MONOID HECKE ALGEBRAS Introduction a Monoid Analogue Of

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 9, September 1997, Pages 3517–3534 S 0002-9947(97)01823-0

MONOID HECKE ALGEBRAS

MOHAN S. PUTCHA

Abstract. This paper concerns the monoid Hecke algebras introduced by Louis Solomon. We determine explicitly the unities of the orbitH algebras associated with the two-sided action of the Weyl group W . We use this to: 1. ﬁnd a description of the irreducible representations of , 2. ﬁnd an explicit isomorphism between and the monoidH algebra of the H Renner monoid R, 3. extend the Kazhdan-Lusztig involution and basis to ,and H 4. prove, for a W W orbit of R, the existence (conjectured by Renner) of generalized Kazhdan-Lusztig× polynomials.

Introduction A monoid analogue of the Iwahori-Hecke algebra [11] was obtained by Solomon [27]–[29]. In an earlier paper [19] the author studied Solomon’s monoid Hecke algebras by studying the associated orbit algebras. These orbit algebras arise from the two-sided action of the Weyl group W on the Renner monoid R. In particular, the coeﬃcients of the unity of the empty level orbit algebra were shown to be Rx,y, where Rx,y are polynomials introduced by Kazhdan and Lusztig [12]. The other orbit algebras were also shown to have unities, but their coeﬃcients were only im- plicitly given. In this paper we give an explicit formula for the unities of all the orbit algebras, thereby obtaining a description of the irreducible representations of monoid Hecke algebras. We also obtain an explicit, but very complicated, isomorphism between the monoid Hecke algebra and the monoid algebra of R,solvinga problem posed by Solomon [28]. We go on to extend to the monoid Hecke algebra the Kazhdan-Lusztig involution and basis for the (group) Iwahori-Hecke algebra. This then immediately yields polynomials Pθ,σ for θ, σ in the same W W orbit of R, partially solving a problem posed by Renner [26]. These polynomials× are still mysterious; however, in the simplest case they are products of relative Kazhdan- Lusztig polynomials introduced by Deodhar [6].

1. Reductive monoids and monoids of Lie type

Consider the general linear group G = GLn(F ) over an algebraically closed ﬁeld F . It is the unit group of the multiplicative monoid M = Mn(F )ofalln nmatrices over F . This monoid has the following structure. The diagonal idempotents× form a Boolean lattice with respect to the natural order of idempotents: f e if ef = fe = f. ≤ Received by the editors December 3, 1993. 1991 Mathematics Subject Classiﬁcation. Primary 20G40, 20G05, 20M30. Research partially supported by NSF Grant DMS9200077.

c 1997 American Mathematical Society

3517

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The G G orbits have a cross-section Λ of idempotents × I 0 e = r . r 00 These idempotents and hence the G G orbits are linearly ordered according to × rank. The orbit Jr = GerG consists of all rank r matrices and gives rise to the orbit semigroup J 0 = Ge G 0 , r r ∪{ } where for a, b J , ∈ r ab if ab J , a b = ∈ r · (0otherwise. We also have the orbit monoid M(J )=G J0. r ∪ r The idempotents in Jr are all conjugate, and hence M(Jr) is completely determined by the parabolic subgroups AB P = g G ge = e ge = A GL , r { ∈ | r r r} 0 C|∈ r A 0 P − = g G e g = e ge = A GL , r { ∈ | r r r} BC|∈ r and the natural homomorphisms δ+ : Pr GLr and δ : Pr− GLr.Wewrite → − → M(Jr)asM(G, Pr,Pr−,GLr) with δ+,δ being understood to be part of the data. There are two deficiencies with the monoid− M. First, only maximal parabolics arise as Pr, and second, the orbit monoids are not submonoids of M.Bothofthese drawbacks can be handled by considering the representation θ of M given by θ(a)= 1(a) 2(a) n(a), ∧ ⊗∧ ⊗···⊗∧ the tensor product of all exterior powers of a. The representation θ has the effect of killing the singular part of M, because θ(M)=θ(G) 0 . However, taking ∪{ } the Zariski closure of θ(M) yields an amazing monoid M. Again there is a G G cross-section Λ of idempotents, but this time Λ 0 is a Boolean lattice isomorphic× \{ } to the power set of the Dynkin diagram of G. Moreover,f any parabolic subgroup is associated with a unique orbit monoid, which now is a submonoid of M.Further, the lattice of diagonal idempotents is the dual of the Coxeter complex. Both M and M are examples of reductive monoids. The theoryf of reductive monoids has been well developed by Renner and the author; cf. [17]. In particular the author [16] hasf proved the existence of a cross-section lattice Λ and has shown that the orbit monoids are of the form M(G, P, P −,L/K) for some pair of opposite parabolics P, P −, K/L=P P−. Renner [24] has shown the existence of the finite inverse monoid ∩ R = W, Λ , h i where W is the Weyl group of G such that M = BσB, σ R G∈ where B is the Borel subgroup G.WecallRthe Renner monoid of M.

