Intersection cohomology of B �B-orbits in groupcompacti�cationsT.A. SpringerIntroduction.An adjoint semi-simple group G has a \wonderful" compacti�cation X ,which is a smooth projective variety, containing G as an open subvariety. Xis acted upon by G� G and, B denoting a Borel subgroup of G, the groupB�B has �nitely many orbits in X . The main results of this paper concernthe intersection cohomology of the closures of the B � B-orbits. Examplesof such closures are the \large Schubert varieties", the closures in X of thedouble cosets BwB in G.After recalling some basic results about the wonderful compacti�cation, wediscuss in no. 1 the description of the B�B-orbits, and establish some basicresults.In no. 2 the \Bruhat order" of the set V of orbits is introduced and de-scribed explicitly. As an application we obtain cellular decompositions ofcertain orbit closures, among which are the large Schubert varieties.Let H be the Hecke algebra associated to G, it is a free module over analgebra of Laurent polynomials Z[u; u�1]. As a particular case of results of[MS], the spherical G � G-variety X de�nes a representation of the Heckealgebra associated to G�G, i.e. HZ[u;u�1 ]H, in a free module M over anextension of Z[u; u�1], with a basis (mv) indexed by V . The de�nition of Mis sheaf-theoretical, working over the algebraic closure of a �nite �eld. Thisis discussed in no. 3. On the model of [LV] a duality map � is introducedon M, coming from Verdier duality in sheaf theory. The matrix coe�cientsof � relative to the basis (mv) are discussed at the end of no. 3, They bearsome resemblance to the R-polynomials of [KL].In no. 4 it is shown that the intersection cohomology of an orbit closure�v leads to \Kazhdan-Lusztig" elements in M. The results about matrixcoe�cients of no. 3 together with results of [MS] imply the evenness of localintersection cohomology, and the existence of Kazhdan-Lusztig polynomials.We also prove evenness of global intersection cohomology of closures �v.The results on intersection cohomology, proved in the �rst instance in posi-tive characteristics, then also follow in characteristic 0, and over C.1

No. 5 contains a brief discussion of the extension of results of the previoussections to intersection cohomology of an orbit closure �v, for certain non-constant sheaves on v.We have formulated the constructions of the paper (e.g. of theHH-moduleM) in such a manner that they also make sense for general Coxeter groups.No. 6 contains some remarks about the constructions for such groups.Computation by hand of our Kazhdan-Lusztig polynomials turns out tobe quite cumbersome, the only manageable case (for the author) beingG = PGL2. The Appendix by W. van der Kallen gives a number of nu-merical examples, obtained by computer calculations.1. Preliminaries.In the sequel G is a connected, adjoint, semi-simple group over the alge-braically closed �eld k. We denote by B and T a Borel group and a maximaltorus contained in it. R is the root system of (G; T ) and R+ the system ofpositive roots of R de�ned by B. The Weyl group of R is W . For w 2 Wwe denote by _w a representative in the normalizer N(T ).We denote by S the set of simple re ections de�ned by R+, and by D theset of simple roots. For I � D let WI � W be the parabolic subgroup of Wgenerated by the re ections in the roots of I . We write SI = S \WI .Denote by W I = fx 2 W j x(I) � R+g the set of distinguished coset repre-sentatives of W=WI and by w0;I the maximal element of WI . On W and itssubsets WI and W I we have the usual Bruhat orders.1.1. We introduce the \wonderful" compacti�cation X of G. We recall anumber of results, established in [DS] and [B1]. (In [DS] it is assumed thatchar(k) 6= 2. This restriction is necessary in the general situation discussedthere, but is unnecessary for the compacti�cation of G.)X is an irreducible, smooth, projective G � G-variety. It contains G asan open G � G-stable subvariety (the action being (g; h):x = gxh�1, forg; h; x 2 G). The G� G-orbits XI in X are indexed by the subsets I of D.Let PI be the standard parabolic subgroup de�ned by I � D, the notationbeing such that the Levi subgroup LI containing T has root system withbasis I . We denote by CI the center of LI , by GI = GI=CI the correspond-ing adjoint group, by BI � GI the image of B \ LI and by TI the image ofT . Then BI is a Borel group of GI and TI a maximal torus. Notice that CIis connected. (Since G is adjoint, D is a basis of the character group of T .2

Since I is a basis of the character group of TI , it follows that CI ' TD�I ,hence is connected.) Let B� � T be the opposite of B and P�I � B� theopposite of PI . Notice that LI is a Levi subgroup of both PI and P�I .The G-orbit XI is a G�G-equivariant �ber space over G=P�I �G=PI , suchthat the �ber over P�I � PI is GI . In fact,XI = (G� G)�P�I �PI GI ; (1)P�I � PI acting on GI via its quotient GI � GI . Similarly,XI = (G� G)�P�I �PI GI :XI contains a unique base point hI such that(a) (B �B�):hI is dense in XI ,(b) there is a cocharacter � of T with hI = limt!0 �(t) (see [B1, Prop. A1]).Under the identi�cation (1), hI is the image in XI of (1; 1; 1). We havehD = 1.If I � D we de�ne I� � D by I� = �w0;D(I). Then (I�)� = I . If A is asubset of W we put A� = w0;DAw0:D. We shall use the following result.1.2. Lemma. (i) The isomorphism g 7! g�1 of G extends to an isomor-phism � of the variety X such that for g; h 2 G; x 2 X�:((g; h):x) = (h; g):(�:x);(ii) �(XI) = XI�;(iii) �:hI = (w0;D; w0;D):hI�.Proof. For (i) see [S, 1.2]. (ii) readily follows from the proof of (i) and (iii)is a consequence of the characterization of the points hI given above.B � B has �nitely many orbits in X . They are described in the followinglemma.For w 2 W I put �I(w) = w0;Dww0;I. Then �I is a bijection of W I andl(�I(w)) = l(w0;D)� l(w0;I)� l(w): (2)1.3. Lemma. (i) The B �B-orbits in XI are of the formO = (B � B):(x; w):hI (3)with unique elements w 2 W;x 2 W I ;(ii) dimO = l(w0;D)� l(x) + l(w) + jI j;3

(iii) �:O = (B � B):(�I�(y�); �I�(x�)))(w0;I�(z�)�1w0;I�):hI�, where y 2W I ; z 2 WI and w = yz;(iv) The isotropy group of (x; w):hI in B�B is the semi-direct product of aconnected unipotent normal subgroup and the isotropy group in T �T , whichconsists of the (t; t0) 2 T � T with x�1(t):w�1(t0)�1 2 CI.Proof. The B � B�-orbits in XI are of the form(B � B�):(v; y):hI = (B �B�):(x; w)where x; y 2 W I ; z 2 WI and v = xz�1; w = yz. (by [B1, 2.1]). Hence theB �B-orbits are of the form(B � B):(x; w0;Dyz):hI :This proves (i).For v = xz�1 as before we haveO1 = (B � B�):(v; y):hI = (B �B�):(x; y):(BI �B�I ):(1; z�1):hI (4)(see [loc.cit. p. 151]). It follows thatdimO1 = dimBxP�I + dimB�yPI + dimBIz�1B�I :Now BxP�I = w0;DB��I(x)P�I , whence dimBxP�I = l(�I(x)). Similarly,dimB�yPI = l(�I(y)). We conclude thatdimO1 = l(�I(x)) + l(�I(y)) + l(w0;I)� l(z) + dimBI :By the proof of (i) we havedimO = dim (B �B�):(vw0;I; �I(y)):hIThe formula of (ii) follows from the previous formula and (2).By 1.2 we have �:O = (B �B)(ww0;D; xw0;D):hI� . Nowww0;D = w0;Dw� = �I�(y�)w0;I�z�;and similarly for xw0;D. The formulas imply (iii), using that (u�; u�) �xeshI� if u 2 WI .Finally, (iv) follows from the description (1) of XI recalling that hI is theimage of (1; 1; 1) (cf. [B1, Prop. A1]).Let V be the set of B � B-orbits in X . For v 2 V we write d(v) = dim v.4

We denote the orbit of (3) by [I; x; w] or [I; x; w]G. Thus, the elements of Vare parametrized by triples I � D; x 2 W I and w(= yz) 2 W . By 1.3 (ii)dim [I; x; w] = l(w0;D)� l(x) + l(w) + jI j: (5)It follows from 1.3 (iii) that�:[I; x; yz] = [I�; y�; x�w0;I�(z�)�1w0;I�]:For I = D, [D; 1; w] is the double coset BwB. Its closure in X is the largeSchubert variety Sw.The combinatorial set-up introduced in [RS] carries over -at least partly- toV and the subsets VI � V of B � B-orbits in XI (I � D).Let M be the monoid M(W �W ) (see [loc.cit., 3.10]). It operates on V . Lett = (s; 1) or (1; s) be a simple re ection of W �W and put Ps = B [ BsB(a minimal parabolic subgroup of G). If v 2 V then m(t):v is the openB � B-orbit in (Ps � f1g):v if t = (s; 1), and similarly for t = (1; s). Thisde�nes an action of M on V , stabilizing all VI (I � D). If m(t):v 6= v thend(m(t):v) = d(v) + 1 (cf. [loc.cit.,7.2]).In [MS, 4.1] an analysis is made of the action of a minimal parabolic groupon the orbits of a Borel group in a spherical variety. This applies to thepresent situation, for the group G = G�G and its spherical variety X . Weuse obvious notations like B = B �B.In general there are four possible cases, labeled I, II, III, IV in [loc.cit.].However, in the present case the situation is rather simple, as follows fromthe next lemma.1.4. Lemma. Let v 2 V and let t 2 S be a simple re ection of W.(i) Pt:v is the union of two B-orbits;(ii) If m(t):v 6= v then m(t):v = BtB:v and Pt:v = v [m(t):vProof. Let t = (s; 1). We havePsxP�I = BxP�I [BsxP�I :BsxP�I is the piece with largest dimension if and only if sx < x. In thatcase BsB:BxP�I = BsxP�I . It follows that then(Ps �B�):(x; w):hI = (B � B�):(sx; w):hI [ (B �B�):(x; w):hI:This implies (i). Also, if m(t):v 6= v, then sx < x and(B � B�)s(B � B�)xP�I = (B �B�)sxP�I :5

These facts imply (i) and (ii) in the case t = (s; 1). The proof in the othercase is similar.From (i) it follows that v and t are in the case II of [MS, 4.1.4].Recall that if x 2 W I an s 2 S there are three possibilities:(a) x > sx and sx 2 W I , (b) sx > x and sx = xt with t 2 SI , sx < x inwhich case sx 2 W I .1.5. Lemma. Let s 2 S and v = [I; x; w] 2 VI .(i) m((s; 1)):v 6= v if and only if we have for x and s case (c) or (b) withwt > t. In these cases we have, respectively, m((s; 1)):v = [I; sx; w] andm((s; 1)):v = [I; x; wt];(ii) m((1; s)):v 6= v if and only if sw > w, in which case m((1; s)):v =[I; x; sw].Proof. This is a consequence of the proof of 1.4.We end this section with some facts on local systems on the (B�B)-orbits,needed in no. 5.1.6. If S is a torus denote by X(S) its character group and by X(S) thetensor product X(T )Z(Z(p)=Z), where Z(p) is the localization at the primeideal (p) of Z, p being the characteristic. The group X(S) parametrizes the(tame) local systems on S which have rank one, see [MS, 2.1].If Y (S) is the group of cocharacters of S we have, similarly, a group Y (S).Between X(S) and Y (S) there is a pairing h ; i, with values in Z(p)=Z.By [MS, 2.2.3] the rank one local systems on v which have a weight forthe B � B-action are of the form c�v� (� 2 X(TI), where c�v is the inducedhomomorphism X(TI) ! X(T � T ).Since G is adjoint the character group X(T ) has basis D. Likewise, X(TI)has basis I , whence an injection X(TI) ! X(T ), which is the homomor-phism induced by T ! TI . Clearly, X(TI) is a direct summand of X(T ),from which we see that the induced homomorphism X(TI) ! X(T ) is injec-tive (cf. [MS, 2.1.5], recall that the kernel CI of the homomorphism T ! TIis connected).We shall identify X(TI) with a subgroup of X = X(T ). In the sequel weshall write X = X(T ) and XI = X(TI).1.7. Let v 2 V . If _v 2 v the isotropy group (B � B) _v is the semi-directproduct of a connected unipotent group and the isotropy subgroup in T �T ,6

