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An inversion formula for relativekazhdan—lusztig polynomialsJ. Matthew Douglass aa Department of Mathematics , University of North Texas , Denton, TexasPublished online: 27 Jun 2007.

To cite this article: J. Matthew Douglass (1990) An inversion formula for relative kazhdan—lusztig polynomials,Communications in Algebra, 18:2, 371-387, DOI: 10.1080/00927879008823919

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COMMUNICATIONS I N A L G E B R A , 18 (2) , 371-387 (1990)

AN INVERSION FORMULA

FOR RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS

J. Matthew Douglass

Department of Mathematics University of North Texas

Denton, Texas

51: Introduction

Let (W, S ) be a Coxeter system, and let 7f be the Hecke algebra of (W,S)

defined over the ring A = Q[u' /~ , u-lI2], where u is an indeterminate. Let

e( . ) be the length function on W, and let 5 be the Bruhat order. Then

?f is free as an A-module, with 'standardn basis, ( Tw 1 w E W }, and

multiplication determined by

for s f S and w E W. Kazhdan and Lusztig construct a family of polyno-

mials, { Pw,z I w, x E W } (see [ 5 ] ) , so that the formulas

for x E W, define a basis of 'H that gives rise to certain modules for 7f,

Copyright @ 1990 by Marcel Dekker, Inc.

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37 2 DOUGLASS

known as "left cell" modules (see 52). Deodhar generalizes these construc-

tions in two parallel theories (see 141) to get various modules for 'H and

"relative" Kazhdan-Lusztig polynomials.

Assume that (W, S) has the property that for every proper subset, I,

of S, the subgroup WI = (I) is finite. This will be the case, for example, if

W is a finite or affine Weyl group.

With the preceding assumption, we will describe a uniform approach

to the parallel constructions in [4] and relate these modules to the left cell

modules of 'H. When W is finite we also prove an inversion formula that

relates the two versions of the relative Kazhdan-Lusztig polynomials. In 52

we construct %-modules and show the relations with left cell modules. In

53, we construct the "R" polynomials, which are used to construct the rel-

ative Kazhdan-Lusztig polynomials. We show that all polynomials are in

fact the polynomials in [4], and we also describe some relations between the

two versions of these "R" polynomials. Finally, in 54 we prove the inver-

sion formula, (4.6), which shows that the inverse of the matrix of one ver-

sion of the relative Kazhdan-Lusztig polynomials is, in a sense, the matrix

of the other version of the relative Kazhdan-Lusztig polynomials.

The author would like to thank the referee for several useful com-

ments, in particular for suggesting an improved version of the proof of The-

orem 2.6.

52: 'H-modules

We begin with a brief summary of the relevant results and notation from

Kazhdan and Lusztig [5], and Lusztig [6].

For w E W, put L(w) = {s E S I sw < w ), and define R(w) to be

~ ( w - l ) . We will write el: for (-1)'(~), and uz for ue(').

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RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS 373

Let w, x E W and suppose w < x. We will denote the coefficient

of u(e(z)-e(w)-1)/2 in Pw,, by j~ (w,x ) and define p(s, w ) to be p(w, s ) . We

further define p(w, x) to be 0 when w $ x or s $ w.

By [6, 5.1.121, the multiplication in the C, basis is determined by

Let be the preorder on W determined by the elementary relation

y < L , s x if and only if Cy appears in C,C,, for x, y E W and s E S. Let

--L be the equivalence relation on W determined by the preorder IL. The

equivalence classes for -L are called left cells. -

Define u1f2 to be and for w E W, define to be ~ ~ 1 1 . -

Then extends to an involution of 7i. By (5, Theorem 1.11, = Cz for

all x E W.

Let be the Q-algebra autornorphisrn of 7i satisfying @ ( u ' / ~ ) = -

-u1l2 and @(Tw) = c ~ u ~ T ~ , for w E W. Then @ is an involution of 'H and

@ commutes with -. As in (6, 5.1.15; 5.121, let CL = iP(C,) for x E W. By the preceding -

paragraph, CL = CL and

Now, let I be a subset of S. Denote by WI the subgroup of W gen-

erated by { s I s E I }, and by wl the longest element in WI. We will use

the letters a , T, and p for left WI cosets.

For the remainder of this paper, we fix a subset, I, of S. .

Let a be a left WI coset. It is well known that u contains a unique

minimal element in the Bruhat order, say x, and it is shown by Curtis [l]

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374 DOUGLASS

that a also contains a unique maximal element, namely XWI. We will define

C(U) = [(x) + e(w1), E, = (-I)+), arid uU = t l e ( ~ ) .

Let X- be the set of all minimal WI coset representatives, and let

X+ be the set of a11 maximal WI coset representatives. Recall that X- can

be characterized as the set of all x in W satisfying

L(xw) = L(x) + L(w) for all w E WI.

It follows from [ l , Theorem 1.21 that X+ can be characterized as the set of

all z E W satisfying

l(.zw) = e(z) - P(w) for all w E WI.

The Bruhat order on W restricts to a partial order on W/WI a s fol-

lows: Let a and T be in W/WI. Then u 5 7 if and only if there exists

y E a and z E T with y 5 z. That this does indeed define a partial order on

W/WI follows by taking J = B in the following general lemma:

(2.2) Lemma. Let I, J C S and let Dl and D2 be (Wj , WI) double

cosets. Suppose that xi (i = 1,2) is the unique minimal element in D;, and

that ri (a' = 1,2) is the unique maximal element in D; (as in [I, Theorem

1.21). Then the following are equivalent:

(a) There exists yl E Dl and y2 E D2 with yl < y2.

