k, e k gives an isomorphism between M −k and M k . The Lefschetz property implies the existence of a unique linear map f : M → M , of degree −2, such that {e, h, f } is a sl 2 -triple, i.e., {e, h, f } span a Lie subalgebra of gl (M ) isomorphic to sl 2 (R). We can explicitly construct f as follows: first we decompose M = k≥0 R[e](P −k ) where P −k = Ker(e k+1 | M −k ), then we define, for p −k ∈ P −k , The uniqueness of f follows from [B,Lem. 11.1.1. (VIII)].
Remark 1. From the construction of f , we also see that if e and h commute with an endomorphism ϕ ∈ gl(M ), then f also commutes with ϕ.
Lemma 1. If h and e belong to a semisimple subalgebra g of gl (M ), then also f ∈ g.
Proof. Since g is semisimple, the adjoint representation of g on gl(M ) induces a splitting g⊕a, Now let V be a finite-dimensional R-vector space. We regard it as a graded abelian Lie algebra homogeneous in degree 2 and we consider a graded Lie algebra homomorphism e : V → gl(M ) (thus the image e(V ) consists of commuting linear maps of degree 2). We say that M is a V -Lefschetz module if there exists v ∈ V such that e v := e(v) has the Lefschetz property. We denote by V L ⊆ V the subset of elements satisfying the Lefschetz property. If e is injective, and we can always assume so by replacing V with e(V ), then Definition 1. Let M be a V -Lefschetz module. We define g(V, M ) to be the Lie subalgebra of gl(M ) generated by e(V ) and f(V L ). We call g(V, M ) the Néron-Severi Lie algebra of the V -Lefschetz module M .
The following simple Lemma is needed in Section 3.2: An element e ∈ gl(M ) has the Lefschetz property on M if and only if e ⊕ e has the Lefschetz property on M ⊕ M . Moreover if {e, h, f } is an sl 2 -triple in gl (M ), then {e ⊕ e, h ⊕ h, f ⊕ f } is an sl 2 -triple in gl (M ⊕ M ). Therefore the algebra g(V, M ⊕ M ) is generated by the elements e(v) ⊕ e(v), with v ∈ V , and by

Polarization of Lefschetz modules
Assume that M is evenly (resp. oddly) graded and let φ : M × M → R be a nondegenerate symmetric (resp. antisymmetric) form such that φ(M k , M l ) = 0 unless k = −l.
We assume for simplicity V ⊆ gl (M ). We say that V preserves φ if every v ∈ V leaves φ infinitesimally invariant: Since the Lie algebra aut (M, φ) of endomorphisms preserving φ is semisimple, if V preserves φ then we can apply the Jacobson-Morozov theorem to deduce that g(V, M ) ⊆ aut (M, φ).
For any operator e : M → M of degree 2 preserving φ we define a form · , · e on M −k , for k ≥ 0, by m, m e = φ(e k m, m ). One checks easily that · , · e is symmetric.
We say that e is a polarization if the symmetric form · , · e is definite on the primitive part P −k = Ker(e k+1 )| M −k . If there exists a polarization e ∈ V , then we call (M, φ) a polarized V -Lefschetz module.
Remark 2. Each polarization e has the Lefschetz property. The injectivity of e k | M −k follows easily from the non-degeneracy of · , · e on P −k . From the nondegeneracy of φ we get dim M −k = dim M k for any k ≥ 0, hence e k | M −k is also surjective. Proof. Since g(V, M ) is generated by commutators, it is sufficient to prove it is reductive. This will be done by proving that the natural representation on M is completely reducible. Let N ⊆ M be a g(V, M )-submodule. It suffices to show that the restriction of φ to N is non-degenerate, so that we can take the φ-orthogonal as a complement of N .
Let e ∈ V be a polarization and let f be such that {e, h, f } is a sl 2 -triple. We can decompose N into irreducible sl 2 -modules with respect to this triple. We for any p −k ∈ P −k , p −h ∈ P −h and any integer a ≥ 0.
We consider now a single summand R[e]P N −k . Because the form · , · e is definite on P N −k ⊆ P −k , it follows that φ is non-degenerate on P N −k + e k P N −k . Since e preserves φ, the restriction of φ to e a P N −k + e k−a P N −k is also non-degenerate for any 0 ≤ a ≤ k. We conclude since the subspaces e a P N −k +e k−a P N −k and Remark 3. The proof of Proposition 3 actually shows that the Lie algebra generated by V and f(e), where e is a polarization, is semisimple. Therefore, by Lemma 1, if e is any polarization in V , then V and f(e) generate g(V, M ).
As in the proof of Proposition 3 one can show that the restriction of φ to N is non-degenerate, so the φ-orthogonal subspace N is a g(V, M )-stable complement of N .
Remark 4. The definitions given above arise naturally in the setting of complex projective (or compact Kähler) manifolds. Let X be a complex projective manifold of complex dimension n and assume that X is of Hodge-Tate type, i.e., if is the Hodge decomposition of X then H p,q = 0 for p = q. In particular the cohomology of X vanishes in odd degrees.
Let M = H(X, R)[n] be the cohomology of X shifted by n and let φ be the intersection form: Notice that φ is symmetric (resp. antisymmetric) if n is even (resp. n is odd). Let ρ ∈ H 2 (X, R) be the first Chern class of an ample line bundle on X. Then the Hard Lefschetz theorem and the Hodge-Riemann bilinear relations imply that ρ is a polarization of (M, φ). It follows that (M, φ) is a polarized Lefschetz module over H 2 (X, R).
We can also replace H 2 (X, R) by the Néron-Severi group N S(X), i.e., the subspace of H 2 (X, R) generated by Chern classes of line bundles on X. We define the Néron-Severi Lie algebra of X as g N S (X) = g(N S(X), H • (X, R)[n]).
In [LL] Looijenga and Lunts consider complex manifolds with an arbitrary Hodge structure. To deal with the general case one needs to modify the definition of polarization given here in order to make it compatible with the general form of the Hodge-Riemann bilinear relations.
However, all the Schubert varieties, the case in which we are mostly interested, are of Hodge-Tate type, so for simplicity we can limit ourselves to this case.