by 1.3 (iv). The latter group group being independent of the choice of _v by[loc.cit., 2.2.5], we denote it by (T � T )v. Denote by �v the homomorphismT � T ! TI sending (x(t); w(t0)) to t(t0)�1CI . Then �v induces an isomor-phism (T � T )=(T � T )v ' TI .Let �v (�v) be the composite of �v and the injection t 7! (t; 1) (respectively,t 7! (1; t)) of T into T � T .1.8. Lemma. Let v = [I; x; w]; � 2 XI . Then c�v(�) = x�1:�, b�v(�) =�w�1:� and c�v = (c�v; b�v).Proof. The proof is straightforward.2. The Bruhat order.On V and its subsets VI we have a \Bruhat order" �, de�ned by the inclu-sion of orbit closures. We discuss it in this section. The order on W and itssubsets WI and W I is the usual one.2.1. Lemma. Let v; v0 2 V and let t 2 S. Then(a) v � m(t):v,(b) If v0 � v then m(t):v0 � m(t):v,(c) If v0 � v then d(v0) � d(v), with equality if and only if v = v0,(d) VI (I � D) has a unique minimal element, viz. BI = [I; w0;Dw0;I ; 1].Proof. The last point follows by using the dimension formula (5). The otherpoints are proved as similar results in [RS, 7.2].From 1.5 we see that VI is an M -set whose order is compatible with theM -action in the sense of [RS, no.5]2.2. Lemma. Let I � D; x; x0 2 W I , w;w0 2 W . Then [I; x0; w0] �[I; x; w] if and only if there exists u 2 WI such that xu�1 � x0; w0u � w.Proof. We havedim[I; x; w]� dimBI = l(xw0;Iw0;D) + l(w):Let (t1; :::; tl) be a reduced decomposition of (xuw0;Iw0;D; w) 2 W, the tibeing simple re ections of W. It follows by repeated application of 1.5 (ii)that [I; x; w] = m(t1):::m(tl):BI = m((xw0;Iw0;D; w)):BI:By familiar arguments (cf. [BT, 3.13]) one shows that[I; x; w] = Pt1 :::Ptl:BI ;7

from which one concludes that [I; x; w] is the union of the orbits m((c; d)):BI,where c � xw0;Iw0;D and d � w.Let [I; x0; w0] � [I; x; w] and take c; d as before with [I; x0; w0] = m((c; d)):BI.Write c = c0uw0;Iw0;D, with c0 2 W I ; u 2 WI . Then c0uw0;I � xw0;I and[I; x0; w0] = m((c; d)):BI = m((c0uw0;Iw0;D; d)):BI = (B�B):(c0u; d):hI = [I; c0; du�1]:We conclude that x0 = c0; w0 = du�1. Hence w0u � w and xw0;I � x0uw0;I .It follows that there exist u1; u2 2 WI such thatxu1 � x0; u2 � uw0;I ; u1u2 = w0;I :Then xu�1 � xw0;Iu�12 = xu1 � x0:So there exists u with the asserted properties. Conversely, if this so, thenxw0;I = xu�1uw0;I � x0uw0;Iand we conclude that there exist c and d as before, whence [I; x0; w0] �[I; x; w].Let J be a subset of D containing I . Let x; y 2 W J ; z 2 WJ ; w = yz.2.3. Lemma. If [J; x; w]\XI 6= ; then I � J. In this is so the intersectionis the union of the orbit closures [I; xv; wv] with v 2 WJ \W I and l(wv) =l(w) + l(v).Proof. The �rst point is immediate. By [B1, 2.1, Theorem] we have forx; y 2 W J ; z 2 WJ(B � B�):(xz�1; y):hJ \XI =[ (B �B�):(xz�1v; yv):hI;where v runs through the elements of WJ such that yv 2 W I and l(xz�1) =l(xz�1v) + l(v). Such a v lies in WJ \W I , from which one sees that alsol(yvu) = l(y) + l(vu) for u 2 WI .Since x 2 W J , we have l(z) = l(z�1v) + l(v). Write xz�1v = x1u1, withx1 2 W I , u1 2 WI and put v1 = z�1vu�11 . Then xv1 2 W I and l(yzv1) =l(yvu�11 ) = l(y) + l(vu�11 ). We can conclude that(B �B�):(x; w):hJ \XI =[ (B �B�):(xv; wv):hI;where v runs through the elements of WJ with xv 2 W I , i.e. v 2 WJ \W Iand l(w) = l(wv) + l(v). Using the relation between B � B�-orbits and8

B �B-orbits of the proof of 1.2 we obtain the assertion of the lemma.2.4. Proposition. Let x0 2 W I ; x 2 W J ; w; w0 2 W . Then [I; x0; w0] �[J; x; w] if and only if I � J and there exist u 2 WI ; v 2 WJ \ W I withxvu�1 � x0, w0u � wv and l(wv) = l(w) + l(v). If this is so we have x0 � xand �l(x0) + l(w0) � �l(x) + l(w).Proof. The preceding lemmas imply the �rst assertions. If u and v with theasserted properties exist, then since x 2 W Jx0 � xvu�1 � x:The last inequality follows the fact that B-orbit closures in XJ intersect XIproperly if I � J (see [B2, 1.4]).2.5. Corollary (i) [I; x0; w0] � [J; x; 1] if and only if I � J, w0 2 WJ andxw0 � x0;(ii) [I; x0; w0] � [D; 1; w] if and only if there exists w1 � w with w�11 w0 � x0;(iii) [I; x; w]� �B if and only if w � x.Proof. By the proposition [I; x0; w0] � [J; x; 1] if and only if I � J and thereexist u 2 WI ; v 2 WJ \W I with xvu�1 � x0 an w0u � v. From the secondrelation we infer that w0 � vu�1. Since vu�1 2 WJ , w0 also lies in WJ .Moreover, xw0 � xvu�1 � x0. We have established the conditions of (i).If, conversely, they are satis�ed write xw0 = xvu�1 with u 2 WI ; xv 2 W I .Then w0 = vu�1. Since w0 2 WJ the same holds for v, so v 2 WJ\W I . It fol-lows that u and v are as required in the proposition, and [I; x0; w0] � [J; x; 1].This proves (i). The special case J = D gives (iii).By the proposition [I; x0; w0] � [J; 1; w] if and only if I � J and there existu 2 W; v 2 W J \ W I with vu�1 � x0 an w0u � wv, l(wv) = l(w) + l(v).There exist w1 � w; v1 � v such that w0u = w1v1 and l(w1v1) = l(w1)+l(v1).Then w�11 w0 = v1u�1 � vu�1 � x0, establishing the condition of (ii).To prove the su�ciency of the condition it su�ces to deal with the casethat w1 = w. So w�1w0 � x0. Write w�1w0 = vu�1, with v 2 W I ; u 2 WI .Then w0u = wv. There exist w1 � w and v1u1 � v (v1 2 W I ; u1 2 WI)such that w0u = w1v1u1 and l(w1v1u1) = l(w1) + l(v1) + l(u1). Thenw0w1v1uu�11 � w1v1, v1u1u�1 � vu�1 � x0 and l(w1v1) = l(w1) + l(v1).Itfollows that [I; x0; w0] � [D; 1:w1] � [D; 1; w]. We have proved (ii). (iii) isalso a special case of (ii).The explicit description of the Bruhat order of V given in 2.4 is a bit cumber-some. We present another description which is somewhat more transparent.De�ne two relations �1 and �2 on V by [I; x0; w0] �1 [J; x; w] if I � J and9

x0 � x; w0 � w and [I; x0; w0] �2 [J; x; w] if I � J and there is z 2 WJ withxz � x0 and w0 = wz; l(wz) = l(w) + l(z).2.6. Lemma. (i) If v; v0 2 V , v0 �i v (i = 1; 2) then v � v0;(ii) �1 and �2 are order relations.Proof. (i) is obvious for �1 (take u = v = 1 in 2.4). If z is as in the de�nitionof �2, write z = vu�1 (v 2 W I ; u 2 WI . Then u and v are as required in2.4.That �1 is an ordering is obvious. The proof that �2 is an ordering isstraightforward.2.7. Lemma. Let v; v0 2 V , v0 � v. There is ~v 2 V with v0 �2 ~v �1 v.Proof. The notations being as in 2.4, there is w1 � w and v1u1 � v(v1 2 W I ; u1 2 WI) with wu = w1v1u1, l(w1v1u1) = l(w1) + l(v1) + l(u1).Then wuu�11 = w1v1, l(w1v1) = l(w1) + l(v1) and xvu1u�1 � vu�1 � x0. Itfollows that v0 �2 [J; x; w1]. Moreover, [J; x; w1] � v. Hence ~v = [J; x; w1] isas required.2.8. Proposition. � is the ordering generated by �1 and �2.Proof. By de�nition, the ordering on V generated by �1 and �2 is suchthat v0 is smaller than v for that order if and only if there exists a chainv0 = v0; v1; :::; vs = v of elements of V such that for i = 1; :::; s eithervi�1 <1 vi or vi�1 <2 vi. By the preceding lemmas it is immediate that thisorder is �.In no. 4 we shall �nd the M�obius function of our oredering, see the Remarkafter 4.6.If w 2 W put I(x) = f� 2 D j x:� 2 R+g and for x; w 2 W putXx;w = [I�I(x)[I; x; w]:2.9. Lemma. (i) [I; x; w]� [I(x); x; w];(ii) Xx;w = [I(x); x; w];(iii) If [I; x0; w0] � [J; x; w] then [I(x0); x0; w0] � [I(x); x; w];(iv) The closure Xx;w is a union of sets Xx0;w0;(v) The sets Xx;w are locally closed and X is their disjoint union.Proof. It is immediate that [I; x; w]�1 [I(x); x; w]. (i) then follows from 2.8and (ii) is a consequence of (i).To prove (iii) it su�ces, using 2.8, to prove that the corresponding assertions10

hold for �1 and �2. But this is immediate from the de�nitions.The closure of a B�B-orbit is a union of such orbits. Hence by (ii), Xx;w is aunion of B�B-orbits. If [I; x0; w0] � Xx;w then by (iii) [I(x0); x0; w0] � Xx;w,whence by (i) Xx0;w0 � Xx;w. This proves (iv). Finally, the �rst part of (v)is a consequence of (iv) and the second part is clear.By 2.8 (v), Xx;w is an algebraic variety.2.10. Proposition. The variety Xx;w is isomorphic to a�ne space ofdimension l(w0;D)� l(x) + l(w)+ j I(x) j.Proof. For � 2 R let U� be the one-parameter additive subgroup of Gde�ned by �. If y 2 W let Uy be the subgroup generated by the U� with� 2 R+; y�1:� 2 �R+. It is a subgroup of the unipotent part U of B.We have [I; x; w] = (B � U):( _x; _w):hI . The isotropy group I of ( _x; _w) inB � U is the set of (tu; u0) 2 B � U with( _x�1tu _x; _w�1u0 _w) = (clv; lv0);where c 2 CI ; l 2 LI ; v 2 Ru(P�I ); v0 2 Ru(PI) (see [B1, Prop. A1]).The morphism (u; u0; tCI) 7! (u; u0):( _x; _w):t:hIof Uxw0;D � Uw � T=CI to [I; x; w] is bijective. This follows from the obser-vation that Uxw0;D � Uw \ I = f1g.We infer that( _x�1; _w�1):Xx;w = ( _x�1Uxw0;D _x; _w�1Uw _w):( [I�I(x)T:hI) � (U � U�):T:hD:But by [DS, 3.8], the last set is isomorphic to an a�ne space and it followsfrom the results of [loc. cit., p. 285] that ( _x�1Uxw0;D _x; _w�1Uw _w):(SI�I(x) T:hI)is an a�ne subspace. The proposition follows, using (5).2.11. By 2.10 and 2.9 the closure Xx;w has a \cellular decomposition" (or\paving by a�ne spaces") with good properties. It is well-known that thisimplies that the odd cohomology of such a variety vanishes and that its 2i-th Betti number equals the number of i- dimensional cells. In the case ofa large Schubert variety Sw = [D; 1; w] = X1;w this leads to the followingresult. If X is an algebraic variety we denote byPX(t) =Xi�0 dimH i(X)11

