SOERGEL BIMODULES, HECKE ALGEBRAS, AND KAZHDAN-LUSZTIG BASIS. Contents. References Introduction

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1 SOERGEL BIMODULES, HECKE ALGEBRAS, AND KAZHDAN-LUSZTIG BASIS BORIS TSVELIKHOVSKY Abstract. These are the notes for a talk at the MIT-Northeastern seminar for graduate students on category O and Soergel bimodules, Fall 207. Contents. Introduction 2. Hecke Algebras 2 3. Soergel bimodules 4 4. Soergel s categorification theorem 8 5. Bott-Samelson varieties 9 References 2. Introduction The main goal of this talk is to explain Soergel s approach to Kazhdan-Lusztig s conjecture [KL79]. This conjecture expresses the multiplicities of simple objects in standard ones in the principal block O 0 of category O in terms of the values of certain polynomials in Z[v ± ] at v =. These polynomials arise from Hecke algebras - certain algebras H over Z[v ± ] with the basis indexed by the elements of a Weyl group W and relations deforming those of Z[W ]. The transition matrix from the standard basis to a certain basis (called Kazhdan-Lusztig s basis) is uni-triangular with non-diagonal entries in v Z 0 [v]. The matrix coefficients evaluated at v = give the multiplicities of simple objects in standard ones in the principal block O 0 of category O. The precise formulations are given in Theorem 2.3.

2 2 BORIS TSVELIKHOVSKY The first proof was provided independently by Beilinston-Bernstein in [BB8] and Brylinski- Kashiwara in [BK8], using the machinery of D-modules and perverse sheaves in the beginning of 980-s. A decade later Soergel in [Soe90] and [Soe92] suggested a different approach via bimodules over the polynomial ring R = R[h], where h is the Cartan subalgebra of g. The independent proof using Soergel s ideas was completed recently by Elias and Williamson in [EW]. The structure of the notes is as follows. In Section 2 we recall the generalities on Hecke algebras associated with finite Weyl groups, introduce the Kazhdan-Lusztig basis and verify its existence and uniqueness. The pivotal point of this section is the statement of Theorem 2.3 (known as the Kazhdan-Lusztig conjecture). Soergel s approach to the conjecture starts to unravel in Section 3, culminating in Soergel s categorification theorem. In Section 5 we explain the connection of Bott-Samelson modules and bimodules to cohomology and equivariant cohomology of Bott-Samelson varieties. 2. Hecke Algebras Definition 2.. Let (W, S) be a Weyl group. The Hecke algebra H is the algebra over the ring Z[v ± ] with the generators given by the symbols {H s s S} and relations (2.) Hs 2 = (v v)h s + (H s + v)(h s v ) = 0 s S (quadratic relations) H t H s H t... = H }{{} s H t H s... s, t S (braid relations). }{{} m st m st For any element x W and a reduced expression x = s i... s ik, define H x := H si... H sik. We set H e to be the unit. Remark 2.. As any two reduced expressions of an element x W can be obtained from one another by a sequence of braid moves, the element H x does not depend on the choice of a reduced expression of x. Remark 2.2. The elements H x x W generate H as Z[v ± ]-module. One can show that they form a basis. Exercise 2.. Check that H s = H s + v v. Therefore, H x is invertible for any x W. There is a ring involution τ on H, given by τ : v v and τ : H x H x := H x. Definition 2.2. Let w, w 2 be in W. Then w w 2 in the Bruhat order if w 2 = s βik... s βi w and l(s βik j... s βi w ) > l(s βik j... s βi w ) for all j {,..., k }, where s βij are some (not necessarily simple) reflections in W. Proposition 2.. There exists a basis (the Kazhdan-Lusztig basis) b x x W characterized by two properties: τ(b x ) = b x ; (2.2) b x = H x + c x,y H y, y W,y x of H uniquely