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The theory of monoids of Lie type is an attempt to accomplish the above for finite groups of Lie type. Let G be a finite group of Lie type with Weyl group W and Borel subgroup B. The most convenient definition of a monoid M of Lie type (with unit group G) is also the most general, and is due to the author [18]. The definition is simply that all the orbit monoids are of the form

M(G, P, P −,L/K),

where P, P − are opposite parabolic subgroups, L = P P −, K/L. These monoids have been classiﬁed by the author [18], [20] using the∩ theory of buildings [30] and the ideas of Renner and the author [21] for reductive monoids. Renner and the author [22] obtained an analogue of the canonical monoid M for any ﬁnite group G of Lie type. Let M be a monoid of Lie type with unit group G. Theref is again a cross- section lattice Λ for the G G orbits. There is also an analogue of Renner monoid R = W, Λ such that × h i M = BσB. σ R G∈ We refer to [18] for details. We will let and ` denote the Bruhat order and the length function, respectively, on the Weyl≤ group W ;cf.[10].Fore Λ, let ∈ W (e)= x W xe = ex , { ∈ | } W = x W xe = e /W(e). e { ∈ | } Then both W (e)andWe are parabolic subgroups of W ,andW(e) is the direct product of We and the Weyl group of the unit group of eMe.Let D(e)= x W xis of minimum length in xW (e) , { ∈ | } D = x W x is of minimum length in xW . e { ∈ | e} Now R = WeW, e Λ G∈ and each element σ WeW can be uniquely written as ∈ 1 σ = xey− ,xD,y D(e). ∈e ∈ We call this the standard form of σ.Ifw0,v0 are respectively the longest elements of W and W (e), then w0v0 is the longest element of D(e). Following Solomon [27], [29] and Renner [25], we let

`(e)=`(w0v0) 1 and, for σ = xey− in standard form, `(σ)=`(x)+`(e) `(y). − It is shown in [25] that `(σ) = 0 if and only if σB = Bσ. 1 1 If σ = xey− and θ = sft− in standard form, then deﬁne (1) σ θ if e f and x sw, tw y for some w W (f)W . ≤ ≤ ≤ ≤ ∈ e

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We have shown in [15] that in the analogous situation for reductive monoids, σ θ if and only if BσB is contained in the Zariski closure of BθB. For this reason≤ we 1 say that σ is triangular if σ 1. Thus, if σ = xey− in standard form, ≤ (2) σ is triangular if and only if x y. ≤ If σ WeW and θ WeW,then ∈ ∈ σ θ implies `(σ) `(θ). ≤ ≤ 2. Hecke algebras

Let G be a Chevalley group deﬁned over Fq, B,B− opposite Borel subgroups of G, T = B B−, B = UT, B− = U−T,andWthe Weyl group of G with generating set S of simple∩ reﬂections. If x W ,let˙xdenote a coset representative in G.Let ∈ 1 = b C[G]. B ∈ b B | |X∈ The Iwahori-Hecke algebra

HC(G)=HC(G, B)=C[G] is a semisimple algebra that is isomorphic to C[W ] by a theorem of Tits (cf. [1], [4]). Clearly HC(G, B) has a basis A = x, x W. x ∈ This basis is normalized as T = q`(x)A ,xW. x x ∈ With respect to this basis, Iwahori [11] showed that the structure constants are integer polynomials in q, depending only on W . In particular,

(3) Txy = TxTy,Axy = AxAy if `(xy)=`(x)+`(y).

One can therefore consider the generic Hecke algebra (W ) with basis Tx (x W ) 1/2 1/2 H ∈ over Z[q ,q− ], where q is now being treated as an indeterminate. Specifying q and tensoring with C, one recovers HC(G, B). Kazhdan and Lusztig [12] introduced an involution on (W )givenby H 1/2 1/2 1 `(y) (4) q =q− , Tx =T−1 = q− Ry,xTy, x− y W X∈ where Ry,x = Ry,x(q) Z[q]. These polynomials are nonzero exactly when y x, and have been studied∈ in detail by Deodhar [5]. We see by (3) that ≤

1 `(x) (5) Ax = A− 1 = q Ry,xAy. x− y W X∈ Now let M be a monoid of Lie type with unit group G, cross-section lattice Λ and Renner monoid R.WeletC[M] denote the complex monoid algebra of M. By a result of Okni´nski and the author [14], C[M] is always semisimple. Following Solomon [27]–[29], we introduce the monoid Hecke algebra

HC(M)=HC(M,B)=C[M].