Lefschetz modules and weight filtrations
Let V be a finite-dimensional R-vector space and (M, φ) a polarized V -Lefschetz module. In this section we show how to each element v ∈ V we can associate a weight filtration and to any such filtration we can associate a subalgebra of g(V, M ). In many situations the knowledge of these subalgebras turns out to be an important tool to study g(V, M ).
Lemma 5. Let e be a nilpotent operator acting on a finite-dimensional vector space M such that e l = 0 and e l+1 = 0. Then there exists a unique non-increasing filtration W , called the weight filtration: Proof. See, for example, [CE+,Prop. A.2.2].
Lemma 6. Let e ∈ V (not necessarily a Lefschetz operator). Then there exists a Proof. This is [LL,Lem. 5.2].
Let {e, h , f } be as is Lemma 6 and W • be the weight filtration of e. Since h is semisimple and part of a sl 2 -triple, we have a decomposition in eigenspaces It is easy to check that W • satisfies the defining condition of the weight filtration of e. In particular, W • = W • and h splits the weight filtration of e, i.e., W k = W k+1 ⊕ M k for all k.
Let h = h − h . Then (h , h ) is a commuting pair of semisimple elements in g(V, M ) and it defines a bigrading M p,q on M such that M n = p+q=n M p,q . Furthermore h and h also act via the adjoint representation on g(V, M ) defining a bigrading g(V, M ) p,q . We have x ∈ g(V, M ) p,q if and only if x(M p ,q ) ⊆ M p+p ,q+q for all p , q ∈ Z. For x ∈ g(V, M ) we denote by x p,q its component in g(V, M ) p,q .
Let V be a subspace of V containing e and such that, for any x ∈ V , we have x(W k ) ⊆ W k+2 for all k. Consider the graded vector space Gr W Let V 2,0 ⊆ g(V, M ) be the span of the degree (2, 0) components of elements of V . The subspace V 2,0 is an abelian subalgebra of g(V, M ). However, notice that in general V 2,0 is not a subspace of V . We denote by M the vector space M with the grading defined by h . Then M is a V 2,0 -Lefschetz module (in fact e = e 2,0 is a Lefschetz operator on M ), so we can define the algebra g( V 2,0 , M ).
Proposition 7. In the setting as above, there exists an isomorphism of Lie algebras g( V , Gr W M ) ∼ = g( V 2,0 , M ). In particular, g(V, M ) contains a subalgebra isomorphic to g( V , Gr W M ).
Proof. Let π k : W k → M k be the projection. Then k π k : Gr W M → M is an isomorphism of graded vector spaces.
Moreover, the isomorphism k π k is compatible with the map V → V 2,0 given by x → x 2,0 , i.e., that for any k ∈ Z the following diagram commutes: Hence, it follows that g( V , The last statement follows from Lemma 1, in fact both V 2,0 and h are contained in g(V, M ), whence g( V 2,0 , M ) ⊆ g(V, M ).

Application to Soergel calculus
Let G be a simply-connected complex reductive Lie group, B be a Borel subgroup of G and T ⊆ B be a maximal torus. We denote by X = G/B its flag variety. Let g be the Lie algebra of G and h ⊆ g be the Lie algebra of T , with dual space h * . Let Φ ⊆ h * be the root system of G and ∆ be the set of simple roots with respect to B. Let W be the Weyl group of G and S ⊆ W be the set of simple reflections. We denote by (· , ·) the Killing form on h * .
Then the projection G × B C λ → G/B is a line bundle L λ on X and the first Chern class c 1 (L λ ) defines an element in H 2 (X) := H 2 (X, R). The map λ → c 1 (L λ ) induces a homomorphism Λ → H 2 (X) which can be extended to a graded algebra This map is surjective, and its kernel is the ideal generated by R W + , the invariants in positive degree under the Weyl group W of G.
Note that in the Hodge decomposition of X only terms of type (p, p) appear.
The R-modules arising as intersection cohomology of Schubert varieties can also be defined purely algebraically. Let w = s 1 s 2 · · · s be a reduced expression for w ∈ W , where := (w) is the length of w and s i ∈ S ⊆ W . We define the Here R si denotes the s i -invariants in R and R has the R-module structure given by Theorem 8 (Soergel,[S1]). We choose any decomposition of BS(w) into indecomposable R-modules and we denote by B w the summand containing 1 ⊗ := 1 ⊗ · · · ⊗ 1. Then: i) Up to isomorphism, B w does not depend on the choice of decomposition, nor on the choice of the reduced expression w of w.
As Soergel pointed out, the definition of the module B w can be easily generalized to any Coxeter group W with h * R replaced by a reflection faithful representation of W (in the sense of [S3, Def. 1.5]). For a general Coxeter group there are no known varieties such that intersection cohomology gives the indecomposable Soergel module. Nevertheless there exists a replacement for the intersection form in this setting.
The degree = (w) component of BS(w) is one-dimensional and it is spanned by c top := α s1 ⊗ α s2 ⊗ · · · ⊗ α s ⊗ 1. Here α s denotes the simple root corresponding with s ∈ S. We define the intersection form φ on BS(w) via where f g denotes the term-wise multiplication, and Tr is the functional which returns the coefficient of c top . The restriction of the intersection form φ to B w is well-defined up to a positive scalar and it is non-degenerate.
Theorem 9 (Elias-Williamson [EW1]). Let η ∈ h * R be in the ample cone, i.e., (η, α) > 0 for any α ∈ ∆. Then left multiplication by η r induces an isomorphism This means that the Néron-Severi Lie algebra can still be defined for Soergel modules as g N S (w) := g(h * R , B w ). We can now apply Corollary 4 to the polarized If w ∈ W and s ∈ S such that ws > w, then We now restrict ourselves to the case where W is the Weyl group of a simplyconnected complex reductive group. We recall some results from [BGG]. The elements [X v ] ∈ H 2 (v) (X), the fundamental classes of the Schubert varieties X v (for v ∈ W ), are a basis of the homology of X. By taking the dual basis we obtain a basis Q v ∈ H 2 (v) (X), for v ∈ W , of the cohomology, called the Schubert basis. Let The following result is due to Carrell-Peterson [Ca]: Corollary 12. For any w ∈ W the following are equivalent: A similar argument works also for a general finite Coxeter group. We explain in the Appendix how to extend the proof of Corollary 12 to that setting.