3 SOERGEL BIMODULES, HECKE ALGEBRAS, AND KAZHDAN-LUSZTIG BASIS 3 where each c x,y vz[v]. Remark 2.3. As the transition matrix from {H x } x W to {b x } x W is upper-triangular with s on the diagonal, the elements {b x } x W, indeed, form a basis of H. Definition 2.3. The polynomials p y,x := v l(x) l(y) c x,y are called the Kazhdan-Luzstig polynomials. Exercise 2.2. The elements b s := {H s + v} s S are self-dual with respect to τ. Check that b 2 s = (v + v )b s. Now we present the proof of Proposition 2.. Proof. We first show the existence of a basis, satisfying the required properties, arguing by induction on the Bruhat order. Thus, we set b e := and b s := H s + v for s S (these are self-dual due to Exercise 2.2) and suppose that b w exist for w x. It is direct to verify that { H sx + vh x, l(x) < l(sx) (2.3) b s H x = H sx + v H x, l(x) > l(sx). Next, to find b x, we use that there exists s S, such that sx x. Hence, using the assumption that b sx exists and formulas (2.3), one can conclude that b s b sx = H x + h y H y y x for some h y Z[v] (the containment follows from the existence of b sx = H sx + h z H z with h(z) v Z[v ± ], formulas (2.3) show that the degrees of monomials in h y are at most one less than the degrees of monomials in polynomials h z it is derived from), i.e. some of the h y s might have constant terms. However, subtracting y x h y (0)b y, we obtain the element b x, which is fixed by τ (as a Z-linear combination of fixed elements), whose coefficients are polynomials in vz[v]. Now we show that b x is unique. Indeed, if we have two elements c = H x +... and c = H x +..., both satisfying (2.2), then c c is also stable under τ and c c vz[v]h y. Now the result follows from Lemma 2.2 below. Lemma 2.2. If h vz[v]h y and τ(h) = h, then h = 0. y W y W,y x Proof. Let z be one of the maximal elements (in the Bruhat order) in the expression of h in the lemma, i.e. we can write h = p z H z + y z p y H y, for some polynomials p z and p y s in vz[v]. Now H z b z + f z Z[v ± ]b f (for some τ-invariant b f s, the existence of which was already established). Hence, τ(h z ) b z + f z Z[v ± ]b f H z + f z Z[v ± ]H f. But then τ(h) = h implies τ(p z ) = p z, and we obtain a contradiction with z sx the assumption p z vz[v].

4 4 BORIS TSVELIKHOVSKY Example 2.. Let us find the Kazhdan-Lusztig basis for the dihedral group W = s, t with s 2 = t 2 = e and sts }{{.. }. = tst }{{.. }.. Clearly, b e = H e, b t = H t + v and b s = H s + v. Next, m m b s b t = H st + v(h s + H t ) + v 2 satisfies the conditions 2.2, so we put b st = b s b t, similarly, b ts = b t b s. Using formulas (2.3), we find b s b ts = H sts + v(h st + H ts ) + vhs 2 + v 2 (H t + 2H s ) + v 3. As vhs 2 = v((v v)h s + ) = v 2 H s + H s + v, we set b sts = b s b ts b s = H sts + v(h st + H ts ) + vhs 2 + v 2 (H t + H s ) + v 3. In general, (2.4) b w = H w + x w v l(w) l(x) H x. Indeed, assume that 2.4 holds for b w, w w. Then, either sw w or tw w. Arguing similarly to the proof of Proposition 2., one can easily verify the formula for b w (w.l.o.g. assume w = sw w): (2.5) b s b w = H w +vh w + v l(w ) l(x) H sx +v l(w ) l(x)+ H x + v l(w ) l(x) H sx +v l(w ) l(x) H x, which is b w + b tw. x w,x=t... x w,x=s... Remark 2.4. In particular, the Weyl groups of types A 2, B 2 and G 2 are dihedral for m = 3, 4 and 6. Hence, the Kazhdan-Lusztig basis is given by Example 2.. The following result and subsequent remark were conjectured in [KL79] and are proved by now. Theorem 2.3 is known as the Kazhdan-Lusztig conjecture. Theorem 2.3. (Kazhdan-Lusztig conjecture) The multiplicity [P (x 0) : (y 0)] is given by the specialization of c x,y at v = (using BGG-reciprocity [ (y 0)] : L(x 0)] equals c x,y v= as well). Remark 2.5. () The polynomials h y,x from 2.2 are in Z 0 [v]. (2) If we write b x b y = µ z x,yb z, then µ z x,y Z 0 [v ± ]. 3. Soergel bimodules Let (W, S) be a Coxeter system. For any two simple reflections s, t S, the order of the element st W will be denoted by m st {2, 3,... }. Definition 3.. An expression of w W is a word w = s i... s ik. The expression w is called reduced if l(w) = k. Next, we fix a vector space h over R, s.t. there exist subsets of linearly independent elements {αs } s S h and {α s } s S h with the following properties: (3.) α s (αt ) = 2Cos( π ) s, t S m st (3.2) s v = v αs (v)α s s S, v h.