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Since C[M] is semisimple, so is HC(M,B). Using this fact, Solomon [27], [29] has proved that H (M,B) [R]. This algebra has been further studied by the author C ∼= C [19]. Clearly, HC(M,B) has a basis, A = σ, σ R. σ ∈ Following Solomon [27], [29], we normalize this basis as T = q`(σ)A ,σR. σ σ ∈ Fix e Λ. Then for some I S we have W (e)=W and ∈ ⊆ I PI = x G xe = exe , (6) { ∈ | } P − = x G ex = exe . I { ∈ | } Let L = P P −.SoP =LU and P − = LU −,whereU and U − are the unipotent I ∩ I I I I I I I radicals of PI ,andPI−, respectively. Then UIe = eUI− = e and BL = B L is a Borel subgroup of L. By [1, Propositions 2.3.3 and 2.5.15],{ } ∩ 1 1 (7) yB y− B,y− By U−B for all y D(e). L ⊆ ⊆ I ∈ Hence, for all y D(e), ∈ 1 A 1 A = ey y = e = A . ey− y − e So 1 (8) A 1 = A A− = A A 1 for all y D(e). ey− e y e y− ∈ Let x W .Thenforb B,be = eb for some b B .Sofory D(e), ∈ ∈ 1 1 ∈ L ∈ 1 1 xbey− = xeb1y− 1 1 = xey− yb y− · 1 · 1 = xey− , by (7).

Thus A A 1 = A 1 .Soby(8), x ey− xey−

(9) A 1 = A A A 1 for x W, y D(e). xey− x e y− ∈ ∈ Now let 1 1 1 = b, 2 = u. BL UI b B u UI | | X∈ L | | X∈ Then = = .Soforz W(e), 1 2 2 1 ∈ AzAe = ze = ze = ez

= ez2 = e2z = 1e2z = e12z

= AeAz. So, by (9), (10) A = A A = A A for z W (e). ze z e e z ∈ 1 Now let e, f Λandx W.Thenx=zy− for some z W (e)andy D(e). So ∈ ∈ ∈ ∈ A A A = A A A 1 A , by (3) e x f e z y− f

= A A A 1 A , by (5), (10) z e y− f

= A A 1 A , by (8) z ey− f

= A A 1 . z ey− f

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So by (5), (9), (10), we have

Theorem 2.1. The structure constants of HC(M,B) with respect to the basis Aσ (σ W ) and hence with respect to the basis Tσ (σ w) are integer Laurent polynomials∈ in q, depending only on R. ∈ Remark 2.2. Solomon [27], [29] has actually proved much more. He showed that

HC(M,B) has a presentation given by

qTσ if `(sσ)=`(σ), TsTσ = Tsσ if `(sσ)=`(σ)+1, qTsσ +(q 1)Tσ if `(sσ)=`(σ) 1, − − `(σ) `(νσ) TνTσ =q − Tνσ for s S, σ, ν R, `(ν)=0. ∈ ∈ Because of Theorem 2.1 we can consider the generic monoid Hecke algebra (R) 1/2 1/2 H with basis Aσ (σ R) or equivalently Tσ (σ R)overZ[q ,q− ], where q is ∈ ∈ now being treated as an indeterminate. Specifying q and tensoring with C,one recovers HC(M,B). 3. Orbit algebras The idea here is very simple and is a key ingredient of semigroup representation theory (cf. [2, Chapter 5]): If is an ideal of a semisimple algebra ,then ∼= / .LetMbe a monoid ofI Lie type with unit group G, cross-sectionA latticeA Λ, I⊕A I Weyl group W and Renner monoid R.SinceC[M] is semisimple by [14], we have 0 (11) C[M] ∼= C0[J ], MJ 0 where the summation is over the G G orbits J and C0[J ] is the contracted semigroup algebra; i.e. the zero of J 0 is the× zero of the algebra. Making the isomorphism in (11) explicit is a diﬃcult open problem. For a G G orbit J,let × 0 HC(J)=HC(J, B)=C0[J ]. Call this an orbit Hecke algebra. As in (11), if we use the natural order on Λ and proceed inductively, beginning with the least element of Λ, we have (12) H (M,B) H (J, B), C ∼= C MJ where the summation is over the G G orbits J. Wewillmaketheisomorphism × in (12) explicit in Theorem 3.3. We note that HC(J, B) is an ideal of HC(M,B)/ , where I

= H (J 0,B). I C JX0

HC(J)=HC(J, B)=HC(J, B)+HC(G, B) is a subalgebra of H (M,B)/ .Wecallittheaugmented orbit Hecke algebra.Now Cb Ib ﬁx e Λ and let J = GeG.LetI S,PI,PI−,L,BL be as in (6). The author [19, Section2]hasshownthatif∈ x, y ⊆D(e), then, in H (J), ∈ C (13) A 1 A = Q A , ex− ye x,y e

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where 1 x− By U−B wB (14) Q = | ∩ I L |A . x,y B w w W (e) ∈X | |