The Néron-Severi algebra of Soergel modules
In [LL] Looijenga and Lunts determined the Néron-Severi Lie algebra g N S (X) of a flag variety X = G/B of every simple group G: it is the complete algebra of automorphisms (H, φ) of the intersection form, i.e., it is a symplectic (resp. orthogonal) algebra if the complex dimension of X is odd (resp. even).
Here we want to extend their results and determine the Lie algebra g N S (w) for an arbitrary w ∈ W . We do not quite succeed; however, we show that g N S (w) is "as large as possible" for many w.

Basic properties of the Schubert basis
Let {Q v } v∈W be the Schubert basis of H(X) introduced in Section 2. The Rmodule structure of H(X) can be described in the basis {Q v } v∈W by the Chevalley formula [BGG,Thm. 3.14]: In particular, if s ∈ S then Q s ∈ H 2 (X) = h * R can be identified with the fundamental weight in Λ corresponding to α s , i.e., we have 2(Q s , α s ) = (α s , α s ) and (Q s , α t ) = 0 for any s = t ∈ S. The following Lemma is an easy application of the Chevalley formula (1): We state here for later reference a preliminary lemma.
Remark 6. The element X is basically (up to a scalar) just the Killing form written in the basis {Q s Q t } s,t∈S of Sym 2 (h * R ). Assume now we have a proper decomposition h * For a subset I ⊆ S, we denote by W I the subgroup of W generated by I and P I ⊇ B the parabolic subgroup corresponding to I. Let π : G/B → G/P I be the projection. Then π * : H(G/P I ) → H(G/B) is injective. We can also characterize the image of π * : it coincides with the set of W I invariants in H(X), i.e., π * (H(G/P I )) = (R/R W , and a basis is given by the For a simple reflection u ∈ S let P u := P {u} be the minimal parabolic subgroup of G containing u. For any element w ∈ W such that (wu) < (w) we can choose a reduced expression w = st · · · u. The projection π : G/B → G/P u is a P 1 -fibration which restricts to a P 1 -fibration on X w since BwB · P u = BwB. The image π(X w ) = X u w is the parabolic Schubert variety of the element w in G/P u . The intersection cohomology IH(X u w ) is a polarized Lefschetz module over (R u ) 2 ∼ = N S(G/P u ), so we can define the Lie algebra g N S (X u w ) := g((R u ) 2 , IH(X u w )).

A distinguished subalgebra of g N S (w)
Let w ∈ W and u be a simple reflection such that wu < w. Let π : G/B → G/P u be the projection as above. We denote by IC w (resp. IC u w ) the intersection cohomology complex for the variety X w (resp. X u w ). Then Rπ * (IC w ) ∼ = IC u w [1] ⊕ IC u w [−1] (not canonically) by the Decomposition Theorem (the use of the Decomposition Theorem here can be avoided using an argument of Soergel [S2,Lem. 3.3.2]). In particular, as graded vector spaces, we have Lemma 15. The Lie algebra g N S (w) contains a Lie subalgebra isomorphic to g N S (X u w ).
Proof. Let η ∈ H 2 (X u w ) be the Chern class of an ample line bundle on X u w . We can apply Lemma 6 to find a sl 2 -triple {π * η, h , f } inside g N S (w) such that h is of degree 0, i.e., h (IH k w ) ⊆ IH k w for all k.

Any choice of a decomposition Rπ
One can easily check that weight filtration of the nilpotent element π * η is W k = (IH(X u w )[1]) k−1 ⊕ n≥k IH n w . Therefore for any x ∈ (R u ) 2 we have x (W k We can now apply Proposition 7, with V = (R u ) 2 , in order to obtain where IH w denotes the vector space IH w with the grading determined by h . In particular g((R u ) 2 , Gr W (IH w )) is a subalgebra of g N S (w). It is easy to see that Gr W (IH w ) ∼ = IH(X u w ) ⊕ IH(X u w ) as graded vector spaces, and the isomorphism is compatible with the action of R u . We conclude using Lemma 2 which implies that g((R u ) 2 , Gr W (IH w )) ∼ = g N S (X u w ).
Example 1. Let G = SL 4 (C) so that W = S 4 is the symmetric group on 4 elements, with simple reflections labeled s 1 , s 2 , s 3 . Let w = s 2 s 1 s 3 s 2 and u = s 2 . Let η be an ample Chern class on X u w . Then we can draw the action of π * η on a basis of IH w and the weight filtration as follows We fix η and h as in Lemma 15 and let h = h − h . Then, as in Section 1.2, h and h define a bigrading on IH w and on g N S (w).
We can now restate and reprove [LL,Prop. 5.6] in our setting: Theorem 16. The Lie algebra g N S (w) contains a Lie subalgebra isomorphic to g N S (X u w ) × sl 2 . Proof. Take ρ to be the Chern class of an ample line bundle on X w . Then by the Relative Hard Lefschetz Theorem [BBD,Thm. 5.4.10] cupping with ρ induces an isomorphism of R u -modules: ]. This means that the (0, 2)-component ρ 0,2 ∈ g N S (w) 0,2 of ρ (thus we have [h , ρ 0,2 ] = 0 and [h , ρ 0,2 ] = 2ρ 0,2 ) has the Lefschetz property with respect to the grading given by h . In particular, because of Lemma 1, we can complete it to an sl 2 -triple {ρ 0,2 , h , f ρ } ⊆ g N S (w). The span of {ρ 0,2 , h , f ρ } is a subalgebra of g N S (w) 0,• . In fact, since both ρ 0,2 and h commute with h so does f ρ (see Remark 1).
Because (R u ) 2 and h commute with ρ 0,2 , so does g((R u ) 2 2,0 , IH w ). Because ρ 0,2 and h commute with g((R u ) 2 2,0 , IH w ), so does f ρ . We obtain a morphism of Lie algebras given by the multiplication. The kernel of J is g N S (X u w )∩sl 2 (R) and it is contained in the center of sl 2 (R), which is trivial. The thesis now follows.