5 SOERGEL BIMODULES, HECKE ALGEBRAS, AND KAZHDAN-LUSZTIG BASIS 5 We choose h of minimal dimension with the above properties. Let R = R[h] be the coordinate ring of h. We define the grading on R by setting deg(α) = 2 for any α h. In case W is a Weyl group, h R is a real part of the Cartan subalgebra, the α s s are the roots and α t s are the coroots. The augmentation ideal (ideal of nonconstant polynomials) of R will be denoted by R +. We consider the abelian category of finitely generated graded R-bimodules. All morphisms preserve the grading (in other words, are homogeneous of degree 0). Definition 3.2. For any simple reflection s S set B s := R R s R(). We denote by (n) the shift of grading by the corresponding number, i.e. R R s R() means that the degree of is, etc. The Bott-Samelson bimodule associated to an expression w = s... s m is BS(w) = B s R... R B sm = R R s R R s 2... R sm R(n). By the Bott-Samelson module we will understand BS(w) R R. Definition 3.3. The operator R R given by s (r) := r s(r) 2α s is called the Demazure operator. Notice that s is R s -linear. Exercise 3.. The elements c id := and c s := 2 (α s + α s ) (of degrees and ) form a basis of B s as a left (or right) R-module. One has relations (3.3) (3.4) c s r = rc s rc id = c id s(r) + s (r)c s, Remark 3.. In general, one can check, that the elements c ɛ := c ɛi... c ɛik, where ɛ = s i... s ik runs through all subexpressions of w form a basis of BS(w) as a left (or right) R- module. Notation 3.. Henceforth we abbreviate B si... B sik := B si R... R B sik We provide an example of an easy calculation of the product of two Bott-Samelson bimodules. Example 3.. Using, R = R s R s α s = R s R s ( 2) (the equality of B s -bimodules), we write which is analogous to the relation in H (see Exercise 2.2). B s B s = R R s R R s R = R R s (R s R s ( 2)) R s R = = B s () B s ( ), b 2 s = (v + v )b s Lemma 3.. In Example 5. (W = A 2 ), the Bott-Samelson bimodule BS(s s 2 s ) decomposes into the direct sum B s s 2 s B s, where B s s 2 s = R R W R(3) is the submodule generated by. Proof. Let us verify this decomposition. The main ingredient of the proof is to produce a nontrivial idempotent of degree 0 in End(BS(s s 2 s )). For this we define some morphisms between bimodules: m s Hom(B s, R) : p q pq

6 6 BORIS TSVELIKHOVSKY m a s Hom(R, B s ) : α s + α s j s Hom(B s B s, B s ) : p h q p s (h) q j a s Hom(B s, B s B s ) : p q p q. Notice, that the morphisms m s and m a s have degree, while the the degree of the morphisms j s and js a is. Next, let us introduce e := m a s 2 js a m s2 j s m s2 End(B s B s2 B s ) : B s B s2 B s B s B s j s Bs j a s Bs B s m a s 2 Bs B s2 B s and claim that e is an idempotent. Indeed, this follows m s2 j s Bs B s from the equality j s m s2 m a s 2 js a js a Hom(B s, B s ) : B m a s s Bs B 2 s Bs B s2 B s B s = id shown below: p q ja s m a s p q 2 p (αs2 + α s2 ) q ms 2 2(p α s2 q) js p q. The last transition follows from the equality s (α s2 ) = α s2 + α s and Definition 3.3. Hence, e is a projector. As the first two maps in the definition of e are surjective and the last - injective and the chain of maps is B s B s2 B s B s B s B s B s B s B s B s2 B s, we see that e is the projector onto B s. Next, the morphism e is a projection as well, so, B s B s2 B s = im(e) im( e). We first show that B s B s2 B s is generates by two elements and x, where R = R[x, x 2, x 3, x 4 ]. Indeed, as (x + x 2 )( ) = (x + x 2 ) (as (x + x 2 ) is invariant under s ), we have x 2 and, thus (x x 2 ) = α s is in the submodule, generated by and x. Next, x = x and x 3 = ( )x 3 and (x x 2 ), thus, (x 2 x 3 ) = α s2 are in the submodule as well. Similarly can be shown that the submodule contains α s α s2 