Let P = PI and P − = PI−, and consider the natural map

ξ : P −P = U −LU L. I I → If w W (e), then ∈ ξ(B−wB)=B−wB = (B−wB B zB ). L L L L ∩ L L z W(e) ∈G Let x, y D(e). Then by (7) ∈ 1 1 1 1 1 (15) B x− ByB = x− xB x− B yB y− y x− By. L L · L · · L · ⊆ 1 Let x, y D(e), z,w W(e), a x− By B−wB be such that ξ(a) B zB . ∈ ∈ ∈ ∩ ∈ L L Then BLzBL = BLξ(a)BL.Letb1,b2 BL be such that b1ξ(a)b2 BL−wBL.Now 1 ∈ ∈ by (15), b ab x− By. Also a = vξ(a)u for some v U −, u U .So 1 2 ∈ ∈ I ∈ I b1ab2 = b1vξ(a)ub2 1 1 = b vb− b ξ(a)b b− ub 1 1 · 1 2 · 2 2 U −B−wB U ∈ I L L I = B−wB and ξ(b ab )=b ξ(a)b .Thus,forx, y D(e)andz,w W(e), 1 2 1 2 ∈ ∈ 1 BLzBL ξ(x− By B−wB) = ∅ ∩ ∩ 6 implies 1 B zB B−wB ξ(x− By B−wB). L L ∩ L L ⊆ ∩ For z,w W(e), let ∈ 1 1 X = x− By ξ− (B zB B−wB ). z,w ∩ L L ∩ L L Then

(16) Xz,w = ∅ implies ξ(Xz,w)=BLzBL B−wBL. 6 ∩ L Suppose Xz,w = ∅, and let a Xz,w.Thenξ(a)=b1zb˙ 2 for some b1,b2 BL.Let 6 ∈ ∈ c X be such that ξ(a)=ξ(c). Then for some v U −, u U , ∈ z,w ∈ I ∈ I c = vb1zb˙ 2u 1 1 = b b− vb z˙ b ub− b 1 · 1 1 · · 2 2 · 2 1 1 = b v zu˙ b with v = b− vb U−,u =bub− U . 1 1 1 2 1 1 1 ∈ I 1 2 2 ∈ I Also by (15) 1 1 1 1 1 1 v zu˙ = b− cb− b− x− Byb− x− By. 1 1 1 2 ∈ 1 2 ⊆ Thus 1 v zu˙ x− By U−zU˙ . 1 1 ∈ ∩ I I 1 Conversely, if v U −, u U with v zu˙ x− By, then by (15), 1 ∈ I 1 ∈ I 1 1 ∈ 1 1 1 c = b v zu˙ b = b v b− b zb˙ b− u b x− By U−ξ(a)U . 1 1 1 2 1 1 1 · 1 2 · 2 1 2 ∈ ∩ I I

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Hence, c Xz,w and ξ(a)=ξ(c). Thus each ﬁbre of ξ restricted to Xz,w has ∈ 1 cardinality x− By U−zU˙ . Hence by (16) | ∩ I I| 1 Xz,w = ∅ implies Xz,w = BLzBL B−wBL U−zU˙ I x− By . 6 | | | ∩ L |·| I ∩ | 1 Now suppose BLzBL BL−wBL = ∅ and UI−zU˙ I x− By = ∅.Thenforsome ∩ 6 1 ∩ 6 v UI−and u UI we have vzu˙ x− By.Alsoforsomeb BL,bz˙ BL−wBL. Then∈ by (15) ∈ ∈ ∈ ∈

1 bvzu˙ = bvb− bz˙ u X . · · ∈ z,w So Xz,w = ∅. Hence in all cases 6 1 X = B zB B−wB U−zU˙ x− By . | z,w| | L L ∩ L L|·| I I ∩ | Thus

1 x− By B−wB = X | ∩ | | z,w| z W(e) (17) ∈X 1 = B zB B−wB U−zU˙ x− By . | L L ∩ L L|·| I I ∩ | z W(e) ∈X Now by [12], [5]

B zB B−wB = R B , | L L ∩ L L| w,z| L| and by (15)

1 1 B U−zU˙ x− By = U−B zU x− By | L|·| I I ∩ | | I L I ∩ | `(z) 1 = q− U−B zB x− By . | I L ∩ | Hence 1 BLzBL B−wBL U−zU˙ I x− By (18) | ∩ L |·| I ∩ | `(z) 1 = q− R U−B zB x− By . w,z| I L ∩ | `(z) `(z)+`(w) `(w) By [12], the inverse of the matrix (q− Rw,z)is(( 1) q Rw,z). Thus by (17), (18) −

1 `(w) `(z)+`(w) 1 (19) U −B wB x− By = q ( 1) R B−zB x− By . | I L ∩ | · − w,z| ∩ | z W(e) ∈X