Irreducibility of the subalgebra and consequences
The goal of the first part of this section is to show the following: Proposition 17. IH(X u w ) is irreducible as a g N S (X u w )-module. We begin with a preparatory lemma: Lemma 18. The cohomology H(G/P u ) is generated as an algebra by the first Chern classes, i.e., by H 2 (G/P u ).
Proof. We can identify H(G/P u ) with R u /(R W + ). The set {Q s } s∈S\{u} forms a basis of H 2 (G/P u ) = N S(G/P u ) = (R 2 ) u . It is enough to show that the map Sym 2 ((R 2 ) u ) → H 4 (G/P u ) is surjective, because all the generators of H(G/P u ) lie in degrees ≤ 4.
The subalgebra R u is generated by Q s , with s ∈ S \ {u}, and α 2 u . Therefore dim(R 4 ) u = dim Sym 2 ((R 2 ) u ) + 1 and, since H 4 (G/P u But since the Killing form is non-degenerate and (R 2 ) u is a proper subspace of R 2 , we have X ∈ Sym 2 ((R 2 ) u ) (as explained in Remark 6).
Proof of Proposition 17. Since g N S (X u w ) is semisimple, it is enough to show that IH(X u w ) is an indecomposable g N S (X u w )-module. In particular it is enough to show that it is indecomposable as a H 2 (X u w )-module (here regarded as an abelian Lie subalgebra of g N S (X u w )). The Erweiterungssatz (in the version proved by Ginzburg [G]) states that taking the hypercohomology (as a module over the cohomology of the partial flag variety) is a fully faithful functor on IC complexes of Schubert varieties. In particular for any w ∈ W we have: This implies, since IC(X u w ) is a simple perverse sheaf on G/P u , that IH(X u w ) is an indecomposable H(G/P u )-module. Now Lemma 18 completes the proof.
Proposition 19. If g C N S (w) := g N S (w) ⊗ C is a simple complex Lie algebra, then we have g N S (w) ∼ = aut(IH w , φ).
In particular this implies that the complexification g C N S (w) is isomorphic to sp IHw (C) if (w) is odd, and is isomorphic to so IHw (C) if (w) is even.
Proof. Proposition 17 shows that the Lie algebra g N S (X u w )×sl 2 (R) acts irreducibly on IH w ∼ = IH(X u w )⊗H(P 1 ). This obviously remains true when one considers, after complexification, the action of g C N S (X u w ) × sl 2 (C) on IH(X w , C). In [D,Thm. 2.3], Dynkin classified all the pairs g ⊆ g (⊆ gl(C N )) of complex Lie algebras such that g acts irreducibly on V and g is simple. From this classification we see that if g = g × sl 2 (C) and sl 2 (C) acts with highest weight 1 then g is one of sl N , so N and sp N .
We apply now this result to the pair g C N S (X u w ) × sl 2 (C) ⊆ g C N S (w) Clearly we cannot have g C N S (w) ∼ = sl(IH(X w , C)) since g N S (w) ⊆ aut(IH(X w , C), φ). This implies g C N S (w) = aut(IH(X w , C), φ), hence g N S (w) ∼ = aut(IH w , φ). Remark 8. We now discuss which real forms of the symplectic and orthogonal groups occur as aut(IH w , φ). If (w) is odd there is, up to isomorphism, only one symplectic form on IH w , hence aut(IH w , φ) ∼ = sp dim(IHw) (R). Now we assume that (w) is even. We want to determine the signature of the symmetric form φ on IH w .
If k > 0 then φ is a perfect pairing between IH k w and IH −k w , hence the signature of φ| IH k . The signature of φ on IH 0 w is determined by the Hodge-Riemann bilinear relations: the dimension of the positive part of φ| IH 0 w is given by

Tensor decomposition of intersection cohomology
We now want to understand for which w ∈ W the Lie algebra g C N S (w) is not simple. The complex Lie algebra g C N S (w) acts naturally on IH(X w , C). To simplify the notation from now on, we will consider in this section only cohomology with complex coefficients and we will denote IH(X w , C) (resp. H(X w , C)) simply by IH w (resp. H w ) and R ⊗ C ∼ = C[h * ] by R.
For any w ∈ W we have H w ⊆ IH w (see Remark 5). In particular H 2 w acts faithfully on IH w and we can regard H 2 w as a subspace of g N S (w). We recall the following lemma from [LL,Lem. 1

.2]:
Lemma 20. Assume there exists a non-trivial decomposition g C N S (w) = g 1 ×g 2 and consider π i : g C N S (w) → g i the projections. Then the decomposition is graded and it also induces a decomposition into graded vector spaces IH w = IH •,0 w ⊗ C IH 0,• w where IH •,0 w (resp. IH 0,• w ) is an irreducible π 1 (H 2 w )-Lefschetz module (resp. π 2 (H 2 w )-Lefschetz module ) with g 1 = g(π 1 (H 2 w ), IH •,0 w ) and g 2 = g(π 2 (H 2 w ), IH 0,• w ). For the rest of this paper we assume that we have a splitting g C N S (w) = g 1 × g 2 and we denote by π 1 : g C N S (w) → g 1 and π 2 : g C N S (w) → g 2 the projections. Let IH w = IH •,0 w ⊗ C IH 0,• w be the induced decomposition. There exist integers a, b ≥ 0 such that IH •,0 w (resp. IH 0,• w ) are not trivial only in degrees between −a and a (resp. between −b and b) with a, b ≥ 0 and a + b = (w). In particular IH −a,0 w and IH 0,−b w are one-dimensional. We define a bigrading on IH w by IH i,j w := IH i,0 w ⊗ IH 0,j w .