its Poincar�e polynomial, with constant coe�cients (in l-adic cohomology, orin classical cohomology if k = C).2.12. Corollary. The Poincare polynomial PSw equalsX[I(a);a;b]�[D;1;w] t2(l(w0;D)�l(a)+l(b)+jI(a)j):2.5 (ii) makes the summation more explicit.In the particular case w = 1 we have S1 = B. Then the formula simpli�esby 2.5 (iii) to PB(t) =Xb�a t2(l(w0;D)�l(a)+l(b)+jI(a)j):For G = PGL2 the right-hand side is 1 + t2 + t4 and for G = PGL3 it is1 + 2t2 + 4t4 + 7t6 + 4t8 + t10:In the particular case w = w0;D one obtains a known formula for PX (see[DP, 7.7]).Another consequence of 2.10 is that the Chow group A�(X) is freely gener-ated by the classes [Xx;w], which is a reformulation of a result due to Brion(see [B1, 3.3]).2.13. From the description of Xx;w given in the proof of 2.10 it is immediatethat there is a cocharacter of T � T wich contracts Xx;w to the �xed point(xB�; wB) 2 X; of T � T in X . Hence our cellular decomposition of X isa Bya linicki-Birula decomposition. It enjoys good properties: any cell is aunion of B � B-orbits, and large Schubert varieties are unions of cells.3. A Hecke algebra representation.3.1. Let H be the Hecke algebra of W . It is a free module over the ring ofLaurent polynomials Z[u; u�1], with a basis (ew)w2W . The multiplication isde�ned by the rulesesew = esw if sw > w= (u2 � 1)ew + u2esw if sw < w;where w 2 W; s 2 S.The variety X is a spherical variety for G = G � G. We now invoke the12

results of [MS], where for any spherical variety a module M over a Heckealgebra is constructed (on the model of the work of Lusztig and Vogan in[LV] in the case of spherical varieties). In our case this is the Hecke algebraassociated to W, i.e. H Z[u;u�1 ] H.There are several technicalities which have to be taken care of. In the �rstplace, one takes the base �eld k to be an algebraic closure of a �nite �eldFq, and one assumes all ingredients of the constructions to be de�ned overFq (which is possible, as there are only �nitely many such ingredients). Themodule M is free, with a basis indexed by set V of orbits v of B in X . Inthe general situation considered in [loc.cit.], the basis elements also involvelocal systems on the orbits. In the present section we consider the case thatall local systems are trivial (that this is possible is a consequence of the factthat, with the notations of [MS, 4.1.4], in X only the case II occurs (wehave already observed that this follows from 1.4). A more general situation,where non-trivial local systems on the orbits will occur, will be taken up inno. 5.In the set-up of [MS] a basis element mv de�ned by v 2 V comes as a classin a Grothendieck group. More precisely, let AX be the category of con-structible Ql-sheaves S on X , provided with an isomorphism � : F �S ! S(where F is the Frobenius morphism). (S;�) and (S;�0) are identi�ed if�n = (�0)n for some n. Some further conditions are imposed, which neednot be spelled out. The pairs are the objects of an abelian category AX ,whose Grothendieck group is denoted by K(AX).Put E = Ql and denote for v 2 V by Ev the sheaf on X which restricts to theconstant sheaf E on v and to 0 on the complement of v. Now mv 2 K(AX)is the class of (Ev; �), where � is the Frobenius map. (Our mv correspondsto �0;v of [MS, 4.3] and our es to �0;s.)Another technicality is that the base ring Z[u; u�1] has to be extended (pro-visionally) to a ring R, the group ring Z[C] of a group C deduced fromthe eigenvalues of Frobenius endomorphisms acting on the stalks of certainsheaves. C is a subgroup of the multiplicative group of non-zero algebraicnumbers modulo roots of unity. Then Z[u; u�1] is the group ring of thegroup generated by the image u of q 12 .To de�ne the Hecke algebra action the extension to R is not needed. Butif one is after more delicate properties of M, such as the existence of aKazhdan-Lusztig basis, the introduction of the ring extension can not beavoided. 13

If v = [I; x; w] (as before) we write mv = mI;x;w. The next lemma describesthe HH-action on M. In (i) we have the three cases for x and s, see 1.5.In case (b) we put sx = x�.3.2. Lemma. Let x 2 W I ; w 2 W; t 2 S.(i) If t = (s; 1), et:mI;x;w equals(a) (u2 � 1)mI;x;w + u2mI;sx;w in case (a),(b1) mI;x;w� in case (b) if w� > w,(b2) (u2 � 1)mI;x;w + u2mI;x;w� in case (b) if w� < w,(c) mI;sx;w in case (c);(ii) If t = (1; s), et:mI;x;w equals mI;x;sw if sw > w and (u2 � 1)mI;x;w +u2mI;x;sw if sw < w.Proof. The formulas follow from the results of [MS, 4.3] and 1.5, taking intoaccount that only case II occurs.3.3. There is an action of W on V (see [RS, no. 2]). Since only case IIoccurs the action can easily be described (for example, using [MS, 4.3.1]).Notice that if t 2 S and m(t):v 6= v we have t:v = m(t):v.Explicitly, the action is given by (s; 1):[I; x;w] equals [I; sx; w] in the cases(a) and (c) and [I; x; w�] in case (b) (notations being as in 3.2 (i)). Also,(1; s):[I; x;w] = [I; x; sw] in all cases.Notice that the formulas of 3.2 can be rewritten aset:mv = mt:v if d(t:v) > d(v)= (u2 � 1)mv + u2mt:v if d(t:v) < d(v): (6)The construction of our representation given in [MS] is non-elementary, ituses l-adic sheaves. One can verify in a more elementary way that theformulas of the proposition de�ne a representation of H H (see 6.1).But we now shall need the sheaf theoretical approach. Verdier duality theoryleads to an involutorial map � of M, which is semilinear in R relative tothe involution de�ned by the inverse in the group C and satis�es�(et:m) = e�1t :�(m) (t 2 S; m 2 M); (7)(see [MS, 3.3.2, 4.4.7], where � is denoted by D).3.4. Lemma. (i) There exist elements bx;v 2 R such that�(mv) = u�2d(v)Xw2V bw;vmx;(ii) bw;v 6= 0 if and only if w � v and bv;v = 1.Proof. A formula like the one of (i) is contained in [MS, 3.4]. However, in14

that formula other terms could appear, corresponding to non-constant localsystems on the orbits w. But by the last line of [MS, 3.4.1] only the constantlocal system on x will appear, since in our situation all mapsc�v are injective(as a consequence of 1.8).For the proof of (ii) we have to go into the de�nition of �. Denote by �(Ev)the Verdier of the sheaf Ev, an object in a derived category. By [MS, 3.3]bw;v =Xi (�1)i(X�i m�i;i�i); (8)where �i runs through the images in C of the eigenvalues of the Frobeniusmap of the stalk Hi(�(Ev))a of the cohomology sheaf Hi(�(Ev)) in a pointa 2 v(Fq), the m�i;i denoting multiplicities.By general facts about Verdier duality, Hi(�(Ev))a is the dual of the localcohomology group H�i[a] (X;Ev). It follows from [B2, 3.1, Theorem], thatthere exists a transversal slice S at a to the orbit w in the sense of [MS,2.3.2]. Then, locally in a for the �etale topology, X is the product of w andS. Hence Hi[a](X;Ev) = Hi�2d(w)[a] (S;Ev):But it follows from [B2, loc.cit.] that there is a cocharacter of T contractingS to a. Then by [MS, Remark after 2.3.1] H i[a](S;Ev) is isomorphic to thecohomology group with proper support H ic(S;Ev) = H ic(S \ v; E). NowS \ v 6= ; if and only if w � �v. If this is so, it follows from (8) that bw;v 6= 0.In fact, up to a power of q the right-hand side of (8) equals the number ofFq-rational points of S \ v which is 6= 0 (after enlarging Fq, if necessary).So we have shown that bw;v 6= 0 if w � v. The converse follows from the factthat the dual �(Ev) is zero outside �v. Finally, that bv;v = 1 follows fromthe fact that �v is smooth in the points of v.3.5. Proposition. (i) If bx;v 6= 0 it is a polynomial Z[u2] with leading term(�u2)d(v)�d(x);(ii) (�u2)d(v)�d(w)bw;v(u�2) = bw;v(u2).Proof. Let v 2 V and assume that there is t 2 S such that d(t:v) < d(v),whence by (7) �(mv) = e�1t :�(mt:v):Writing this out in terms of the b's, using thate�1t = u�2(et � u2 + 1);15

we obtain for d(t:v) < d(v)bw;v = bt:w;t:v if d(t:w) < d(w);= (1� u2)bw;t:v + u2bt:w;t:v if d(t:w) > d(w): (9)Using these formulas, a straightforward induction shows that the proof ofpart (i) is reduced to the case that d(t:v) � d(v) for all t 2 S. Then v is ofthe form [J; w0;Dw0;J ; 1].Put ~bw;v = u�d(v)+d(w)bw;v. Then (9) shows that~bw;v = ~bt:w;t:v if d(t:w) < d(w);= (u�1 � u)~bw;t:v + ~bt:w;t:v if d(t:w) > d(w): (10)Using these formulas, the induction will give that ~bw;v is a polynomial inu�1 � u, hence is invariant under the change u 7! �u�1. Then (ii) willfollow from (i).It remains to deal with the case v = [J; w0;Dw0;J ; 1].In the sequel the R-polynomials of Kazhdan-Lusztig (see [KL, x2]) will ap-pear. They lie in Z[u2]. They are de�ned in terms of the Hecke algebra Hby e�1x�1 = u�2l(x)Xy (�1)l(x)�l(y)Ry;x(u2)ey ;where x; y 2 W: From (7) we deduce thatb[D;1;y];[D;1;x] = (�1)l(x)�l(y)Ry;x:We have Ry;x = 0 if y 6� x and Rx;x = 1. The R-polynomials satisfy thefollowing recursive relations (where x; y 2 W; s 2 S). Together with theboundary conditions Ry;1 = �y;1 these relations de�ne the R-polynomialsuniquely. Ry;sx = Rsy;x if sx > x; sy < y= (u2 � 1)Ry;x + u2Rsy;x if sx > x; sy > y: (11)We have similar relations relative to the right action of a simple re ection.Moreover, if Ry;x 6= 0 it has leading term u2(l(x)�l(y)).We return to the determination of the bw;v for v = [J; w0;Dw0;J ; 1]. Firstlet J = D. Then v = [D; 1; 1] is the B-orbit B � G. By 3.5 (i) and 2.5(iii), bw;v 6= 0 if and only if w = [I; a; b] with I � J and b � a. Write16

�Ia;b = b[I;a;b];[D;1;1].3.6. Lemma. �Ia;b = (�1)l(a)+l(b)(1� u2)jD�IjRb;a(u2):Proof. Let s 2 S. Thene(s;1):mD;1;1 = e(1;s):mD;1;1 = mD;1;s:This implies, using (7),e(s;1):(X�Ia;bmI;a;b) = e(1;s):(X�Ia;bmI;a;b):Fix a 2 W I and assume s 2 S is such that sa > a. Using 3.2 we determinethe coe�cients of mI;a;b in both sides of the preceding equation. We obtain�Isa;b = �Ia;sb if sb < b= (1� u2)�Ia;b + u2�Ia;sb if sb > b:Moreover, we see from 3.4 (ii) that �I1;b = �b:1�I1;1. Comparing the precedingformulas with the inductive formulas (11) for the R-polynomials we concludethat �Ia;b = (�1)l(a)+l(b)Rb;a(u2)�I1;1:It remains to determine �I1;1 = b[I;1;1];[D;1;1]. Let �T be the closure of T inX . It contains the point a = hI of [I; 1; 1]. We use again that by [B2,3.1,Theorem] there exists a contractible transverse slice S at a to [I; 1; 1]. Itfollows from [loc.cit.] that S can be taken to be a transverse slice in �T at ato TI :a. From the proof of 3.4 (ii) we see how to determine �I1;1: we have tostudy S \ B and the action of the Frobenius map on its cohomology. NowS \ B = S \ T is a torus isomorphic to TD�I . From familiar results aboutthe Frobenius action on the cohomology of Fq-split tori we then obtain that�I1;1 = (1� u2)jD�Ij, �nishing the proof of 3.6.Fix J and put vJ = [J; w0;Dw0;J ; 1]. Let I � J .3.7. Lemma. Let a 2 W I ; b 2 WJ and w0;Dw0;Jb � a Then b[I;a;b];vJ =(�1)l(a)+l(b)+l(w0;Dw0;J )(1� u2)jJ�IjRw0;Dw0;J b;a.Proof. By 2.5 (i) we know that b[I;a;b];vJ 6= 0 if and only if I � J , b 2 WJand w0;Dw0;Jb � a. Write a = a1c, with a1 2 W J ; c 2 WJ . The precedinginequality can only hold if a1 = w0;Dw0;J and then b � c. Since a 2 W I wehave c 2 WJ \W I . By [KL, 2.1 (iv)], applied for W and WJ ,Rw0;Dw0;J b;a = Rw0;Dw0;J b;w0;Dw0;J c = Rb;c:17