Let z W (e), U − = U − B−, U = U B ,and ∈ L ∩ L L ∩ L 1 X= v U− zvz− U− , { ∈ L| ∈ L } 1 Y = v U− zvz− U . { ∈ L | ∈ L} Then 1 1 x− By B−zB = x− By U−U−zB ∩ ∩ I L 1 = x− By U−zXB ∩ I 1 = x− By zU−XB ∩ I

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and 1 1 x− By zB−B = x− By zU−U−B | ∩ | | ∩ I L | 1 = x− By U−zU−B | ∩ I L | 1 = x− By U−zY XB | ∩ I | 1 1 = x− By U−zY z− zXB | ∩ I · | 1 1 = x− By zY z− U−zXB | ∩ · I | 1 1 = zY z− x− By U−zXB , by (15) | |·| ∩ I | `(z) 1 = q x− By zU−XB | ∩ I | `(z) 1 = q x− By B−zB . | ∩ | Hence 1 `(z) 1 (20) x− By zB−B = q x− By B−zB . | ∩ | | ∩ | By [19, Theorem 2.4] 1 1 1 x− By zB−B = z− x− By B−B | ∩ | | ∩ | `(xz) `(y) = B q − R . | |· xz,y Hence, by (20)

1 `(x) `(y) (21) x− By B−zB = B q − R . | ∩ | | |· xz,y So by (14), (19), (21),

`(w) `(z)+`(w) `(x) `(y) (22) Q = q ( 1) q − R R A . x,y − w,z xz,y · w w W (e) z W (e) ∈X ∈X Now consider the generic monoid Hecke algebra (R). Fix e Λ and let H ∈ 1/2 1/2 (23) = Z[q ,q ]Aσ. I σ WfW ∈f

1/2 1/2 (e)= Z[q ,q− ]Aσ H σ WeW ∈X is an ideal of (R)/ . Again we call H I (e)= (e)+ (W) H H H the augmented orbit Hecke algebra. b Since (13), (22) are valid for all q, they are valid in (e). Now in (e), H H `(x) `(y) `(z)+`(w) `(w) Q = q − R ( 1) q R A x,y xz,y − b w,z bw z W (e) w W (e) ∈X ∈X `(x) `(y) `(z) = q − Rxz,y q Rw,zAw z W(e) w W (e) ∈X ∈X `(x) `(y) = q − Rxz,yAz, by (5). z W(e) ∈X

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By [19, Theorem 3.1], the unity of (e) is obtained by inverting the matrix (Qx,y). Now for y D(e)wehave H ∈ `(y) Ay =q− Rw,yAw, by (5) w W X∈ `(y) = q− Rxz,yAxz x D(e) z W (e) ∈X ∈X `(y) = q− Ax Rxz,yAz, by (3) x D(e) z W (e) ∈X ∈X `(x) = q− AxQx,y. x D(e) ∈X So, for x D(e), ∈ `(w) Ax = q AwQw,x. w D(e) ∈X Hence `(w) `(x) Ay = q − AwQw,xQx,y x D(e) w D(e) ∈X ∈X

`(w) `(x) = A q − Q Q . w  w,x x,y w D(e) x D(e) ∈X ∈X Thus, for y,w D(e),   ∈ `(w) `(x) q − Qw,xQx,y = δw,yAe. x D(e) ∈X `(x) `(y) Hence (q − Qx,y)istheinverseof(Qx,y). Now `(x) `(y) q − Qx,y = Rxz,yAz. z W (e) ∈X Hence by [19, Theorem 3.1] the unitye ˆ of the orbit algebra (e)isgivenby H (24) eˆ= R A 1 . xz,y xzey− x,y D(e) z X∈W (e) ∈ 1 For σ = xey− WeW in standard form, let ∈ (25) Rσ = Rxz,y. z W X∈ e K Remark 3.1. (i) If We = WK , K S, then by Deodhar [6], Rσ = Rx,y. We also note that the condition xz y has⊆ been studied by Deodhar [7]. (ii) By (2), R = 0 if and≤ only if σ is triangular. σ 6 By (24), we now have Theorem 3.2. The unity eˆ of the generic orbit Hecke algebra (e) is given by H `(σ) eˆ = RσAσ = q− RσTσ. σ WeW σ WeW ∈X ∈X

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Now considere ˆ as an element of (R) and let H (26) e˜ =ˆe (1 fˆ). · − f Λ f

Theorem 3.3. (i) The elements e˜, e Λ are independent of the order of the product in (26). ∈ (ii)e ˜ (e Λ) form an orthogonal set of central idempotents of (R), and hence ∈ H (R)= e Λe˜ (R). H ∈ H (iii) The map Aσ eA˜ σ, σ WeW, extends to an isomorphism between (e) and e˜ (RL). → ∈ H (iv) If isH as in (23), then the natural homomorphism from (R) to (R)/ restrictsI to an isomorphism between e˜ (R) and (e). ThisH isomorphismH isI the inverse of the isomorphism in (iii).H H