and, therefore, by Remark 3. generates the module. The calculations above, in particular, show that dim(r R B s B s2 B s R R) = 2. The fact that dim(r R B s B s2 B s R R) = 2 implies that there are only two indecomposable summands in the decomposition of BS s s 2 s. Now define a map of R-bimodules γ : R R W R(3) BS s s 2 s by p q p q. Since as left R-modules BS s s 2 s = R( 3) R( ) 3 R() 3 R(3) (see Remark 3.) and R R W R(3) = R( 3) R( ) 2 R() 2 R(3), im( e) and R R W R(3) have the same graded dimensions as vector spaces over R, it suffices to show that the map γ is surjective. As ( e)( ) = (follows from s () = 0), and im(γ) as well, it suffices to show that im( e) is generated by. For this we need to show that the submodule, generated by contains im( e)( x ) First we compute e( x ): x ms 2 x js 2 ( ) ja s 2 ( ) ma s 2 2 ( (x 2 x 3 ) + (x 2 x 3 ) ).

7 SOERGEL BIMODULES, HECKE ALGEBRAS, AND KAZHDAN-LUSZTIG BASIS 7 So, ( e)( x ) = x + ( (x 2 2 x 3 ) + (x 2 x 3 ) ). Now we show that this element lies in the submodule generated by. For this we write x = ( x 2 + x ) and show that ( x 2 + (x 2 x 3 ) ) is in the submodule (for ( x 2 + (x 2 x 3 ) ) the computation is completely analogous): 2 ( x + (x 2 x 3 ) ) = 2 ( (x +x 2 x 3 ) = 2 (x +x 2 x 3 ). This concludes the proof. One should notice the resemblance between the decomposition B s B s2 B s = B s s 2 s B s and the relation b s b s2 b s = b s s 2 s + b s derived in Example 2.. Remark 3.2. More generally, it can be shown that if W is a dihedral group generated by simple reflections (s, t) and l(w ) < l(w), where w = sw, then B s B w = B w B tw (compare to (2.5)). Definition 3.4. The category of Soergel bimodules SBim is the full subcategory of Z-graded R-bimodules, where the objects are the direct sums of direct summands of graded shifts of BSbimodules. The morphisms are grading preserving morphisms of R R-bimodules. Similarly, we define the category of Soergel modules SM od to be the full subcategory of Z-graded left R-modules, where the objects are the direct sums of direct summands of graded shifts of BS- modules. The morphisms are grading preserving morphisms of R-bimodules. Remark 3.3. Notice that BS w BS w2 = BS w w 2 implies that the category SBim is closed w.r.t the tensor product. As fg s R i g 2 s R i2... R s in g n = g s R i g 2 s R i2... R s in g n f for f R W, every Soergel bimodule is actually an R R W R-module. Definition 3.5. An additive category is said to be Krull-Schmidt if every object is isomorphic to a direct sum of indecomposable objects and such decomposition is unique up to isomorphism and permutation of summands. Proposition 3.2. The category of Soergel bimodules is Krull-Schmidt. Proof. We notice that the category SBim is closed under taking direct summands (by its definition). Since the bimodule Hom R R (M, N) between any two finitely generated graded bimodules M and N is graded and finitely generated, the degree 0 part is a finite-dimensional space. Thus, the additive category SBim is closed under taking direct summands and has finite-dimensional Hom-spaces. It is a standard fact that such categories are Krull-Schmidt. Next we explain what we mean by the split Grothendieck group K 0 (SBim) of the category SBim. This is the abelian group generated by symbols [B] for all objects B SBim subject to the relations [B] = [B ] + [B ] whenever B = B B in SBim. We make K 0 (SBim) into a Z[v ± ]-module via v i [M] = [M](i) and [M] K 0 (SBim). The tensor product on SBim endows K 0 (SBim) with multiplication, thus, making it a Z[v ± ]-algebra. Moreover, K 0 (SBim) is a free Z[v ± ] - module, whose basis consists of indecomposable objects (we take one up to a grading shift). We can now formulate the main theorem.