Now [I; w0;Dw0;Jc; b] lies in XJ and is, in fact, the BJ �BJ -orbit [J; c; b]GJ .The statement of the lemma then asserts thatb[I;w0;Dw0;J c;b]G;[J;w0;Dw0;J ;1]G = b[I;c;b]GJ ;[J;1;1]GJ : (12)We have the Hecke algebra HJ of WJ and the HJ HJ -module MJ , withbasis (mJv ), where v runs through the BJ � BJ -orbits in Y = XJ , i.e. theB �B-orbits in Y . There is an injective module homomorphism� : MJ !M;with �(mJI;a;b) = mI;w0;Dw0;Ja;b.Denote by �J the duality map of MJ . The equality (12) will follow from� ��J = � � �: (13)To prove (13) notice that under the �bration map X ! G=P�J �G=PJ , theorbit [J; w0;dw0;J ; 1] is mapped onto a point. The �ber over that point isisomorphic to Y , whence a B �B-equivariant closed embedding i : Y ! X ,with i(BJ) = [J; w0;dw0;J ; 1].Denote by �X and �Y Verdier duality in the derived category of l-adicsheaves on X , respectively Y . We then havei� ��Y = �X � i�;because i is a proper morphism. (13) is a consequence of this equality,observing that � comes from i�.Lemma 3.7 provides the �nishing touch to the proof of 3.5.3.7 is a particular case of the formula of the following lemma (which waspointed out to me by W. van der Kallen). Notations are as before.3.8. Lemma. Let x 2 W J , I � J, b 2 WJ . Thenb[I;a;b];[J;x;1] = (�1)l(a)+l(b)+l(x)(1� u2)jJ�IjRxb;a(u2): (14)Proof. Recall that by 3.4 (ii) and 2.5 (i) the left-hand side is 6= 0 if and onlyif I � J , b 2 WJ and xb � a. We prove (14) by descending induction onl(x). If x is the maximal element of W J the formula holds by the previouslemma.Assume (14) holds for x and that s 2 S; sx < x. Then sx 2 W J . From (9)we see that b[I;a;b];[J;sx;1] equalsb(s;1):[I;a;b]:[J;x;1] if d((s; 1):[I; a; b])< d([I; a; b]);(1� u2)b[I;a;b];[J;x;1] + u2b(s;1):[I;a;b];[J;x;1] if d((s; 1):[I; a; b])> d([I; a; b]):18

For the action of (s; 1) see 3.3. We putb[I;a;b];[J;x;1] = (�1)l(a)+l(b)+l(x)(1� u2)�1cx;a;b:We then have to prove that cx;a;b = Rxb;a. In applying the formulas thereare four cases to be dealt with.(1) sa > a and sa 2 W I . Then csx;a;b = cx;sa;b = Rxb;sa by induction. Sincesxb < xb; sa > a this equals Rsxb;a by the �rst formula (11).(2) sa = at with t 2 WI and bt < b. Now csx;a;b = cx;a;bt = Rxbt;a. By the�rst formula (11) for right action of t and left action of s,Rxbt;a = Rxb;at = Rxb;sa = Rsxb;a:(3) sa < a. In this casecsx;a;b = (u2 � 1)cx;a;b + u2cx;sa;b = (u2 � 1)Rxb;a + u2Rxb;sa:This equals Rsxb;a by the formulas (11).(4) sa = at with t 2 WI and bt > b. Nowcsx;a;b = (u2 � 1)cx;a;b + u2cx;a;bt = (u2 � 1)Rxb;a + u2Rxbt;a = Rxb;atby the second formula (11) for right action of t. Moreover,Rxb;at = Rxb;sa = Rsxb;aby the �rst formula (11).The lemma follows.3.9. Remark. The arguments of the proof of 3.8 can be used to show that(14) holds for all x 2 W I as soon as it holds for one particular x.We claim that if x 2 W I and s 2 S are such that sx < x, validity of (14)for sx implies validity for x. By the lemma, validity of (14) for x impliesvalidity for x = 1. The claim will then imply validity for any x 2 W I .In proving the claim we use the notations of the proof of 3.8. Assume that(14) holds for sx. We consider the four cases of the proof of 3.8. In case(1), csx;a;b = cx;sa;b. Assuming that csx;a;b = Rsxb;a, we have cx;sa;b = Rxb;sa,whence cx;a;b = Rxb;a for sa < a.In case (2) a similar argument shows that cx;a;bt = Rxbt;a, whence cx;a;b =Rxb;a if bt < b.In case (3) we �nd, using the result of case (1)csx;a;b = Rsxb;a = (u2 � 1)cx;a;b + u2cx;sa;b = (u2 � 1)Rxb:a + u2cx;sa;b:19

From (11) we see that cx;sa;b = Rxb;sa, whence cx;a;b Rxb;a if sa > a. Case(4) is dealt with in a similar manner.We conclude this section with some additional results about about the poly-nomials bw;v (v; w 2 V ).3.10. Lemma. (i) bw;v(1) = �w;v;(ii) b0w;v(1) = �1 if there is a re ection r 2W (not necessarily simple) withw = r:v � v;(iii) Let v = [J; c; d]; w = [I; a; b]. Then b0w;v(1) = �1 if I � J; jJ � I j = 1and there exists f 2 WJ with a = cf; b = df ;(iv) In the cases not covered by (ii) and (iii) we have b0w;v(1) = 0.Proof. It follows from (9) that if t 2 S we have bw;v(1) = bt:w;t:v. Letv = [J; c; d], I = [I; a; b]. Using the preceding formula the proof of (i) isreduced to the case that c = d = 1. In that case (14) shows that bw;v = 0unless I = J and a = b = 1. (i) follows.We prove the other assertions by induction on d(v). It is a bit easier to workwith the ~bw;v. Using (i) one sees that ~b0w;v(1) = 2b0w;v(1).Let s 2 S, t = (1; s). It follows from (10) that if d(t:v) < d(v)~b0w;v(1) = ~b0t:w;t:v(1) if d(t:w) < d(w);= �2~bw;t:v(1) + ~b0t:w;t:v(1) if d(t:w) > d(w):If w = t:v < v we have bt:w;t:v = 0 by 3.4 (ii), and then (i) shows that~bw;v = �2, proving (ii) in that case. If w 6= t:v the preceding formulas implythat b0w;v(1) = b0t:w;t:v(1):By induction the proof of (ii), (iii) and (iv) is then reduced to the case thatd = 1. In that case we have the explicit formula (14) for bw;v. It impliesthat for I = J b0w;v(1) = R0cb;a(1):It is known (see [GJ, 2.2]) that for u; z 2 W we have R0u;z(1) = 0 exceptwhen there is a re ection r 2 W with u = r:z < z, in which case R0u;z(1) = 1.This implies that for I = J , d = 1, ~bw;v = 0 unless there is a re ection r 2 Wwith rc = ba�1, which means that (r; 1):v = w, proving (ii) and (iv) if I = J .In the case that I � J; I 6= J , d = 1 it is clear from (14) that b0w;v(1) = 0if jJ � I j > 1. If jJ � I j = 1, (14) gives that b0w;v(1) = 0 unless a = cb, inwhich case it equals �1, in accordance with (ii) and (iv). This concludesthe proof of 3.10. 20

4. Intersection cohomology of B � B-orbits and Kazhdan-Lusztigpolynomials.4.1. The notations are as in the preceding section. For v 2 V let I = Ivbe the intersection cohomology complex of the closure �v, i.e., the irreducibleperverse sheaf on X supported by �v whose restriction to v is E[d(v)]. Itde�nes an element cv = u�d(v) Xw2V cw;vmxof M, with cx;v =Xi (�1)i(X�i mi;�i�i);where the �i run through the images in C of the eigenvalues of the Frobeniusmap of the stalk Hi(Iv)a in a 2 x(Fq), the mi;�i denoting multiplicities, see[MS, 3.1.2]. We have cv;v = 1, and cw;v = 0 if w 6� v.4.2. Theorem. (i) The cx;v are polynomials in u2 with positive integralcoe�cients;(ii) I is even, i.e. Hi(I) = 0 if i + d(v) is odd.Proof. Again we use the result from [B2, 3.1, Theorem] that in a point ofan orbit v there is a contractible transversal slice. By [MS, 2.3.3] it thenfollows that I is punctually pure, i.e. that all eigenvalues of the Frobeniusmap of a stalk Hi(Iv)a (a 2 �v(Fq)) have absolute values q 12 (i+d(v)). The factthat the bx;v are polynomials, proved in 3.4, then implies (i) and (ii). Werefer to [loc.cit., 3.4.3 and proof of 7.1.2 (ii)].Now assume that we work over an arbitrary algebraically closed �eld.4.3. Corollary. I is even.Proof. Since this is true when k is the algebraic closure of a �nite �eld by thetheorem, it is true for any k by a familiar reduction procedure (see [BBD,no. 6]). It also follows that the result holds over C, relative to the classicaltopology.We can now discard the big ringR. View the cw;v as polynomials in u2. Theyare the \Kazhdan-Lusztig polynomials" for M, characterized by propertiesof the usual kind:4.4. Proposition. The (cw;v)x;v2V are the uniquely determined polynomialswith the following properties:(a) cv;v = 1 and cw;v = 0 if w 6� v; 21

(b) if w < v the u-degree of cw;v(u2) is < d(v)� d(w)(c) u�d(v)Pw cw;v(u2)mw is invariant under �.Proof. Our polynomials cw;v have these properties. For (a) this is clear, and(b), (c) re ect properties of the perverse sheaf I viz. the support conditionsand duality.The uniqueness follows from the following identity (cf. [KL, 2.2])u�d(v)+d(w)cw;v(u2)� ud(v)�d(w)cw;v(u�2) == Xw<y�v ud(v)+d(w)�2d(y)cy;v(u�2)bw;y(u2):This can be written in a somewhat less cumbersome form. Write~cw;v(u) = ud(v)�d(w)cw;v(u�2); ~bw;v(u) = u�d(v)+d(w)bw;v(u2):For w 6= v, ~cw;v is a polynomial in u without constant term, an integrallinear combination (with coe�cients � 0) of powers uj with j + d(v) + d(w)even. Moreover, ~bw;v is a Laurent polynomial in u, a linear combination ofpowers of u satisfying the previous parity condition. The preceding formulacan be rewritten as~cw;v(u�1)� ~cw;v(u) = Xw<y�v ~cy;v(u)~bw;y(u): (15)It follows from 3.5 (ii) that the Laurent polynomial ~bw;v(u) is of the formf(u� u�1), where f 2 Z[T ] is divisible by T jD�Ij.4.5. The formulas (15) lead to an inductive procedure to determine theKazhdan-Lusztig polynomials cw;v, via the Laurent polynomials ~cw;v. Namely,the right-hand side of (15) is a Laurent polynomial without constant termand �~cw;v is its polynomial part. One needs to know the Laurent polynomi-als ~bw;y . They can be determined inductively, as in the proof of 3.5: reduceto the case that v = [J; x; 1] by using (10) and then apply 3.8.The procedure has been implemented by W. van der Kallen in a Mathemat-ica program (see the Appendix). Among other things, he computed the cw;vin the case that G is simple of rank two and that v = [D; 1; 1], i.e. the Borelsubgroup B of G. By 2.5 (ii) and 4.4 (a) (for arbitrary G) cw;B 6= 0 impliesthat w = [I; x; y] with y � x.The computations give that in type A2 (i.e. for G = PGL3) we have c[I;x;y];B= 1 unless I = ; and either x = st, y = t; 1 or x = sts, y = s; t; 1 (where sand t 6= s are simple re ections). In these cases c[;;x;y];B = 1 + u2, except22