Remark 3.4. (i) By Theorem 3.2 and [19, Theorem 3.1] we have an explicit isomorphism betwen (e)andMd( (WK)), where d = D(e) and W (e)=We WK, K S. We thereforeH have a sequenceH of homomorphisms| | × ⊆ (R) (R)/I (e) M ( (W )). H →H →H → d H K Upon specializing q, we see that an irreducible representation of degree n of (W ) H K yields explicitly an irreducible representation of degree nd of HC(M,B). Moreover every irreducible representation of HC(M,B) is obtained in this manner for some e Λ. ∈ 1/2 1/2 (ii) Let Λ = Z[q ,q− ]. Then as in Theorem 3.3 (but much more easily) Γ[R] 0 is explicitly isomorphic to a direct sum of orbit algebras Γ0[J ]. For J = WeW, 0 the unity e of Γ0[J ]isgivenby

1 e= xex− . x D(e) ∈X 0 If d = D(e) ,thenΓ0[J ] is isomorphic to Md(Γ[WK ]), where W (e)=We WK, K S. Now| by| Lusztig [13] (or see [3]) we have an explicit isomorphism between× ⊆1/2 1/2 Q(q )[WK ]and (WK) ΓQ(q ). Combining with (i), we have an explicit but H ⊗ 1/2 1/2 very complicated isomorphism between (R) Γ Q(q )andQ(q )[R], solving a problem posed by Solomon [28]. H ⊗

4. Involution We wish to extend the Kazhdan-Lusztig involution (4) on (W )to (R). By H H Theorem 3.3 it suﬃces to do this for the augmented orbit Hecke algebras (e). We accomplish this by considering the order (1) on WeW and rememberingH that, by 1 (5), Ax,(x W) is a linear combination of Ay, y x.Ifσ=xey− inb standard form, then,∈ by (1), ≤

σ e if and only if x W (e)andx y. ≤ ∈ ≤

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Theorem 4.1. There is a unique extension of the involution on (W ) to (e) such that H H b A = R A 1 . e z,y zey− z W (e) y∈XD(e) ∈ 1 If σ = set− WeW in standard form, then ∈ `(t) (27) A = q− A R A 1 . σ s · tz,y zey− z W (e) y∈XD(e) ∈

Proof. Let w0 and v0 denote the longest element in W and W (e), respectively. Then u0 = w0v0 is the longest element in D(e). Also (28) `(xw )=`(w x)=`(w ) `(x) for all x W. 0 0 0 − ∈ Hence by (3) (29) A = A A and A = A A for all x W. xw0 x w0 w0x w0 x ∈ 1 If σ = set− in standard form, then `(t) A = A q− R A 1 σ s · · tz,y zey− z W (e) y∈XD(e) ∈ `(t) = A q− R 1 1 1 A 1 s · · z− t− ,y− zey− z W(e) y∈XD(e) ∈ `(t) = As q− Rv z z 1t 1,v z y 1 Azey 1 , by (28) · · 0 · − − 0 · − − z W(e) y∈XD(e) ∈ `(t) = As q− R 1 1 A 1 · v0t− ,v0zy− zey− z W (e) y∈XD(e) ∈ `(t) = A q− A R 1 1 A A 1 , by (9), (10) s · e · v0t− ,v0zy− z y− z W(e) y∈XD(e) ∈ `(t) = A q− A R 1 1 A A 1 s · e · v0t− ,zy− v0z y− z W (e) y∈XD(e) ∈ `(t) = A q− A A R 1 1 A A 1 , by (29) s · e v0 · v0t− ,zy− z y− z W (e) y∈XD(e) ∈ `(t) = A q− A A R 1 A , by (3) s · e v0 · v0t− ,x x x W X∈ `(t) = A q− A A R 1 A s · e v0 · v0 t− ,w0x w0x x W X∈ `(t) = A q− A A R 1 A s · e v0 · x,u0t− w0x x W X∈ `(t) = A q− A A A R 1 A , by (29) s · e v0 w0 · x,u0t− x x W X∈

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`(t) `(e)+`(t) = A q− A A A q− A 1 , by (5), (28) s · e v0 w0 · u0t− `(e) = q A A A A A 1 − s e v0 w0 u0t− `(e) = q− AsAeA 1 Au t 1 , by (29) u0− 0 − `(e) 1 1 = q− AsAeA Aw0 Av t , by (29) u0− 0 − `(e) 1 1 = q− AsAeA Aw0 Av0 At , by (3) u0− − `(e) 1 1 = q− AsAeA Au0 At , by (29). u0− − Hence, `(e) 1 1 (30) Aσ = q− AsAeA Au0 At . u0− − In particular, `(e) 1 (31) Ae = q− AeA Au0 . u0− By (13), (14), (22), we see that