8 8 BORIS TSVELIKHOVSKY Theorem 3.3. (Soergel s categorification theorem) There is an isomorphism of Z[v ± ]-algebras H K 0 (SBim), sending b s to [B s ]. Corollary 3.4. (Weak form of Soergel s categorification theorem). There exists a unique homomorphism of rings c : H K 0 (SBim), s.t. c(b s ) = B s. Proof. The quadratic relation was checked in Example 3. and the braid relations - in Lemma 3. (for the simply laced case) and stated in Remark 3.2 for the general case. The uniqueness of c is obvious, since it is defined on a generating set. 4. Soergel s categorification theorem Next we would like to present the classification of indecomposable Soergel bimodules and give a prove of the main theorem. We will use the following proposition (see Section 4 of [Soe92]). Proposition 4.. For two Soergel bimodules B, B 2, the canonical map G : Hom R R (B, B 2 ) R R Hom R (B R R, B 2 R R) is an isomorphism. Corollary 4.2. The map δ : M M R R induces an embedding of indecomposable objects in SBim into indecomposable objects in SM od. Proof. We first show that the image of an indecomposable module M is indecomposable. Indeed, if δ(m) would decompose as M M 2 there would be a degree zero idempotent e M End R (M R R), but since End R (δ(m)) = End R R (M) R R, this implies the existence of a degree zero idempotent (there exists a lift - this is a standard fact, which can be shown by constructing the lifts modulo (R + ) n for every n N and (R + ) n End R R (M) has no degree 0 elements for n large enough) ẽ End R R (M) R R, which (as M = ẽm ( ẽ)m and ẽ ) contradicts our assumption that M is indecomposable. Next we check that δ maps non isomorphic indecomposables to non isomorphic ones. Assume the contrary and let M, M 2 SBim be indecomposable and δ(m ) = δ(m 2 ) = M SMod. Then there exist a α Hom R R (M, M 2 ) and β Hom R R (M 2, M ), s.t. G(α β) is invertible. The application of graded Nakayama s lemma implies α β is invertible: α β(m 2 ) + R + M 2 = M 2 M 2 R + α β(m 2 ) = M 2 α β(m 2 ) gives M 2 = α β(m 2 ). So α β is surjective, hence invertible. Theorem 4.3. Each Bott-Samelson bimodule BS(w) contains a unique indecomposable summand B w which does not appear in BS(x) for x w depends only on w, but not on the reduced expression. Proof. Recall from Dmytro s talk (Corollary 5.9) that, for a reduced w, the module BS(w) R R contains a unique graded indecomposable summand, S w, that does not appear in BS(w ) R R for shorter w and that depends only on w. In fact, in Dmytro s talk the claim was proved over C but one can show it holds over R as well. So BS(w) R R = S w w S w (d i ) n i where the sum is taken over w w. Let BS(w) = B... B k be the decomposition into

9 SOERGEL BIMODULES, HECKE ALGEBRAS, AND KAZHDAN-LUSZTIG BASIS 9 indecomposables. By Corollary 4.2, there is a unique index i (say i= to be definite) such that B R R = S w and then B i R R = S w (d w ) for i >. We set B w := B. Our claim follows from the induction on the length of w and Corollary 4.2. Corollary 4.4. The indecomposable Soergel bimodules are in bijection with the elements of W Z. Proof. The result follows from the observation that grading shifts preserve indecomposability. The above results allow us to prove the main theorem (Theorem 3.3). Proof. We choose one reduced expression w = s... s m for every element w W, then it follows from Theorem 4.3 that the classes of the corresponding BS(w) s form a basis of K 0 (SBim) (each [BS(w)] contains the indecomposable [B w ] as a summand with coefficient and it is not hard to show by induction on the Bruhat order that there exists a Z[v ± ]-linear combination of [BS(w )], w w that, being subtracted from [BS(w)], gives [B w ]). Then the corresponding elements b s... b sm H are also a basis (again, using induction on the Bruhat order analogously to the proof of Proposition 2., we show that b w is b s... b sm minus a Z[v ± ]-linear combination of b sj... b sjk for w = s j... s jk w). 