when x = sts; y = 1, in which case this polynomial is (1 + u2)2.In general, Py;x = c[;;x;y];B (y � x) is a polynomial in u2 whose u-degree is< 2(r + l(x)� l(y)), where r is the rank of G (by 4.4 (b) and (5)). Thesepolynomials remind one of the usual Kazhdan-Lusztig polynomials, intro-duced in [KL]. One can wonder whether our Py;x also have some bearing onrepresentation theory. It would be interesting to have a more direct combi-natorial de�nition of these polynomials.We next give some other consequences of (15).4.6. Proposition. (i) If w � v then cw;v has constant term 1;(ii) cw;v 6= 0 if and only if w � v;(iii) If w < v then Pw�y�v (�1)d(y)�d(w) = 0.Proof. (i) is equivalent to the statement that the polynomial (in u�1)~cw;v(u�1) has leading term (u�1)d(v)�d(w). Now ~cy;v is a polynomial in uwithout constant term if y < v and ~bw;y is a Laurent polynomial in u withlowest term (u�1)d(y)�d(w). It follows that for w < y � v, ~cy;v~bw;y is a Lau-rent polynomial in u whose lowest term is (u�1)m with m � (d(y)� d(w)),equality occurring only if y = w. This implies that the left-hand side of (15)has lowest term (u�1)d(v)�d(w). This lowest term must occur in ~cw;v and (i)follows.(ii) is a consequence of (i) and 4,4 (a).To prove (iii) consider the leading term in u in both sides of (15). By (i)this is �1 in the left-hand side. In the right-hand side the leading term isPw<y�v(�1)d(v)�d(y), as follows from 3.5. The formula of (iii) follows.Remark. (iii) implies that the M�obius function of the ordered set V isgiven by �(w; v) = (�1)d(v)�d(w) (w � v). This is similar to a result ofVerma for the Bruhat order of W . In [KL, 3.3 b)] Verma's result is deducedfrom properties of Kazhdan-Lusztig polynomials. It can also be proved alongthe lines of the proof of 4.6 (iii).If w = [I; a; b] and I � J we put �I;J(w) = [J; c; d], where c 2 W J issuch that a 2 cWJ and d = bc�1a. We denote by R(W) the set of re ec-tions in W.4.7. Proposition. Let v 2 V and assume that �v is rationally smooth. Thenfor any w < vjfr 2 R(W) j w < r:w � vgj+ jfJ � D j J � I; jJ � I j = 1; �I;J(w) � vgjequals d(v)� d(w). 23

Proof. That �v is rationally smooth means that the intersection cohomologycomplex Iv is E�v[dim v] or, equivalently, that cw;v = 1 for w � v. Then~cw;v(u) = ud(v)�d(w) for w � v. Inserting this into (15) and taking derivativesof both sides for u = 1 one obtains for w < v�2(d(v)� d(w)) = Xw<y�v((d(v)� d(y))bw;y(1) + ~b0w;v(1)):The asserted equality is now a straightforward consequence of 3.10.An application of 4.7 is the following result due to Brion, proved (in an-other manner) in [B2, 3.3].4.8. Corollary. Assume that G is simple. If w 2 W the large Schubert va-riety Sw = BwB is rationally smooth if and only if G ' PGL2 or w = w0;D.Proof. Take in the proposition v = BwB = [D; 1; w] and w = [;; w0;D; 1].By 2.5 (iii), w � [D; 1; 1] � v (w is the unique B � B-�xed point in Sw).We have d(v) = d+ r+ l(w), where d = l(w0;D) and r is the rank of G. If sis any re ection in W then (s; 1):w = [;; sw0:D; 1] and (1; s):w = [;; w0:D; s].From 2.5 (iii) we then see that all r 2 R(W) satisfy w < r:w � [D; 1; 1]� v.Further, if � 2 D and s 2 S is the corrresponding simple re ection then�;;f�g(w) = [f�g; wO;Ds; s]. By 2.5 (iii) we have �;;f�g(w) � v if s � w0;Ds.If this is not the case and r > 1, we must have w0;Ds 2 WD�f�g. Sincel(w0;Ds) = d � 1, w0;Ds has to be the longest element of WD�f�g. Thisimplies that s 2 WD�f�g whence s:� 2 R+ for all simple roots � 6= �, whichis impossible since r > 1 and R is irreducible (G being simple).It follows that if r > 1 the number given by the displayed formula in theproposition is at least 2d + r. Since d(v)� d(w) = d + r + l(w), 4.7 givesthat if r > 1 rational smoothness of Sw impliesr + d + l(w) � 2d+ r;which can only be if w = w0;D.Conversely, if w = w0;D then Sw = X is smooth. To �nish the proof of4.8 it remains to be shown that if G = PGL2, �B is rationally smooth. Thewonderful compacti�cation X of PGL2 is isomorphic to projective space P3,viewed as the set of lines in the space of 2� 2-matrices. It follows that �B isisomorphic to P2, hence is smooth. (Rational smoothness of �B in this casecan also be proved by hand, using (15).)4.9. Global intersection cohomology. The results of this section alsoimply parity of global intersection cohomology of orbit closures.24

If X is any irreducible variety de�ne its global intersection cohomologygroups by IH i(X) = Hi(X; IX[� dimX ]);the hypercohomology of a shift of the intersection cohomology complex Iof X . The shift is added in order to recover ordinary cohomology if X issmooth.As before, we work over the algebraic closure of a su�ciently large �eld Fq,over which X is de�ned. Let X0 be the underlying Fq-variety. Then IXcomes from a perverse sheaf I0 on X0. Moreover, I0 is pure of weight 0 (see[BBD, 5.3.2]). We have a Frobenius endomorphism F of the intersectioncohomology groups. If X is projective and IH i(X) 6= 0, all absolute valuesof the eigenvalues of F on a IH i(X) are q 12 i, as follows from [loc. cit., no.5].Now let X = �v, where v 2 V .4.10. Lemma. The eigenvalues of F on a nonzero intersection cohomologygroup are integral powers of q.Proof. Let A be the shifted intersection cohomology complex IX [� dimX ].Put Xn = [w2V;w�v;dimw�nw;this is a closed subset of X , coming from an Fq-subvariety (Xn)0 of X0.Clearly, Xdimv = X . We show by induction on n that for each n theeigenvalues of F on a nonzero hypercohomology groupHi(Xn; A) are integralpowers of q. We have exact sequences of hypercohomology groups:::! Hi�1(Xn�1; A)! Hic(Xn�Xn�1; A)) ! Hi(Xn; A)! Hi(Xn�1; A) !;the arrows commuting with the respective Frobenius endomorphisms. More-over, Xn �Xn�1 = adimw=nw:A straightforward argument now shows that it su�ces to prove that theeigenvalues of F on a nonzero group Hic(w;A) are integral powers of q.We have a spectral sequenceH ic(w;Hj(A))) Hi+jc (w;A);from which we conclude that it su�ces to prove a similar assertion for thegroups H ic(w;Hj(A)). The restriction of the locally constant sheaf Hj(A)25

to w is B �B-equivariant. Since the isotropy groups in B �B of the pointsof w are connected (see 1.3 (iv)) this restriction is constant. By 4.2 (ii)Hj(A) = 0 if j is odd and it follows from 4.2 (i) (cf. the description of cw;vgiven in 4.1) that all eigenvalues of F on the stalk Hj(A)x (x 2 w(Fq)) areqj . This reduces us, �nally, to proving that the eigenvalues of F on H ic(w;E)are powers of q. This follows from the fact that w is a isomorphic over Fqto the product of a torus and an a�ne space.4.11. Theorem. IH i(X) = 0 if i is odd.Proof. All absolute values of an eigenvalue of F on a nonzero group IH i(X)are q 12 i. By 4.10 this must be an integral power of q, which can only be if iis even.Again, the parity result is true true in any characteristic and over C, in theclassical context.The arguments of the proof can be extended a bit, so as to give a descriptionof the intersection cohomology Poincar�e polynomial of X = �v.4.12. Corollary. Pi ui dim IH i(X) =Pw�v u2dimw(1� u�2)jIwjcw;v(u2):Proof. It follows from the theorem that all eigenvalues of F on IH 2i(X) areqi. A being as in the proof of 4.10, it also follows thatXi�0 qi dim IH 2i(X) =Xi (�1)iTr(F;Hi(X;A)):By a result of Grothendieck, the right-hand side equalsXx2v(Fq);i(�1)iTr(F;H i(A)x):Now v(Fq) = `w�v w(Fq), and the number of points of w(Fq) equalsqdimw(1� q�1)jIwj:Moreover, for all x 2 w(Fq)Xi (�1)iTr(F;H i(A)x) = cw;v(q):We conclude that the di�erence of the two sides of the asserted formulavanishes for t = q 12 . But it then also vanishes for all powers (q 12 )n, hencethe di�erence is identically zero. 26

Example. Let G = PGL3, v = B. The Kazhdan-Lusztig polynomials weredescribed in 4.5. Cut the sum of 4.12 into pieces corresponding to the cells of2.10. A straightforward computation gives for the intersection cohomologyPoincar�e polynomial of �B1 + 4t2 + 9t4 + 9t6 + 4t8 + t10:It was pointed out by M. Brion that the description of global intersectioncohomology can also be obtained as an application of the results of [BJ].5. Nonconstant local systems.In this section we discuss the generalization of Theorem 3.2 to the case ofintersection cohomology complexes for not necessarily constant local systemson the B � B-orbits v in X . These are the local systems having a weightfor the B �B-action (see [MS, 2.2]).If v = [I; x; v] (as before) we write I = Iv.5.1. Assume that k is the algebraic closure of a �nite �eld Fq, over whichall objects which occur are de�ned. Let R be as in no. 3. Following[MS] we introduce the free R-module N with basis m�;v where 2 V and� 2 X(T � T=(T � T )v) = X(TIv) = XIv � X (see 1.7). Then m�;v is theclass in the Grothendieck group K(AX) (see 3.1) of (S�;v; �), where S�;v isthe sheaf which restricts to � on v and to 0 on the complement of v, � beinga Frobenius map. The module M of no. 2 is the submodule with basismv = m0;v (v 2 V ).Let K be the algebra over Z[u; u�1] with basis (e�;w), where w 2 W and� 2 X = X(T ), the multiplication being de�ned by the following rules:e�;xe�;y = 0 if � 6= y:�;and for s 2 S; � = y:�e�;se�;y = e�;sy if sy > y;= (u2 � 1)e�;y + u2e�;sy if sy < y and hy�; ��i = 0;= u2e�;sy if sy < y and hy�; ��i 6= 0;see [loc.cit., 4.3.2 and 3.2.3]. �� is the element �_ 1 of Y (T ) = Y (T ) (Z(p)=Z), where �_ is the cocharacter de�ned by �.Moreover, e�;1e�;y = ��;y:�e�;y ;27