A 1 A =0 ifx, y D(e),y

(33) A A 1 A A =0 ifx D(e),y W, y < x. e x− y e ∈ ∈ By (31), (32) (34) A A = A . e · e e By (29) we see that for all x W , ∈

Aw0 AxAw0 = Aw0 Axw0 = Aw0xw0 = Aw0xAw0 = Aw0 AxAw0 . Hence for all x W , ∈ (35) A = A A A = A A A and A2 A = A A2 . w0xw0 w0 x w0 w0 x w0 w0 x x w0 By (29), (35) we see that for all z W (e), ∈ 2 2 1 A Au0 Az = Av0 Aw0 Aw0 Av0 Az = A A Az u0− w0 v0 2 2 1 = AzA A = AzAv0 Aw0 Aw0 Av0 = AzA Au0 . w0 v0 u0− Hence by (5), (10), (31) (36) A A = A A , A A = A A for all z W (e). e z z e e z z e ∈ If x D(e), then by (28), ∈ 1 1 1 `(u x− )=`(w v x− )=`(w ) `(v x− ) 0 0 0 0 − 0 =`(w ) `(v ) `(x)=`(u ) `(x). 0 − 0 − 0 − So, by (3), A = A 1 A . Hence u0 u0x− x

AeAx 1 = AeA 1 Au x 1 for all x D(e). − u0− 0 − ∈ So, by (33),

A A 1 A =0 forx D(e),x=1. e x− e ∈ 6

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So, by (3), (36), A A A =0 forallx W W(e). e x e ∈ \ By (3), (5), (10), (33), A A A =0forx W W(e). Hence, by (5), (34), (36), e x e ∈ \ (37) A A A = A A A for all x W. e x e e x e ∈ By (3), (5), (30), (31), (36), A A = A A , A A = A A for all x W, σ WeW. x σ x σ σ x σ x ∈ ∈ It follows from (5), (9), (37) that − : (e) (e) is a homomorphism. To complete H → H the proof we need to show that Ae = Ae. Now, by (5), (31), b b `(e) 1 Ae = q AeA Au0 u0− `(e) `(e) 1 1 = q q− AeA Au0 A Au0 u0− u0− = Ae. This completes the proof.

If x, z W , then we see by induction on `(z)that ∈ 1 (38) AxAz Z[q− ]Ayw. ∈ y x wX≤z `(yw)=`≤(y)+`(w) By (3), (5), (38) and Theorem 4.1, we have

Corollary 4.2. Let σ WeW. Then there exist Rθ,σ Z[q], θ WeW,such ∈ ∈ ∈ that, in (e), H `(σ) `(e) (i) Aσ = q − θ WeW Rθ,σAθ, b ∈ (ii) Rθ,σ =0only if θ σ, 6 P ≤ (iii) Rσ,σ =1.

Problem 4.3. Determine the polynomials Rθ,σ explicitly for σ, θ WeW.Does θ σimply R =0? ∈ ≤ θ,σ 6 When W (e)=We, Theorem 4.1 can be simpliﬁed by the use of the relative R-polynomials of Deodhar [6, Proposition 2.12].

1 Corollary 4.4. Suppose W (e)=We =WI,I S.Letσ=set− WeW in standard form. Then s, t D(e) and ⊆ ∈ ∈ `(s) `(t) I I A = q R R A 1 . σ − x,s t,y xey− x,y D(e) X∈ In [12], Kazhdan and Lusztig introduced the now-famous polynomials Px,y Z[q] to deﬁne a basis ∈ (39) C =( 1)`(y)q`(y)/2 ( 1)`(x)P A ,yW, y − − x,y x ∈ x W X∈ 1/2 1/2 of the Z[q ,q− ]-algebra (W ) such that Cy = Cy for all y W . We refer to [10, Chapter 7] for details. H ∈

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Let e Λ, and let w0 and v0 denote the longest element of W and W (e), ∈ 1 respectively. For σ = set− in standard form, let

`(t) `(t)/2 `(yz) (40) C =( 1) q− C ( 1) P A 1 1 . σ − s · − w0yzv0,w0tv0 z− ey− y D(e) z ∈XW (e) ∈

Theorem 4.5. Let e Λ.ThenCσ=Cσfor all σ WeW, and they form a 1/2 1/2 ∈ ∈ Z[q ,q− ]-basis of (e). H 1 Proof. Let σ = set− in standard form. Then it is easy to see that

(`(t) `(s))/2 q − C A σ − σ 1/2 1/2 1 is a Z[q ,q− ]-linear combination of Aθ such that if θ = xey− in standard form, then either y

`(t) C 1 =( 1) A 1 . et− − Pt,y ey− y D(e) ∈X Hence by (27)

`(t) `(y) C 1 =( 1) q− R A 1 . et− − Pt,y · yz,x zex− y D(e) x D(e) ∈X z ∈XW (e) ∈ Let