Splitting of H 2 w
We can assume from now on H 2 w = H 2 (G/B). In fact, we can replace G by its Levi subgroup corresponding to the smallest parabolic subgroup of G containing w. This does not change the Schubert variety X w , the cohomology H w and the Lie algebra g N S (w). In particular we have R = Sym(H 2 w ). In general H w = IH w , so it is not clear a priori that a tensor decomposition for IH w descends to one for H w . Still, this holds in our setting: Proposition 21. Assume we have a decomposition g C N S (w) = g 1 × g 2 . Then Proof. It is enough to show that dim H 2 w ≥ dim π 1 (H 2 w ) + dim π 2 (H 2 w ). We define We can define a T -module structure on IH w via (x ⊗ y)(a) = x(a) ⊗ y(a) for any x ∈ π 1 (H 2 w ), y ∈ π 2 (H 2 w ) and a ∈ IH w . We have a bigrading T p,q := Sym p (π 1 (H 2 w )) ⊗ Sym q (π 2 (H 2 w )) on T compatible with the bigrading of IH w , i.e., T p,q (IH i,j w ) ⊆ IH p+i,q+j w . The subspace T 2,0 ∼ = π 1 (H 2 w ) ⊆ g 1 acts faithfully on IH •,0 w , while T 0,2 ∼ = π 2 (H 2 w ) ⊆ g 2 acts faithfully on IH 0,• w . Hence T 2,2 ⊆ g 1 ⊗g 2 ⊆ gl(IH •,0 w )⊗gl(IH 0,• w ) = gl(IH w ) acts faithfully on IH w , i.e., if t ∈ T 2,2 acts as 0 on IH w , then t = 0.
Let Ψ : R → T the inclusion induced by Ψ(x) = π 1 (x) + π 2 (x) for any x ∈ R 2 . We observe that the T -module structure on IH w extends the R-module structure.
We can decompose Q s = L s +R s where L s = π 1 (Q s ) ∈ g 1 and R s = π 2 (Q s ) ∈ g 2 for all s ∈ S. Now we consider the element X ∈ (R 4 ) W defined in Lemma 14. The R-module structure on IH w factorizes through H(X, C) = R/(R W + ), therefore Ψ(X ) ∈ T acts as 0 on IH w . In particular also the component Ψ(X ) 2,2 ∈ T 2,2 acts as 0 on IH w . Since the action is faithful on T 2,2 we obtain Ψ(X ) 2,2 = s,t∈S c st (L s ⊗ R t + L t ⊗ R s ) = 0 ∈ T 2,2 . Since c st is symmetric we can rewrite it as follows: for any s ∈ S L . Since (c st ) s,t∈S is a non-degenerate matrix, it follows that we have #(S L ) linearly independent equations vanishing on (R s ) s∈S , hence dim π 2 (H 2 It also follows that Ψ : R → T is an isomorphism, so we have a bigrading on R compatible with the one on IH w . Hence H w is also bigraded as a subspace of IH w , since it is the image of the map of bigraded vector spaces R → IH w induced by x → x(1 w ), where 1 w is a generator of the one-dimensional space IH In the next sections we provide a sufficient condition for the Lie algebra g N S (w) to be maximal. However, there is a case where the proof is considerably easier and we provide it here for convenience and to motivate the reader.
Recall that for any w ∈ W , the set {Q st } st≤w is a basis of H 4 w . In particular, if st ≤ w for any s, t ∈ S, we have H 4 w ∼ = H 4 (X). In this case from Lemma 14 we have also Ker(R 4 → H 4 w ) = (R W + ) 4 = RX . Corollary 22. Assume that the root system of G is irreducible and suppose that whenever s i , s j ≤ w then s i s j ≤ w. Then g N S (w) ∼ = aut(IH w , φ).
Proof. We assume for contradiction that we have a non-trivial decomposition g C N S (w) = g 1 × g 2 . From Proposition 21 we know that H 4 w splits as H 4,0 w ⊕ H 2,2 w ⊕ H 0,4 w . This implies that also K := Ker(R 4 → H 4 w ) splits as K = K 4,0 ⊕K 2,2 ⊕K 0,4 where K i,j = Ker(R i,j → H i,j w ). But K is one-dimensional and generated by X , thus X belongs to either R 4,0 , R 2,2 or R 0,4 , which is impossible since X is nondegenerate (see Remark 6). Hence the Lie algebra g C N S (w) must be simple. We can now apply proposition 19 to deduce g N S (w) ∼ = aut(IH w , φ).

A directed graph associated to an element
Let w ∈ W . We construct an oriented graph I w as follows: the vertices are indexed by the set of simple reflections S and we put an arrow s → t if ts ≤ w and ts = st (i.e., if ts ≤ w and s and t are connected in the Dynkin diagram).
Recall that we assumed, by shrinking to a Levi subgroup, that s ≤ w for any s ∈ S. It follows that for any pair s, t ∈ S we have either st ≤ w, ts ≤ w or both. Hence the graph I w is just the Dynkin diagram where each edge s − t is replaced by the arrow s ← t, by the arrow s → t, or by both s t. In particular, if the Dynkin diagram is connected, then also I w is connected. In this case we call w connected.
Remark 9. Since the Dynkin diagram has no loops, then also I w has no nonoriented loops (we only consider loops in which for any pair s, t ∈ S at most one of the arrows s → t and t → s occurs).
We call a subset C ⊆ S closed if any arrow in I w starting in C ends in C. Union and intersection of closed subsets are still closed. We call a closed singleton in S a sink.
Example 2. Let W be the Coxeter group of D 5 . We label the simple reflections as follows: Consider the element w = s 1 s 2 s 4 s 3 s 5 s 2 s 1 . Then the diagram I w associated to w is: Here the coloured lines describe all the non-empty closed subsets of I w .
As we show in the following sections, the graph I w determines H 4 w , and we can make use of it to provide obstructions for the algebra g N S (w) to not admit a decomposition, hence find sufficient conditions for the algebra g N S (w) to be simple. More specifically, we prove in Theorem 28 that, if I w is connected and has no sinks, then g N S (w) is maximal.
Proposition 23. If w = w 1 w 2 as above, then we have decompositions IH w ∼ = IH w1 ⊗ C IH w2 and g N S (w) ∼ = g N S (w 1 ) × g N S (w 2 ).

Proof. In this case
where H w1 acts on the factor IH w1 while H w2 acts on IH w2 . Since the Lie algebra g N S (w 1 ) × g N S (w 2 ) is semisimple and both h and H 2 w are contained in g N S (w 1 ) × g N S (w 2 ), from Lemma 1 we have g N S (w) = g N S (w 1 ) × g N S (w 2 ).