see [loc.cit., 3.2].If � 2 X we denote by R� the closed subsystem of R consisting of the roots� with h�; ��i = 0. Its Weyl group is W�. It is a normal subgroup of theisotropy group W 0� of � in W .By [loc.cit., 3.2] we have a structure of K Z[u;u�1 ] K-module on N . Thenext lemma, which generalizes 3.2, describes the module structure.Let v = [I; x; w]2 V . For � 2 XI we put m�;I;x;w = m�;v. In (ii) we use thenotations introduced before 3.2.5.2. Lemma. Let x 2 W I ; w 2 W; � 2 XI ; � 2 X and s 2 S.(i) e�;(s;1):m�;I;x;w = 0 if � 6= x�1:� and e�;1m�;I;x;w = ��;x:�m�;I;x;w;(ii) If � = x�1:� the product of (i) equals(a) (u2 � 1)m�;I;x;w + u2m�;I;sx;w in case (a) if x:� 2 R� and u2m�;I;sx;wotherwise,(b1) m�;I;x;w� in case (b) if w� > w,(b2) (u2 � 1)m�;I;x;w + u2m�;I;x;w� in case (b) if w� < w, w:� 2 R� andu2m�;I;x;w� if w� < w, w:� 62 R�,(c) m�;I;sx;w in case (c);(iii) e�;(1;s):m�;I;x;w = 0 if � 6= �w�1:�;(iv) If � = �w�1:� the product of (iii) equals m�;I;x;sw if sw > w. If sw < wit equals (u2�1)m�;I;x;w+u2m�;I;x;sw if w:� 2 R� and u2m�;I;x;sw otherwise.Proof. The �rst point of (i) follows from [loc.cit., 3.2.3] and the second pointis an easy consequence of the de�nitions of [loc.cit., 3.2.1].The formulas of [MS, 4.3.1] for the cases IIa and IIb give formulas like thoseof (ii), except that at �rst sight on the right-hand side other elements of XImight appear. Consider, for example, case (a) with x:� 2 R�. Then [loc.cit.]shows, using 1.8, that there is �0 such thatex�1:�;s:m�;I;x;w = m�0;I;sx;w:It follows from the de�nition of �v (see 1.8) that �(s;1):v = �v �(s; 1), whence\�(s;1):v = (s; 1) �c�v : By the results of [loc.cit.] \�(s;1):v(�0) = (s; 1):c�v(�). By1.11 y�1:�0 =\�(s:1):v(�0) = b�v(�) = y�1:�;whence �0 = �: In the other cases the proof that only � will occur is similar.This will prove (ii). The proofs of (iii) and (iv) are similar to the proof of(i), respectively (ii).5.2 shows that the m�;v with a �xed � 2 X span a submodule M� of N28

which is stable under the action of H H. Clearly, N is the direct sum ofthe M�.5.3. As for the module M, there exists a semilinear involutorial endomor-phism � of N , coming from Verdier duality, see [MS, 3.3]. It will followfrom 4.5 (ii) that � maps M� onto M��.For s 2 S; � 2 X;m 2 N we have�(e�;(s;1):m) = u�2(e��;(s;1) + (1� u2)e��;1):�(m) if s 2 W�= u�2e��:(s;1):�(m) if s 62 W�; (16)and similarly for e�;(1;s). See [loc.cit., 5.1].5.4. Lemma. (i) There exist elements b�;w;�;v 2 R such that�(m�;v) = u�2d(v) Xw2V;�2XIw b�;w;�;vm�;w;(ii) If b�;w;�;v 6= 0 then w � v and c�w:� = �c�v:�. Moreover, b��;v;�;v = 1.Proof. The proof of (i) is like the proof of 3.4 (i). The �rst point of (ii)follows from the fact that the Verdier dual �(S�;v) is zero outside �v. Therestriction of �(S�;v) to v is �� shifted by 2d(v), which implies the laststatement of (ii). The second one follows from [MS, 3.4].5.5. Proposition. (i) The b�;w;�;v are polynomials in Z[u2];(ii) If b�;w;�;v 6= 0 then � = ��, in particular � 2 XIw .Proof. The proof of is along the lines of the proof of 3.5 (i). We shall notspell out the details. Let v = [I; x; w] 2 V and assume that there is t 2 Swith d(t:v) < d(v). Using 5.2 and (16) one reduces the proof to the casethat v = [J; w0;Dw0;J ; 1]. (The important fact for the proof of (ii) is that bythe formulas of 5.2, e�;t:m�;v lies in M�.)Assume that v has this form, and assume also that J = D. So v = B.We then have to compute the b�;[I;1;1];�;B. First note that by 5.4 (ii) and1.8, b�;[I;1;1];�;[D;1;1] = 0 if � 6= ��.So assume that � = ��. Then � 2 XI . Proceeding now as at the end of theproof of 3.6, we see that we have to compute the cohomology of the torusTD�I with values in the restriction to TD�I of the local system � on T . Butthe restriction map X ! X(TD�I) has kernel XI , so contains �. It followsthat the restriction of � is the trivial local system, and the computation isthen as in 3.8, showing that b��;[I;1;1];�;B is a nonzero polynomial in u2 if29

� 2 XI .It remains to deal with the case that v = [J; w0;Dw0;J ; 1] with J arbitrary.This is done by using an analogue of (13).5.6. Intersection cohomology. For v 2 V and � 2 XIv denote by I�;vthe intersection cohomology complex of the closure �v, for the local system� on v, i.e., the irreducible perverse sheaf on X which is zero outside �v andwhose restriction to v is �[d(v)]. As in 3.1, it de�nes an element of Nc�;v = u�d(v) Xw2V;�2XIw c�;w;�;vm�;w;the c�;w;�;v being elements of R.5.7. Theorem. (i) The c�;w;�;v are polynomials in u2 with positive integralcoe�cients;(ii) If c�;w;�;v 6= 0 then � = ��, in particular � 2 XIw ;(iii) I�;v is even.Proof. The proof is of (i) and (iii) is similar to the proof of the analogousresults of no. 4.(ii) follows from 5.5 (ii), using the inductive description of the c�;x;�;v con-tained in [MS, 3.4.2, 3..4.2]. It follows from (iii) that the restriction of I�;vto XI is zero if � 62 XI .(ii) remains true over any algebraically closed �eld (cf. 3.3).6. Arbitrary Coxeter groups.This section deals with a tentative extension to arbitrary Coxeter groups ofthe constructions of the preceding sections.6.1. We �rst give a more intrinsic description of the module M of no. 2.We now identify D and S. For I � S let MI be the submodule of Mspanned by the mv with Iv = I . Denote by HI the Hecke algebra of thesubgroup WI generated by I . It is a subalgebra of HS = H. Denote by jIthe isomorphism of HI sending ew (w 2 wI) to ew0;Iww0;I .Let i be the endomorphism of M sending mI;x;w to mI;x;w�1. Then((h; h0); m) 7! h:m:h0 = i((h h0):i(m)) (h; h0 2 H; m 2 M)30

de�nes a structure of (H;H)-bimodule on H. In particular, we can view Mas an (HI ;H)-bimodule.If N is a (left) HI -module, denote by jIN the HI -module N twisted by jI .6.2. Lemma. (i) M = �I�SMI ;(ii) There is an isomorphism � of left H-modules of MI onto the twistedinduced module JS (H HI (jIH));(iii) � commutes with right H-actions.Proof. For (iii) notice that the induced module N of (ii) has a natural rightH-action.(i) is clear. For the proof of (ii) notice that H is a free right module over HIwith basis ex (x 2 W I). It follows that (ex m)x2W I ;m2M is a basis of N .Now de�ne � by �(mI;x;w) = e�I (x) ew;where �I(x) = w0;Dxw0;I , as in 1.3. A straightforward check shows that fors 2 S �(e(s;1):mI;x;w = e(s�;1):�(mI;x;w) if sx 2 W I ;= e�I(x) ew0;I�w0;I ew if � = x�1sx 2WI :This proves (ii). The proof of (iii) is easy.Now assume that (W;S) is an arbitrary Coxeter group (with a �nite set ofgenerators S). As in no. 3, we write W = W �W; S = S � S:For I � S let, as before, WI be the subgroup generated by I and W I theset of distinguished coset representatives for WI , i.e. the set of x 2 W withxs > x for all s 2 I .Let V be the set of triples [I; x; w] with I � S; x 2 W I ; w 2W . We introducethe free Z[u; u�1]-module M with basis (mv)v2V . For v = [I; x; w] we writemv = mI;x;w.As before, M is a direct sum of submodules MI (I � S).6.3. Proposition. The formulas of 3.2 de�ne a structure of HH-moduleon all MI .Proof. It is straightforward to verify that for t 2 S; v 2 Ve2t :mv = (u2 � 1)et:mv + u2mv:To prove 6.3 it remains to verify that for t; t0 2 S, the endomorphisms etand e0t of M verify the appropriate braid relations. This is immediate ifone of t; t0 is of the form (1; s) with s 2 S. So assume that t = (s; 1); t0 =31

(s0; 1) (s; s0 2 S). We may assume that s 6= s0. If a braid relation is to beveri�ed, ss0 has �nite order. Putting J = fs; s0g the group WJ is �nite.Fix v = [I; x; w] 2 V . Let N be the submodule of M spanned by themI;x0;w0 with x0 2 WJx, w0 2 wWI . The same submodule is obtained bytaking x to be the unique element of W I \ (W J)�1. One knows that thenWI \ x�1WJx = WK , where K = I \ x�1:J (see [C, p. 65]). Now the w0which occur will lie in a �xed coset modulo WK . Taking w 2 WK we seethat N can be viewed as a module like MI , for the Hecke algebra HJ of the�nite Coxeter group WJ . But for such a group we have again the result of6.2, which implies that the formulas of 3.2 de�ne a representation of HJ inN . This implies that the braid relations hold for the action in M of e(s;1)and e(s0;1).6.4. The set V . Suggested by the results of no. 2 we introduce somestructure on the set V .(a) Suggested by (5) we de�ne a dimension function d on V byd([I; x; w]) = �l(x) + l(w)+ j I j :Note that if W is in�nite the value of d can be any integer.(b) We have an action on V of W = W �W , described as in 3.3. That wedo have a W-action follows from 6.3, specializing u to 1.We also have an action of the monoid M(W �W ) on V .(c) We next introduce an order relation on V . A simple way of doing this issuggested by 2.8. Let �1 and �2 be de�ned as in 2.6. These are orderings.Denote by � the ordering generated by �1 and �2.6.5. Lemma. (i) If [I; x0; w0] � [J; x; w] then x � x0 and �l(x0) + l(w0) ��l(x) + l(w);(ii) Segments in V for the ordering � are �nite.Proof. To prove (i) it su�ces that the properties of (i) hold for �1 and �2.For �1 this is immediate. If [I; x0; w0] �2 [J; x; w] there is z 2 WJ withw0 = wz; l(w0) = l(w) + l(z); xz � x0: Moreover I � J . Then�l(x0) + l(w0) = �l(x0) + l(xz)� l(x) + l(w) � �l(x) + l(w):Also, x � xz � x0.To prove (ii), let [H; I 00; J 00] � [I; x0; w0] � [J; x; w]:32

By (i) we have x0 � x00 andl(w0) � �l(x) + l(w) + l(x0) � l(w) + l(w00):These inequalities imply that the segments in V for � are �nite, proving(ii).We prove that 2.5 (i) carries over.6.6. Lemma. Let x 2 W J , a 2 W I ; b 2 W . Then [I; a; b]� [J; x; 1] if andonly if I � J; b 2 WJ and xb � a.Proof. If these conditions are satis�ed it is immediate that [I; a; b]�2 [J; x; 1]whence [I; a; b]� [J; x; 1]. Conversely, let this be the case and assume thatv0 = [I; a; b]; v1; :::; vs = [J; x; 1]is a sequence of elements of V such that for i = 1; :::; s either vi�1 �1 vior vi�1 �2 vi. If s = 1 the condition of the lemma is easily seen to hold.So assume that s > 1. By induction we may assume that v1 is of the form[K; c; d], with K � J , d 2 WJ and xd � c. If v0 �1 v1 then xb � xd � c � a.Since d 2 WJ and b � d we have b 2 WJ . Moreover, I � K � J . Thisshows that the conditions of the lemma hold.If v0 �2 v1 there is z 2 WK with b = dz; l(b) = l(d)+ l(z) and cz � a. Thenxb = xdz � cz � a. Moreover I � K � J and b = dz 2 WJ , since d 2 WJand z 2 WK � WJ . The lemma follows.The next lemma is a partial analog of 2.1 (b).6.7. Lemma. Let [I; x0; w0] � [J; x; w] and assume that s 2 S is such thatsw > w, sw0 > w0, Then [I; x0; sw0] � [J; x; sw].Proof. Take a chain v0 = [I; x0; w0]; v1; :::; vs = [J; x; w] such that for i =1; :::; s either vi�1 �1 vi or vi�1 �2 vi.First let s = 1. If v0 �1 v1 it is immediate that [I; x0; sw0] �1 [J; x; sw].If v0 �2 v1 and z is as in the de�nition of �2 (before 2.6) then z may alsoserve to show that [I; x0; sw0] �2 [J; x; sw]. Now let s > 1. Assume that v1 =[K; c; d]. If sd > d we may assume by induction that [K; c; sd] � [J; x; sw].By the case s = 1 we know that [I; x0; sw0] � [K; c; sd] and the assertionfollows.If sd < d and v0 � v1 we have w0 � d. But then we must have sw0 � d,whence [I; x0; sw0] �1 [K; c; d]� [J; x; w]�1 [J; x; sw]:If sd < d we cannot have v0 �2 v1. For if w0 = dz; l(w0) = l(d) + l(z) thensw0 > w0 implies sd > d. 33