`(y) (43) = q− R A . Ry,x yz,x x z W(e) ∈X So

`(t) C 1 =( 1) A 1 . et− − Pt,yRy,x ex− x,y D(e) X∈ Thus we need to show that, for all x D(e), ∈ (44) = . Pt,x Pt,yRy,x y D(e) ∈X

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Now by (3), (39), (41),

`(tu0) (`(u0) `(t))/2 `(x) C 1 =( 1) q − ( 1) P A 1 v0t− w0 − − x,w0tv0 x− x W X∈

`(tu0) (`(u0) `(t))/2 `(yu0z) =( 1) q − ( 1) P A 1 A 1 −  − w0yv0z,w0tv0 z−  · v0y− w0 y D(e) z W (e) ∈X ∈X  

`(t) `(u0)/2 `(t)/2 `(yz) =( 1) q q− ( 1) P A 1 A 1 . −  − w0yv0z,w0tv0 z−  · v0y− w0 y D(e) z W (e) ∈X ∈X  

Substituting v0zv0 for z and using (35), (42), we see that

`(t) `(u0)/2 (45) C 1 =( 1) q A A A 1 . v0t− w0 − v0 Pt,y v0 · v0y− w0 y D(e) ∈X

`(t) `(u0)/2 (46) C 1 =( 1) q− Av t,yAv A 1 . v0t− w0 − 0 P 0 · v0y− w0 y D(e) ∈X

By (3), (5), (41),

`(u0) `(y) A 1 = q R A 1 A 1 v0y− w0 − w0xv0z,w0yv0 z− v0x− w0 x D(e) z ∈XW (e) ∈ `(u0) `(y) 1 = q R A A 1 − yv0,xv0z z− v0x− w0 x D(e) z ∈XW(e) ∈ `(u0) `(y) = q R 1 A 1 A 1 , − yv0z− v0,x z− v0x− w0 x D(e) z ∈XW (e) ∈ 1 since `(yv0)=`(yv0z− v0)+`(v0z)

and `(xv0z)=`(x)+`(v0z)

`(u0) `(y) =q R A A 1 − yz,x v0zv0 v0x− w0 x D(e) z∈XW(e) ∈

`(u0) `(y) = q − A R A A A 1 , by (35) v0  yz,x z v0 v0x− w0 x D(e) z W(e) ∈X ∈X  

`(u0) = q A A A 1 , by (43). v0  Ry,x v0 v0x− w0 x D(e) ∈X  

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Hence, by (46),

`(t) `(u0)/2 C 1 =( 1) q A A A 1 v0t− w0 − v0 Pt,yRy,x v0 v0x− w0 x D(e) y∈XD(e) ∈

`(t) `(u0)/2 =( 1) q A A A 1 , − v0  Pt,yRy,x v0 · v0x− w0 x D(e) y D(e) ∈X ∈X   by (35). Since C 1 = C 1 , we see by (45) that (44) is valid. This com- v0t− w0 v0t− w0 pletes the proof. Using (3), (5), (38), we have

Corollary 4.6. Let σ WeW. Then there exist Pθ,σ Z[q], θ WeW, such that in (e) ∈ ∈ ∈ H `(σ) (`(σ) `(e))/2 `(θ) (i) Cσ =( 1) q − θ WeW( 1) P θ,σAθ, − ∈ − (ii) Pθ,σ =0if 0 σ, P (iii) Pσ,σ =1. Problem 4.7. Determine P explicitly for θ, σ WeW.IsP =0forθ σ? θ,σ ∈ θ,σ 6 ≤ Remark 4.8. Renner [26] has posed the problem of ﬁnding the correct polynomials P for θ, σ R. Thus Corollary 4.6 provides a partial solution. θ,σ ∈ In the special case W (e)=We the polynomials Pθ,σ are related to the relative I Kazhdan-Lusztig polynomials Px,y studied by Deodhar [6], [8] and Douglass [9]. 1 1 Corollary 4.9. Let W (e)=We =WI,I S, and let σ = set− ,θ = xey− e ⊆ I I ∈ WeW in standard form. Then s, t, x, y D(e) and Pθ,σ = P P . ∈ x,s w0yv0,w0tv0 Combining Theorems 3.3, 4.1 and 4.5, we have e e Theorem 4.10. For σ WeW, let Aσ,Cσ (R)be given by (27), (40) respectively. Then: ∈ ∈H

(i) The map eA˜ σ e˜Aσ yields an involution on (R) extending the involution on (W ). → H H (ii) If Cσ =˜eCσ,thenCσ is invariant under the involution in (i),andCσ (σ R) 1/2 1/2 ∈ forms a Z[q ,q− ]-basis of (R). H e e e References

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Department of Mathematics, North Carolina State University, Raleigh, North Car- olina 27695-8205 E-mail address: [email protected]

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