5. Bott-Samelson varieties Let G be a connected reductive complex algebraic group, T B G a maximal torus and a Borel subgroup. Every element w in the Weyl group W = N G (T )/T has a reduced expression w = s i... s il with s ij - simple reflections. We also denote by P ij the minimal parabolic subgroup generated by B and a representative of s ij W (the Lie algebra p ij := Lie(P ij ) = b f ij with f ij - a negative simple root). Definition 5.. The Schubert cell C w is the variety BwB/B and the closed Schubert subvariety C w is BwB/B. Definition 5.2. The Bott-Samelson variety BS w is defined as P i B P i2 B... B P il /B, where the action of B l(w) on P i P i2... P il is given by (b,..., b l ) (p i,..., p il ) = (p i b, b p i2 b 2,..., b l p il b l ). As P ik /B P, one can think of a Bott-Samelson variety as a tower of P -bundles. particular, these varieties are smooth. Moreover, there is a surjective map ϕ : BS w C w ϕ(p i,... p il ) = p i... p il, which restricts to an isomorphism on the preimage of C w. Hence this map is birational and, therefore, a resolution of singularities. However, the following example illustrates that even if C w is smooth ϕ is not necessarily an isomorphism. In

10 0 BORIS TSVELIKHOVSKY Example 5.. We consider the element s s 2 s S 3, the corresponding Schubert variety C w is the flag variety, hence, smooth. However, the map ϕ : BS s s 2 s C s s 2 s is not an isomorphism. One reason is that the Poincare polynomial for the cohomology of BS s s 2 s is ( + q 2 ) 3, while the one for C s s 2 s is + 2q 2 + 2q 4 + q 6 = ( + q 2 )( + q 2 + q 4 ) (as C s s 2 s is the flag variety GL 3 /B). Alternatively, one can see, that the preimage of any point in relative position e or (2) with respect to the standard flag is isomorphic to P. Before stating and proving the result, explaining the connection of Bott-Samelson varieties and Bott-Samelson bimodules, we need some definitions. Definition 5.3. The variety X is called equivariantly formal if H T (X) (the equivariant cohomology with respect to the algebraic torus T = (C ) n -action) is a free R-module. From now on we assume that X is a complex projective algebraic variety, which is equivariantly formal and has finitely many T - fixed points and one-dimensional orbits. Definition 5.4. The moment graph of a T -variety X is a triple (V, E, χ E ), where V is the set of vertices and is in bijection with the points of X T ; E is the set of edges and is in bijection with the one-dimensional orbits of T on X, each edge connects the two vertices of V, corresponding to the points in its closure; For any edge in E the closure of the corresponding orbit is isomorphic to P with two fixed points. The action of T on this P produces a character T C. We denote the differential of this character by χ E and put it as a label on the corresponding edge. Theorem 5.. () There is an isomorphism of R - bimodules H T (BS s...s m ) = R R s R R s 2... R sm R (2) H (BS s...s m ) = R R s R R s 2... R sm R R R Proof. We verify () using the localization theorem (Theorem.2.2. of [GKM98]), which says that if X is equivariantly formal and has finitely many T - fixed points and one-dimensional orbits, then there is an embedding i : H T (X) H (X T ) R (this embedding comes from the map i : X T X and becomes an isomorphism after tensoring with the quotient field Quot(R), but we will not need this). Furthermore, one can give a precise description of the image of i. By Theorem.2.2. of [GKM98] the T -equivariant cohomology of X are given by { (f x ) x V R x χ E divides f x f y whenever x and y are connected by the edge E }. In our case, there are 2 m fixed points. These are the points (p i,..., p im ) BS w = P i B P i2 B... B P im /B with each p ik B or s ik B. Two vertices are connected, whenever the corresponding sequences of elements differ only in one entry. The example of the moment graph for BS s s 2 s is given on Figure 5. Number the fixed points so that the corresponding vertices of the moment graph are arranged from bottom to top and from left to right on each row.