We have proved the lemma.6.8. The map �. The results of no. 3 suggest the introduction of a map�, semilinear with respect to the automorphism of Z[u; u�1] sending u tou�1, such that for v 2 V�(mv) = u�2d(v)Xw2V bx;vmw;satisfying(A) �(et:m) = e�1t :�(m)(t 2 S; m 2 N ),(B) b[I;a;b];[J;1;1] = (�1)l(a)+l(b)(1�u2)jJ�IjRb;a(u2), for a 2 W I ; b 2 WJ andb � a.(A) is formula (7) and (B) is the particular case x = 1 of (14). The targetspace ~M of � should be a completion of M, as in�nite sums arise.6.9. Proposition. (i) There exists a unique � satisfying (A) and (B);(ii) bw:v = 0 if w 6� v;(iii) If w � v then bw;v is a polynomial in Z[u2] with leading term (�u2)d(v)�d(w).Proof. The formulas of 3.2 imply thatmJ;x;w = e�1(x�1;1):e(1;w):mJ;1;1:If � exists it follows that�(mJ;x;w) = e(x;1):e�1(1;w�1):�(mJ;1;1); (17)showing that � is uniquely determined by (A) and (B).We de�ne � by (17). Then we have to prove (A).If s 2 S, sw > w then by 3.2�(e(1;s):mJ;x;w) = �(mJ;x;sw) = e(x;1):e�1(1;w�1s):�(mJ;1;1) = e�1(1;s)�(mJ;x;w);establishing part of (A). The case that sw < w is dealt with similarly, aswell as the cases of t = (s; 1) with sx 2W J .There remains the case that t = (s; 1) and sx = x� with � 2 WJ . Then�(e(s; 1):mJ;x;w) = �(mJ;x;w�) = e�1(1;(w�)�1):�(mJ;x;w):For (A) to hold this should be equal to e�1(s;1):�(mJ;x;w): Using (17) we seethat it su�ces to deal with the case that w = 1. In that case we have toprove e(s;1):�(mJ;x;1) = e(1;�):�(mJ;x;1):34

Now the arguments of 3.9 show that (B) implies the validity of (14) in thepresent situation. The equality to be proved then follows by using propertiesof R-polynomials, as in the proof of 3.8.We prove (ii) by induction on l(w). For w = 1 the assertion is true by 6.6.Let w 6= 1 and take s 2 S with sw < w. Formula (9) holds in our situationand impliesb[I;a;b];[J;x;w] = b[I;a;sb];[J;x;sw] if sb < b= (1� u2)b[I;a;b];[J;x;sw] + u2b[I;a;sb];[J;x;sw] if sb > b:If b[I;a;b];[J;x;w] 6= 0 and sb < b then we can conclude by induction that[I; a; sb]� [J; x; sw]. Application of 6.7 shows that [I; a; b]� [J; x; w].If b[I;a;b];[J;x;w] 6= 0 and sb > b then induction gives that [I; a; b] � [J; x; sw]or [I; a; sb] � [J; x; sw]. In the �rst case we have [J; x; sw] �1 [J; x; w] andin the second case we must have [I; a; b]�1 [I; a; sb]� [J; x; sw] �1 [J; x; w].In both cases [I; a; b]� [J; x; w]. This establishes (ii).The proof of (iii) is like the proof of 3.5 (i).6.10. From 6.5 (ii) and 6.9 (ii) we infer that �2 is a module homomorphismM! ~M, given by�2(mv) = Xw2V ( Xw�z�v u2d(v)�2d(z)bw;z(u�2)bz;v(u2))mw:The results of no. 3 suggest the conjecture that �2 = 1. But so far I havenot been able to prove this.If �2 = 1 one can show that there exist Kazhdan-Lusztig polynomials cw;vwith the properties of 4.4. In fact, the existence of such polynomials, for allv; w 2 V , is equivalent with the involutive property of �.Some support for the conjecture is provided by calculations made by W.van der Kallen. He did some experimentation with formula (15) in the caseof a�ne Weyl groups of small rank, with his program for computing theKazhdan-Lusztig polynomials of no. 4. The experiments produced againpolynomials cw;v with positive integral coe�cients.It is natural to ask whether there is some geometric background to theconstructions of the present section.AppendixWilberd van der Kallen35

We computed the cx;v with a Mathematica program. We got them for all x,v when G is of rank two, and for v = B = [D; 1; 1] when G is of rank three.After minor changes in the program, we could also explore a few a�ne Weylgroups and a few Coxeter groups that are not Weyl groups.The set V of B �B orbits has size 1800 for type A3 and it has size 7056 fortype B3. As memory becomes a problem we had to give up on computingand storing the entire partial order on V . But note that ~bv;w 6= 0 exactlywhen v � w. Therefore we replace in recursion (15) the sumXw<y�v ~cy;v~bw;ywith a sum over the y with w 6= y and ~by;v 6= 0.We also need an e�cient solution of the word problem in our Weyl groups.We are primarily interested in very small Weyl groups, so we can simplystart from a faithful representation. (Matrix coe�cients have to lie in a ringfor whose elements a normal form has been implemented.) We might havestored the multiplication table, but we prefer manipulating words. As Math-ematica is very good at replacement rules in terms of pattern matching, wehave it look for rules that make a word go down in length or lexicographicorder. Five rules su�ce for B3. In the main computation these rules arethen applied automatically when reduce is called. Then it is straightfor-ward to introduce things like distinguished coset representatives, the lengthfunction, the R polynomials. We use the R polynomials also for storing theBruhat order.As the partial order on V is not readily available, we do not use directly that~bv;w = 0 when v 6� w. Instead we look if the criteria in 2.4, 2.5 guaranteevanishing. To be speci�c, suppose we want to compute ~bv;w with v = [I; a; b]and w = [J; x; y]. We �rst check if v = w. Then we check if I � J , a � x,l(b)� l(a) � l(y)� l(x). If not, then ~bv;w certainly vanishes. If y = 1 wecheck if b 2 WJ . Apart from these checks, the procedure is as described inthe paper: To compute ~bv;w one �rst uses (10) to reduce to y = 1, then oneapplies the formula in 3.8.As we recompute the ~bv;w each time they are needed, it should not be asurprise that the program is slow. It took about a week to compute the cwBfor type B3. (Actually at that time we did not yet use R polynomials.) Ona machine with more memory one could speed things up.The mathematica �les are available on our web site. Seehttp://www.math.uu.nl/people/vdkallen/kallen.html36

There one also �nds more of the output, some of it in PostScript, most of itin Mathematica InputForm.

37

TablesWe put q = u2. In the tables we have left out all cases where cwv equalszero or one.Case A2, cwv for v = [D; 1; 1]Observe there is some duplication caused by the symmetry which inter-changes s1 with s2.cwv w1 + q [;; s1s2; 1]1 + q [;; s1s2; s2]1 + q [;; s2s1; 1]1 + q [;; s2s1; s1]1 + q [;; s1s2s1; s1]1 + q [;; s1s2s1; s2]1 + 2 q + q2 [;; s1s2s1; 1]

38

Case B2, cwv for v = [D; 1; 1]cwv w1 + q [;; s1s2; 1]1 + q [;; s1s2; s2]1 + q [;; s2s1; 1]1 + q [;; s2s1; s1]1 + q [;; s1s2s1; s2s1]1 + q [;; s2s1s2; s1s2]1 + q [;; s1s2s1s2; s1s2]1 + q [;; s1s2s1s2; s2s1]1 + q [f1g; s2s1s2; 1]1 + q [f1g; s2s1s2; s1]1 + q [f2g; s1s2s1; 1]1 + q [f2g; s1s2s1; s2]1 + 2 q [;; s1s2s1; s1]1 + 2 q [;; s1s2s1; s2]1 + 2 q [;; s2s1s2; s1]1 + 2 q [;; s2s1s2; s2]1 + 3 q + q2 [;; s1s2s1; 1]1 + 3 q + q2 [;; s2s1s2; 1]1 + 3 q + q2 [;; s1s2s1s2; s1]1 + 3 q + q2 [;; s1s2s1s2; s2]1 + 4 q + 3 q2 [;; s1s2s1s2; 1]39

Case G2, cwv for v = [D; 1; 1]cwv w1 + q [;; s1s2; 1]1 + q [;; s1s2; s2]1 + q [;; s2s1; 1]1 + q [;; s2s1; s1]1 + q [;; s1s2s1; s2s1]1 + q [;; s2s1s2; s1s2]1 + q [;; s1s2s1s2; s2s1s2]1 + q [;; s2s1s2s1; s1s2s1]1 + q [;; s1s2s1s2s1; s2s1s2s1]1 + q [;; s2s1s2s1s2; s1s2s1s2]1 + q [;; s1s2s1s2s1s2; s1s2s1s2]1 + q [;; s1s2s1s2s1s2; s2s1s2s1]1 + q [f1g; s2s1s2; 1]1 + q [f1g; s2s1s2; s1]1 + q [f1g; s1s2s1s2; s2]1 + q [f1g; s1s2s1s2; s2s1]1 + q [f1g; s2s1s2s1s2; s1s2]1 + q [f1g; s2s1s2s1s2; s1s2s1]1 + q [f2g; s1s2s1; 1]1 + q [f2g; s1s2s1; s2]1 + q [f2g; s2s1s2s1; s1]1 + q [f2g; s2s1s2s1; s1s2]1 + q [f2g; s1s2s1s2s1; s2s1]1 + q [f2g; s1s2s1s2s1; s2s1s2]1 + 2 q [;; s1s2s1; s1]1 + 2 q [;; s1s2s1; s2]1 + 2 q [;; s2s1s2; s1]1 + 2 q [;; s2s1s2; s2]1 + 2 q [;; s1s2s1s2; s1s2]1 + 2 q [;; s1s2s1s2; s2s1]1 + 2 q [;; s2s1s2s1; s1s2]1 + 2 q [;; s2s1s2s1; s2s1]1 + 2 q [;; s1s2s1s2s1; s1s2s1]1 + 2 q [;; s1s2s1s2s1; s2s1s2]1 + 2 q [;; s2s1s2s1s2; s1s2s1]1 + 2 q [;; s2s1s2s1s2; s2s1s2] 40

G2 continuedcwv w1 + 2 q [f1g; s1s2s1s2; 1]1 + 2 q [f1g; s1s2s1s2; s1]1 + 2 q [f1g; s2s1s2s1s2; s2]1 + 2 q [f1g; s2s1s2s1s2; s2s1]1 + 2 q [f2g; s2s1s2s1; 1]1 + 2 q [f2g; s2s1s2s1; s2]1 + 2 q [f2g; s1s2s1s2s1; s1]1 + 2 q [f2g; s1s2s1s2s1; s1s2]1 + 3 q [f1g; s2s1s2s1s2; 1]1 + 3 q [f1g; s2s1s2s1s2; s1]1 + 3 q [f2g; s1s2s1s2s1; 1]1 + 3 q [f2g; s1s2s1s2s1; s2]1 + 3 q + q2 [;; s1s2s1; 1]1 + 3 q + q2 [;; s2s1s2; 1]1 + 3 q + q2 [;; s1s2s1s2s1s2; s1s2s1]1 + 3 q + q2 [;; s1s2s1s2s1s2; s2s1s2]1 + 4 q + q2 [;; s1s2s1s2; s1]1 + 4 q + q2 [;; s1s2s1s2; s2]1 + 4 q + q2 [;; s2s1s2s1; s1]1 + 4 q + q2 [;; s2s1s2s1; s2]1 + 4 q + q2 [;; s1s2s1s2s1; s1s2]1 + 4 q + q2 [;; s1s2s1s2s1; s2s1]1 + 4 q + q2 [;; s2s1s2s1s2; s1s2]1 + 4 q + q2 [;; s2s1s2s1s2; s2s1]1 + 5 q + 3 q2 [;; s1s2s1s2; 1]1 + 5 q + 3 q2 [;; s2s1s2s1; 1]1 + 5 q + 3 q2 [;; s1s2s1s2s1s2; s1s2]1 + 5 q + 3 q2 [;; s1s2s1s2s1s2; s2s1]1 + 6 q + 3 q2 [;; s1s2s1s2s1; s1]1 + 6 q + 3 q2 [;; s1s2s1s2s1; s2]1 + 6 q + 3 q2 [;; s2s1s2s1s2; s1]1 + 6 q + 3 q2 [;; s2s1s2s1s2; s2]1 + 7 q + 5 q2 [;; s1s2s1s2s1; 1]1 + 7 q + 5 q2 [;; s2s1s2s1s2; 1]1 + 7 q + 5 q2 [;; s1s2s1s2s1s2; s1]1 + 7 q + 5 q2 [;; s1s2s1s2s1s2; s2]1 + 8 q + 7 q2 [;; s1s2s1s2s1s2; 1]41