The connected case
In view of Proposition 23 we can restrict ourselves to the case of a connected w.
Lemma 24. Let w be connected and let K = Ker(Sym 2 (H 2 w ) → H 4 w ). Then the elements X C := s,t∈C c st Q s Q t , with C closed, generate K.
Proof. We know that dim K = #{(s, t) ∈ S 2 | st ≤ w} + 1 because Sym 2 (H 2 w ) → H 4 w is surjective. Since w is connected, if st ≤ w then s and t are connected by an edge in the Dynkin diagram and ts ≤ w.
Let (a, b) be any pair of elements of S such that ab ≤ w, i.e., such that there is no arrow b → a. We can define a proper closed subset C ab by taking the connected component of b in I w after erasing the arrow a → b. From Remark 9 it follows that a ∈ C ab . It is easy to see that X C ab together with X = X S are linearly independent in Sym 2 (H 2 w ): in fact when we write them in the basis {Q s Q t } s,t∈S we have X C ab ∈ c bb Q 2 b + R ab , where R ab = span Q s Q t | (s, t) = (a, a), (b, b) , while all the other X C a b are either in R ab or in c aa Q 2 a + c bb Q 2 b + R ab . By the formula for the dimension of K given above, it remains to show that all the X C , for C closed, lie in K. Let y denote the projection to H 4 (G/B) of an element y ∈ Sym 2 (H 2 w ). Let C be a closed subset and let E : a(i) ∈ C and b(i) ∈ C} be the set of arrows starting outside C and ending in C. Applying Lemma 13, on one hand we obtain: On the other hand we have Since X = 0 in H 4 (G/B), projecting from R 4 to H 4 (G/B) we obtain Then (2) together with (3) implies that the projection For a closed C let N S(C) := span Q s | s ∈ C ⊆ H 2 w . The proof of Proposition 21 applies also to N S(C) if we replace X by X C = s,t∈C c st Q s Q t . This means that whenever we have a decomposition g C N S (w) = g 1 × g 2 , then N S(C) splits compatibly.
Lemma 25. Let K C := K ∩ Sym 2 (N S(C)). Then K C is generated by X D , with D closed and D ⊆ C.
Proof. Let i a i X Di ∈ K ∩ Sym 2 (N S(C)) with D i closed and a i ∈ C. Then it is easy to see that i a i X Di = i a i X Di∩C ∈ Sym 2 (N S(C)).
Lemma 26. Let C be a connected and closed subset of S. Assume that there exists a non-empty closed subset D ⊆ C such that N S(D) = π 1 (N S(C)). Then if D does not contain any sink we have D = C.
be the set of arrows starting in U and ending in D. The set {L s } s∈D = {Q s } s∈D is a basis of N S(D) = π 1 (N S(C)), therefore the set {R u } u∈U is a basis of π 2 (N S(C)). We assume for contradiction that U = ∅. By writing the (2, 2)-component of X C −X D we have u∈U s∈C c su L s ⊗ R u = 0 ∈ g 1 ⊗ g 2 from which we get s∈C c su L s = 0 for any u ∈ U .
Let U be a connected component of U and let E = {a(i) i → b(i) | a(i) ∈ U and b(i) ∈ D} ⊆ E. Since C is connected we have E = ∅. Since U is connected and there are no loops in the Dynkin diagram, we have b(i) = b(j) for any i = j ∈ E, and moreover there are no arrows between b(i) and b(j). Then for any u ∈ U Since the set {L b(i) } i∈ E is linearly independent, this can be thought of as a non-degenerate system of linear equations in L s , with s ∈ U and it has a unique solution In particular Claim 1. We have y(s, i) > 0 for any s ∈ U and any i ∈ E.
Proof of the claim. From Equation (4) it is easy to see that Hence s∈ U y(s,i) (αs,αs) α s is (up to a positive scalar) equal to the fundamental weight of a(i) in the root system generated by the simple roots in U . Now the claim follows from the fact that in any irreducible root system all the fundamental weights have only positive coefficients when expressed in the basis of simple roots.
For any s ∈ U we have R s = Q s − i∈ E y(s, i)Q b(i) ∈ g 2 . Now consider the element Let p : R 4 → H 4 w denote the projection. The previous equation implies that We can write Θ = Θ 1 + Θ 2 with Since there are no edges between b(i) and b(j), we have that Thus, by Lemma 13, we have } is the set of arrows in I w starting in b(i). It is easy to see that all the terms in p(Θ 1 ) and p(Θ 2 ) are linearly independent, whence p(Θ 1 )+ p(Θ 2 ) = 0 if and only if all their terms vanish. Recall that y(a(i), i)c a(i)b(i) < 0 for all i ∈ E. Hence p(Θ 1 ) + p(Θ 2 ) = 0 forces E i = ∅ for any i ∈ E. But this is a contradiction because there are no sinks in D, whence U = ∅ and C = D.
Lemma 27. Let C be a closed and connected subset of S. Assume that there are no sinks in C. Then N S(C) ⊆ g 1 or N S(C) ⊆ g 2 .
Proof. We work by induction on the number of vertices in C. There is nothing to prove if C = ∅.
Let D ⊆ C be a maximal proper closed subset. The kernel K C := K ∩ Sym 2 (N S(C)) is generated by X C and X D with D ⊆ D. In fact if D ⊆ C is a proper closed subset and D ⊆ D, then by maximality D ∪ D = C and X D = X C − X D + X D∩ D . In particular we have dim K C = dim K D + 1.
By induction on the number of vertices we can subdivide D into two subsets D L and D R , each consisting of the union of connected components of D, such that N S(D L ) ⊆ g 1 and N S(D R ) ⊆ g 2 .
Theorem 28. For w ∈ W , if the graph I w is connected and without sinks, then g N S (w) = aut(IH w , φ).
Proof. Applying Lemma 27 to C = S we see that any decomposition of g C N S (w) must be trivial, hence by Proposition 19 we get g N S (w) = aut(IH w , φ).
Example 3. It is in general false that g N S (w) is simple for any connected w.
Let W be the Weyl group of type A 3 (i.e., W = S 4 ) where S = {s, t, u}. We consider the element usts ∈ W whose graph I usts is u t s The closed subsets in I usts are S, {u} and ∅. Then g N S (usts) ∼ = g N S (u) × g N S (sts) ∼ = sp 2 (R) × sp 6 (R) ∼ = sl 2 (R) × sp 6 (R). The splitting induced on H 2 w is We have a similar behaviour more generally: for any w ∈ S n+1 , with S = {s 1 , . . . , s n }, such that w = s 1 w where w is the longest element in W {s2,...,sn} the Lie algebra g N S (w) is isomorphic to sl 2 (R) × g N S (w ).
Example 4. The following example demonstrates that having no sinks in I w is not a necessary condition for the algebra g N S (w) to be simple.
Let W be the Weyl group of type B 3 , where we label the simple reflections as follows: s t u Then for w 1 = usts we get again g N S (w 1 ) ∼ = g N S (u) × g N S (sts) ∼ = sl 2 (R) × sp 6 (R), but for w 2 = stut the Lie algebra g N S (w 2 ) is simple (hence it is isomorphic to so 6,6 (R)). Notice that the graphs I w1 and I w2 are isomorphic.
Remark 10. The results given in this section work in the same way, replacing the cohomology of X with the coinvariant ring R/R W + and the intersection cohomology of Schubert variety by indecomposable Soergel modules whenever is needed, for a finite Coxeter group W : if there are not sinks in the diagram of w ∈ W then g N S (w) is maximal, i.e., it coincides with aut (B w , φ). To complete the proof one needs to generalize Proposition 17 in this setting. A possible way to achieve this is to extend the results in [EW1] to the setting of singular Soergel bimodules [W].
For a general Coxeter group W our methods do not apply directly. In fact in general a reflection faithful representation of W is not irreducible, thus Lemma 14 does not hold and the kernel of the map R → B w seems harder to compute.