11 SOERGEL BIMODULES, HECKE ALGEBRAS, AND KAZHDAN-LUSZTIG BASIS (s, s 2, s ) α s s α s2 s s 2α s (s, s 2, id) (s, id, s ) (id, s 2, s ) α s2 α s s 2α s α s s α s s α s2 (s, id, id) (id, s 2, id) (id, id, s ) α s α s2 α s (id, id, id) Figure. Moment graph of BS(s s 2 s ) Define the map θ : BS w H T (BS w) as follows: θ maps the element g f... f m to g(f... f m,..., s im (f m (s im (f l... s i (f ))))) R 2l(w), where the ith coordinate is given by the following procedure: apply the leftmost element of the sequence corresponding to the vertex with number i (i.e. id or s ) to f, then the second element from the left to the product of f 2 with the result of the first application, etc. For instance, the image of θ : BS s s 2 s H T (BS s s 2 s ) is given by θ(g f f 2 f 3 ) = g(f f 2 f 3, s (f )f 2 f 3, s 2 (f f 2 )f 3, s (f f 2 f 3 ), s 2 s (f )s 2 (f 2 )f 3, s (s (f )f 2 f 3 ), s (s 2 (f f 2 )f 3 ), s (s 2 (s (f )f 2 )f 3 )). It remains to show that θ is an isomorphism. For this purpose, it is more convenient to rewrite BS s...s m using the method of Example 3. multiple times (starting from the rightmost copy of R as we look at BS s...s m a left R-module) as R R( 2)... R( 2) }{{} m... R( 2m) (m). Now the key observation is that we can recover the decomposition of F = g f... f m in terms of the basis provided in Remark 3. from the image of θ. As every f R can be written as f = s+ s s+ (f)+ (f), with (f) Rs and s (f) 2 Rs α s, the coefficient of the basic element c ( ( ɛ in the decomposition of F equals to g s ± f... s m ± s fm m±(f m ) )... ), where the sign is a ( ( minus, if c ɛk = c sk and a plus in case c ɛk = c id. The expression g s ± f... s m ± s fm m±(f m ) )... ) is nothing else, but the sum of the monomials in θ(f ) with coefficients equal to ± 2 m.

12 2 BORIS TSVELIKHOVSKY The second assertion is a consequence of the basic fact that H (X) = H T (X)/(R+ H T (X)), whenever X is equivariantly formal (see remarks after Theorem.2.2. of [GKM98]). References [BB8] A. Beilinson and J. Bernstein. Localisation de g-modules. C. R. Acad. Sci. Paris SÂťer. I Math., 292(), 98. [BK8] J.-L. Brylinski and M. Kashiwara. Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math., 64(3), 98. [EW] B. Elias and G. Williamson. The Hodge theory of Soergel bimodules. arxiv:22.079v2. [GKM98] M. Goresky, R. Kottwitz, and R. MacPherson. Equivariant cohomology, Koszul duality and the localization theorem. Invent. Math., 3():25 83, 998. [KL79] D. Kazhdan and G. Lusztig. Representations of Coxeter groups and Hecke algebras. Invent. Math., 53(2):65 84, 979. [Leb] N Lebedinsky. Gentle introduction to Soergel bimodules I: the basics. arxiv: v. [Soe90] W. Soergel. Kategorie O, perverse Garben und Moduln uber den koinvarianten zur Weylgruppe. JAMS, 3(2):42 445, 990. [Soe92] W. Soergel. The combinatorics of Harish-Chandra bimodules. J. Reine Angew. J. Reine Angew. Math., 429:49 74, 992. Department of Mathematics, Northeastern University, Boston, MA, 025, USA address: tsvelikhovskiy.b@husky.neu.edu