Type A3, sample of large cwv for v = [D; 1; 1]cwv w1 + 7 q + 12 q2 + 4 q3 [;; s1s2s1s3s2s1; s2]1 + 7 q + 12 q2 + 4 q3 [;; s1s2s3s2s1; 1]1 + 7 q + 13 q2 + 4 q3 [;; s1s2s1s3s2s1; s1]1 + 7 q + 13 q2 + 4 q3 [;; s1s2s1s3s2s1; s3]1 + 7 q + 13 q2 + 4 q3 [;; s1s2s1s3s2; 1]1 + 7 q + 13 q2 + 4 q3 [;; s2s1s3s2s1; 1]1 + 8 q + 19 q2 + 10 q3 [;; s1s2s1s3s2s1; 1]Type B3, sample of large cwv for v = [D; 1; 1]cwv w1 + 18 q + 71 q2 + 73 q3 + 11 q4 [;; s2s1s3s2s1s3s2s3; s3]1 + 18 q + 71 q2 + 73 q3 + 11 q4 [;; s1s2s1s3s2s1s3s2; s1]1 + 18 q + 74 q2 + 75 q3 + 11 q4 [;; s1s2s1s3s2s1s3s2; s3]1 + 18 q + 74 q2 + 76 q3 + 13 q4 [;; s1s2s3s2s1s3s2s3; s2]1 + 18 q + 71 q2 + 78 q3 + 15 q4 [;; s2s1s3s2s1s3s2s3; s2]1 + 18 q + 71 q2 + 78 q3 + 15 q4 [;; s1s2s3s2s1s3s2s3; s1]1 + 18 q + 72 q2 + 79 q3 + 15 q4 [;; s1s2s1s3s2s1s3s2s3; s1s3]1 + 18 q + 72 q2 + 79 q3 + 15 q4 [;; s2s1s3s2s1s3s2; 1]1 + 18 q + 75 q2 + 81 q3 + 16 q4 [;; s1s2s3s2s1s3s2s3; s3]1 + 18 q + 75 q2 + 81 q3 + 16 q4 [;; s1s2s1s3s2s1s3s2; s2]1 + 19 q + 81 q2 + 107 q3 + 29 q4 [;; s1s2s1s3s2s1s3s2s3; s1]1 + 19 q + 81 q2 + 107 q3 + 29 q4 [;; s2s1s3s2s1s3s2s3; 1]1 + 19 q + 85 q2 + 113 q3 + 34 q4 + q5 [;; s1s2s1s3s2s1s3s2s3; s2]1 + 19 q + 85 q2 + 113 q3 + 34 q4 + q5 [;; s1s2s3s2s1s3s2s3; 1]1 + 19 q + 86 q2 + 116 q3 + 36 q4 + 2 q5 [;; s1s2s1s3s2s1s3s2s3; s3]1 + 19 q + 86 q2 + 116 q3 + 36 q4 + 2 q5 [;; s1s2s1s3s2s1s3s2; 1]1 + 20 q + 96 q2 + 153 q3 + 67 q4 + 6 q5 [;; s1s2s1s3s2s1s3s2s3; 1]42

Poincar�e polynomials for intersection cohomology.Put IPw(q) =Xi�0 qi dim IH i( �w):Poincar�e polynomials IPw for type A1.IPw w1 [;; s1; 1]1 + q [;; 1; 1]1 + q [;; s1; s1]1 + q + q2 [f1g; 1; 1]1 + 2 q + q2 [;; 1; s1]1 + q + q2 + q3 [f1g; 1; s1]Poincar�e polynomials IPv for v = [D; 1; 1]IPv Type1 + q + q2 A11 + 4q + 9q2 + 9q3 + 4q4 + q5 A21 + 6q + 17q2 + 24q3 + 17q4 + 6q5 + q6 B21 + 10q + 33q2 + 64q3 + 80q4 + 64q5 + 33q6 + 10q7 + q8 G21 + 11q + 56q2 + 154q3 + 250q4 + 250q5 + 154q6 + 56q7 + 11q8 + q9 A31 + 23q + 181q2 + 770q3 + 2046q4 + 3610q5 + 4350q6+ B33610q7 + 2046q8 + 770q9 + 181q10 + 23q11 + q12Experiments with other Coxeter groupsDihedral group of order 10, cwv for v = [D; 1; 1]As usual we deleted all cases where cwv equals zero or one. We also removedall duplication caused by the symmetry which interchanges s1 with s2 andthus f1g with f2g.43

cwv w1 + q [;; s1s2; 1]1 + q [;; s1s2; s2]1 + q [;; s1s2s1; s2s1]1 + q [;; s1s2s1s2; s2s1s2]1 + q [;; s1s2s1s2s1; s1s2s1]1 + q [;; s1s2s1s2s1; s2s1s2]1 + q [f1g; s2s1s2; 1]1 + q [f1g; s2s1s2; s1]1 + q [f1g; s1s2s1s2; s2]1 + q [f1g; s1s2s1s2; s2s1]1 + 2 q [;; s1s2s1; s1]1 + 2 q [;; s1s2s1; s2]1 + 2 q [;; s1s2s1s2; s1s2]1 + 2 q [;; s1s2s1s2; s2s1]1 + 2 q [f1g; s1s2s1s2; 1]1 + 2 q [f1g; s1s2s1s2; s1]1 + 3 q + q2 [;; s1s2s1; 1]1 + 3 q + q2 [;; s1s2s1s2s1; s1s2]1 + 3 q + q2 [;; s1s2s1s2s1; s2s1]1 + 4 q + q2 [;; s1s2s1s2; s1]1 + 4 q + q2 [;; s1s2s1s2; s2]1 + 5 q + 3 q2 [;; s1s2s1s2; 1]1 + 5 q + 3 q2 [;; s1s2s1s2s1; s1]1 + 5 q + 3 q2 [;; s1s2s1s2s1; s2]1 + 6 q + 5 q2 [;; s1s2s1s2s1; 1]A1 a�ne, a sampleWe restricted the lengths of elements of W to four and looked for large cwv.Again we removed duplicates. One recovers them by interchanging s1 withs2. cwv w v1 + 4 q + 3 q2 + q3 [;; s2s1s2s1; 1] [f1; 2g; 1; s1s2s1]1 + 4 q + 3 q2 + q3 [;; s2s1s2s1; s1] [f1; 2g; 1; s1s2s1]1 + 4 q + 3 q2 + 2 q3 [;; s1s2s1s2; 1] [f1; 2g; 1; s1s2s1s2]1 + 4 q + 3 q2 + 2 q3 [;; s1s2s1s2; s1] [f1; 2g; 1; s1s2s1s2]1 + 4 q + 3 q2 + 2 q3 [;; s1s2s1s2; s2] [f1; 2g; 1; s1s2s1s2]1 + 4 q + 3 q2 + 2 q3 [;; s1s2s1s2; s1s2] [f1; 2g; 1; s1s2s1s2]44

Type A2 a�ne, sample of large cwvWe restricted the lengths of elements of W to three.cwv w v1 + 6 q + 5 q2 [;; s2s1s3; s1] [f1; 2g; 1; s1s2s3]1 + 3 q + 3 q2 + q3 [;; s3s2s1; 1] [f1; 2; 3g; 1; s1s3s2]1 + 3 q + 3 q2 + q3 [;; s3s2s1; s1] [f1; 2; 3g; 1; s1s3s2]Type H3, sample of intermediate size cwv for v = [D; 1; 1]For some w the computation of cwv was not feasible on our machine. There-fore we just present a few cwv that were still within reach. (It gets moredi�cult as dim(v)� dim(w) increases.)cwv w1 + 21 q + 62 q2 + 33 q3 [f1g; s1s2s1s2s3s2s1s2s1s3s2; s1s2s1s2s1]1 + 23 q + 65 q2 + 33 q3 [f3g; s1s2s3s2s1s2s1s3s2; s2s3s2]1 + 23 q + 65 q2 + 33 q3 [f3g; s1s2s1s2s3s2s1s2s1s3s2s1; s2s1s2s3s2s1]1 + 22 q + 69 q2 + 33 q3 [;; s1s2s1s2s3s2s1s2s1s3; s1s2s3s2s1]1 + 22 q + 69 q2 + 33 q3 [;; s2s1s2s3s2s1s2s1s3s2; s2s3s2s1s2]1 + 24 q + 70 q2 + 34 q3 [f3g; s1s2s3s2s1s2s1s3s2s1; s2s3s2s1]1 + 24 q + 70 q2 + 34 q3 [f3g; s1s2s1s2s3s2s1s2s1s3s2; s2s1s2s3s2]1 + 26 q + 85 q2 + 40 q3 [;; s2s1s2s3s2s1s2s1s3s2; s1s2s1s2s1]1 + 26 q + 85 q2 + 40 q3 [;; s3s2s1s2s1s3s2s1s2s3; s1s2s3s2s1]1 + 26 q + 81 q2 + 44 q3 [f3g; s2s1s2s3s2s1s2s1s3s2; s1s2s3s2]1 + 26 q + 81 q2 + 44 q3 [f3g; s2s1s2s3s2s1s2s1s3s2s1; s1s2s3s2s1]References[BBD] A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers,Ast�erisque, vol. 100, Soc. Math. de France, 1982.[B1] M. Brion, The behaviour at in�nity of the Bruhat decomposition,Comm. Math. Helv. 73 (1998), 137-174.[B2] M. Brion, Rational smoothness and �xed points of torus actions,Transformation Groups 4 (1999), 127-156.[BJ] M. Brion and R. Joshua, Vanishing of odd intersection cohomol-ogy II, preprint. 45

[BT] A. Borel and J. Tits, Compl�ements �a l'article\Groupes r�eductifs",Publ. Math. I.H.E.S. 41 (1972), 253-276.[C] R. W. Carter, Finite groups of Lie type. Conjugacy classes andcomplex characters, Wiley, 1985.[DP] C. De Concini and C. Procesi, Complete symmetric varieties,in: Invariant Theory, p. 1-44, Lect. Notes in Math., vol. 996,Springer, 1983.[DS] C. De Concini and T.A. Springer, Compacti�cation of symmetricvarieties, Transformation Groups 4 (1999), 273-300.[GJ] O. Gabber and A. Joseph, Towards the Kazhdan-Lusztig conjec-ture, Ann. Scient.�Ec. Norm. Sup., 14 (1981), 261-302.[KL] D. Kazhdan and G. Lusztig, Representations of Coxeter groupsand Hecke algebras, Inv. Math. 53 (1979), 165-184.[LV] G. Lusztig and D.A. Vogan, Singularities of closures of K-orbitson ag manifolds, Inv. Math. 71 (1983), 365-379.[MS] J.G.M. Mars and T.A. Springer, Hecke algebra representationsrelated to spherical varieties, Journal of Representation Theory2 (1998), 33-69.[RS] R.W. Richardson and T.A. Springer, The Bruhat order on sym-metric varieties, Geom. Dedic. 35 (1990), 389-436.[S] T.A. Springer, Combinatorics of B-orbit closures in a wonderfulcompacti�cation, in preparation.46