A. Appendix: Extension of Corollary 12 to a general Coxeter group
The goal of this Appendix is to extend Corollary 12 to a general Coxeter group W . In the general case we cannot use the geometry of the Schubert varieties to construct a graded R-submodule H w of B w such that dim(H w ) k = #{v ∈ W | v ≤ w and 2 (v) = k + (w)}. In this section we construct an algebraic replacement of such a module.

A.1. A basis of the Bott-Samelson bimodule
We use the diagrammatic notation for morphisms between Soergel bimodules from [EW2].
For any word w = s 1 . . . s we have the Bott-Samelson bimodule BS(w) = R ⊗ R s 1 R ⊗ R s 2 R ⊗ R s 3 . . . ⊗ R s −1 R ⊗ R s R and for any w ∈ W let B w denote the corresponding indecomposable Soergel bimodule. We have BS(w) ⊗ R R = BS(w) and B w ⊗ R R = B w .
Let w = s 1 s 2 · · · s be a (not necessarily reduced) word of length and e ∈ {0, 1} be a 01-sequence. As explained in [EW2,Section 2.4], to a 01-sequence we can associate a sequence of elements of {U 0, U 1, D0, D1}. Let def(e) be the defect of e, i.e., the number of U 0's minus the number of D0's of e. We define downs(e) to be the number of D's (both D1's and D0's) of e. We denote by w e the element s e1 1 s e2 2 · · · s e . We have def(e) = (w) − (w e ) − 2 downs(e).
For any k, 0 ≤ k ≤ , let w ≤k = s 1 s 2 · · · s k and w e ≤k = s e1 1 s e2 2 · · · s e k k . We say x ≤ w if there exists e such that w e = x. For any element x ∈ W we denote by R(x) its right descent set, i.e., R = {s ∈ S | xs < x}.
Lemma 29. Let w be a word. For any x ≤ w there exists a unique 01-sequence e such that w e = x and e has only U 0's and U 1's. Moreover such e is the unique 01-sequence of maximal defect with w e = x, and def(e) = (w) − (x).
Proof. We first prove the existence. Let w = s 1 · · · s . We start with x = x and we define recursively, starting with k = l and down to k = 1, It follows that s k ∈ R(x k−1 ) for any 1 ≤ k ≤ , hence e has only U 1 s and U 0 s. At any step we have x k ≤ w ≤k , therefore x 0 = Id and w e = x.
To show the uniqueness, assume that there are two 01-sequences e and f with only U 's and satisfying w e = x = w f . If e = f we can conclude that e = f by induction on . Otherwise we can assume e = 1 and f = 0. Now we get w f ≤ −1 = x, and xs < x because the last bit of e is a U 1. But this means that the last bit of f is a D0, hence we get a contradiction.
The last statement follows from (5).
Definition 2. Let w be a word and x ≤ w. We call the unique 01-sequence e without D's such that w e = x the canonical sequence for x.
In [EW2,Chap. 6] Libedinsky's Light Leaves are introduced in the diagrammatic setting. We make use of Elias and Williamson's results.
Let w be a word and e a 01-sequence with w e = x. The Light Leaf LL w,e is an element in Hom(BS(w), BS(x)), for some choice of a reduced expression x of x. For any light leaf LL w,e , let L L w,e ∈ Hom(BS(x), BS(w)) be the morphism obtained by flipping the diagram of LL w,e upside down. If w e = w f let LL w,e,f = L L w,e • LL w,f . We know from [EW2,Thm. 6.11] that the set {LL w,e,f } w e =w f is a basis of End(BS(w)) as a right R-module.
Lemma 30. Let w be a word and e be a 01-sequence. Then if e has (at least) one D.
Proof. The statement easily follows from the definitions when e has only U 's. By induction on (w) we can assume that e has only one D at the right end. Then LL w,e looks like The box labelled by "braid" contains only 2m st -valent vertices. By induction LL w ≤k−1 ,e ≤k−1 ⊗ Id Bs (w) 1 ⊗ w = 1 ⊗ x . Notice that every 2m st -valent vertex preserves 1 ⊗1⊗· · ·⊗1. It follows from the definition that a trivalent vertex applied to 1 ⊗1⊗1 returns 0, thus LL w,e (1 ⊗ w ) = 0.
Corollary 31. Let w be a word. The set {ll w,e } with e ∈ {0, 1} (w) is a basis of BS(w) as a right R-module.
Then clearly also the span of all the LL w,e,f (1 ⊗ w ) with w e = w f generates BS(w). Applying Lemma 30 we see that LL w,e,f (1 ⊗ w ) = ll w,e if f is canonical and 0 otherwise. It follows that {ll w,e } e∈{0,1} spans BS(w). Since the rank of BS(w) as a right R-module is 2 (w) the thesis follows, cf. [M,Thm. 2.4].
Remark 11. The results of this section are, at least to my knowledge, still unpublished. However, Geordie Williamson and Ben Elias explained canonical subexpression and how to construct the basis {ll w,e } in a master class at the QGM in Aarhus in 2013. Videos and notes of the lectures are available at http:/ /qgm.au.dk/ video/mc/soergelkl/.
A.2. The "homology" submodule of an indecomposable Soergel module Recall from [EW1,Sect. 3.5] that for any Soergel bimodule B we have ] is a shift of the standard bimodule R x and v denotes the degree shift. In particular, if BS(w) is a Bott-Samelson bimodule then h x (BS(w)) = e : w e =x v def(e) , while if B w is an indecomposable bimodule, then h x (B w ) is equal to the polynomial h x,w (v). The polynomials h x,w (v) are related to the usual Kazhdan-Lusztig polynomials via the formula In particular, h x,w ∈ Z [v] and h x,w = v (w)− (x) +"lower terms," for any x ≤ w.
The basis {ll w,e } is compatible both with the filtration support and with the degree grading of BS(w). In other words, for any x and any k ∈ Z ≥0 , the set {ll w,e | w e = x, def(e) = k} induces a basis on the summand ∇ x [k] ⊕c k ⊆ Γ ≤x BS(w)/Γ <x BS(w), where c k is the coefficient of v k in h x (BS(w)).
Let us consider the following right R-submodules of BS(w): C w = e canonical ll w,e R and D w = e not canonical ll w,e R.
In general C w is not a left R-module.
Lemma 32. Let D w as above. Then D w is a R-subbimodule of BS(w).
Proof. It suffices to show that, for any non-canonical e and for any f ∈ R, we have f · ll w,e = i ll w,ei g i , with e i not canonical and g i ∈ R. Since R is generated in degree 2 we can assume f to be homogeneous of degree 2. Let x = w e . The element f · ll w,e is contained in Γ ≤x (BS(w)). Using repeatedly the nil-Hecke relation [EW2,(5.2)] on the bottom of the diagram we see that f · ll w,e = ll w,e · x −1 (f ) + Θ, with Θ ∈ Γ <x (BS(w)). Therefore we can write Θ = i ll w,fi h i , with h i ∈ R and w fi < x. Furthermore, since the equation (6) is homogeneous, if h i = 0 we have deg(h i ) + deg(ll w,fi ) = deg(f ) + deg(ll w,e ) = deg(ll w,e ) + 2 for all i, whence deg(ll w,fi ) ≤ deg(ll w,e ) + 2 ≤ (w) − 2 (x) < (w) − 2 (w fi ) and f i must be not canonical.
Let now w be a reduced word. Fix a decomposition of BS(w) into indecomposable bimodules and let E w ∈ End(BS(w)) be the primitive idempotent corresponding to B w , i.e., BS(w) = Ker(E w ) ⊕ Im(E w ) and Im(E w ) ∼ = B w . Since, for any x, the map Γ ≤x BS(w)/Γ <x BS(w) → Γ ≤x B w /Γ <x B w induced by E w is surjective, it follows that the projection of the set {E w (ll w,e ) | w e = x, def(ll w,e ) = k} spans the summand ∇ In particular, because of Lemma 29, for any x ≤ w the summand ∇ x [ (w) − (x)] ⊆ Γ ≤x BS(w)/Γ <x BS(w) is spanned by ll w,e , where e is the canonical sequence for x. Moreover, we have h x,w = v (w)− (v) +"lower terms," hence the summand ∇ x [ (w) − (x)] ⊆ Γ ≤x B w /Γ <x B w has as a basis the projection of {E w (ll w,e )}. Therefore, the map is an isomorphism in degree v (w)−2 (x) . Let C w = C w ⊗ R R, D w = D w ⊗ R R and let us denote by E w : BS(w) → B w the induced morphism of left R-modules. For any e, let ll w,e denote the projection of ll w,e to BS(w).
Lemma 33. The kernel of E w is contained in D w .
Proof. Let i ll w,ei g i ∈ Ker E w , with g i ∈ R. Since E w is homogeneous we can assume the sum to be homogeneous. Assume that a canonical sequence e j appears in the sum with g j = 0. Then w ej = w ei for any i = j with g i = 0 and, in addition, x := w ej must be of maximal length among X := {w ei | g i = 0}.
We can also choose a refinement of the Bruhat order into a total order of W such that x is maximal inside X. We label the elements of W as w 1 < w 2 < . . . in order.
For an integer k ≥ 1 let us denote by Γ ≤k B the submodule of elements supported on {w 1 , . . . , w k }. Then by Lem. 6.3] we have for any Soergel bimodule B, Let h be the index of x, i.e., x = w h . We have ll w,ei g i ∈ Γ ≤h BS(w) and projects to ll w,ej g j ∈ Γ ≤h BS(w)/Γ ≤h−1 BS(w). But the map is an isomorphism in degree v (w)−2 (x) . Hence ll w,ei g i , or equivalently ll w,ej g j , is sent to 0 if and only if g j = 0. We obtain a contradiction, whence ll w,ei g i ∈ D w . It follows that B w = E w (C w ) ⊕ E w (D w ) as R-vector spaces. Moreover, E w (D w ) is a R-submodule of B w and the restriction of E w to C w is injective. We now have all the tools to generalize Corollary 12 to the setting of a general finite Coxeter group.
Corollary 34. For any w ∈ W the following are equivalent: ii) #{v ∈ W | v ≤ w and (v) = k} = #{v ∈ W | v ≤ w and (v) = (w) − k} for any k ∈ Z. iii) All the Kazhdan-Lusztig polynomials p v,w are trivial.
Notice that ii) holds if and only if we have dim(E w (C w )) k = dim(E w (C w )) −k for any k ∈ Z, hence if and only if dim(E w (D w )) k = dim(E w (D w )) −k for any k ∈ Z.
If ii) holds, then we can apply Corollary 10 to the R-submodule E w (D w ) ⊆ B w . Since B w is indecomposable it follows that E w (D w ) = 0. Hence ii) implies i).
The rest of the proof continues just as in Corollary 12, where IH w is replaced by B w and H w [ (w)] by E w (C w ).
Remark 12. One could also define H w := E w (D w ) ⊥ , where the orthogonal is taken with respect to the intersection form of B w , and check that H w coincides with H w if W is the Weyl group of some reductive group G.