(b) X I 5 x2.

(c) Z l 5 22.

Proof. Clearly, (b) implies (a), and (c) implies (a).

We next prove that (a) implies (b). It follows from the proof of [I,

Theorem 1.21 that we can write y2 = w'x2w", where C(y2) = L(wl) + C(x2) + 4(wU), w' E W j , and w'' E WI. Let s E J and suppose sy2 < y2. Then by

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RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS 375

Property "Zn of Deodhar [3], either yl 5 sy2 or syl < s ~ 2 . Hence we can

multiply on the left by elements of J to get vtyl x2w", for some v' E W j .

Similarly, we can multiply on the right by elements of I to get vtylv" 5 22,

for some v'' E W I . Then ~ ' ~ ~ v ' ' E D l , so $1 _< vtylv". Therefore, x l 5 22.

Finally, we will prove that (a) implies (c). Let w E W, and suppose

that w < 22. Let s E J and suppose sw > w. Again, by Property "2" in

[3], we have sw 5 22. Similarly, s E I and ws > w imply ws < 22. By

assumption, yl 5 z2, and the preceding shows that v'ylv" 5 22 whenever

v' E W j and v" E WI. Hence 21 < 22. This completes the proof of (2.2).

Let a E { - I , u }. For s E S , define E,Q to be the a-eigenspace of

right multiplication by Ts on "H. Then E,Q = { h E 7i ( hTs = a h ). Recall

that we have fixed a subset, I, of S. Define Ea to be nsEIE,O. Thus, Ea is

a left ideal in 'If.

For a subset X & W, write TX for CzEX Tz. For example, we have

TWI = CwEWI Tw. Define 'HI to be the left 'If-module, 'IfTwI. We can

now start to make some sense of ail this notation.

(2.3) Proposit ion. Let s E S, and let a E W/WI. Then

uTsu + ( U - l)Tu if s u < u

i f s a > u

if su = u .

Proof. It follows from [4, Lemma 2.11 that

(i) s u < u if and only if sx < x, in which case s z E X - ;

(ii) so > a if and only if sx > x and sx E X - ;

(iii) sa = o if and only if s x > z and 3s 4 X - .

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Suppose sa < a. Then the argument used by Curtis, Iwahori, and

Kilmoyer in the proof of [2, Lemma 2.41 gives the desired formula for TsTu.

Next, suppose that sa > a. Then it follows easily from (ii) and (1.1)

that TsTu = Tsu.

Finally, suppose that sa = a. Then by [4, Lemma 2.11, there is an

sl E I with sx = xsl. Also, by [2, Lemma 2.41, TslTWI = uTWI. There-

fore,

TsTo = TsTzTwI = TzTslTwI = uTZTwI = u T ~ .

This completes the proof of (2.3)

(2.4) Proposit ion.

(a) EUhasA-bases, { T , I U E w/WI) and{C: I z E X + ) .

(b) EU = 'HI.

(c) With the obvious notation, = 311.

Proof. (a) and (b) are a restatement of [I, Lemma 1.9 and Theorem 1.101,

where (I, J) is replaced by (0, I).

Since a = CL for all s E W, (c) follows directly from (a). This

completes the proof of (2.4).

(2.5) Proposit ion.

(a) @(Eu) = E-' .

(b) E-' has bases{F, l a c W/WI) and {C, J z E x ' ) .

(c) -= E-l.

Proof. It follows from the definitions that @(E-l) = EU and @(Fu) =

E , u , ~ . Combining these with (2.4a) and the fact that @(CI) = C, for

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RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS 377

z E W proves (a) and (b). Finally, (c) follows from (a), (2.4b), and (2 .4~) .

This completes the proof of (2.5).

For o E W/ WI, define Fu to be euuu EYE, E ~ ~ ~ - ~ T ~ . The Tu1s play

the role of the Tu7s when u is replaced by -1.

(2.6) Proposition. Let s E S and a E W/WI. Then

Proof. It is easily checked that @(Tc) = E ~ u ~ ~ , and hence @(TsT,) =

- u - ~ E ~ u ; ~ T ~ T ~ . Therefore, applying (2.3) we have

- Now (2.6) follows by applying and then cP to both sides of (2.7) and us-

ing the fact that @ ( E u u ~ ~ ) = Fu.

Combining (2.3) through (2.6) with [4, Corollary 2.31, we see that

EU is the 'H-module, M I , of [4], with "u = q," and E - I is the 3-I-module,

M I , of [4], with "u = -1."

Let r be a left cell in W, and choose y in r. Define 7fr to be the A-

submodule of 3.1 spanned by the C, with xSLy. Notice that the definition

of Er does not depend on the choice of y. It follows from the definition of

Sr, that 'Hr is actually an 'H-submodule. Let 'H', be the A-submodule of 3-1

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378 DOUGLASS

spanned by the Cz with ILLY and x f L y. Then 3.1; is an 'H-submodule

of Hr. Define Mr to be the quotient 'Hr/Hb. Thus, Mr is an 'H-module,

called a left cell module.

By [5, Proposition 2.41, x s ~ y implies R(y) s R(x). In particular,

for a left cell I?, we will define R(F) to be the common value of R(x), for

x E r. Recall that EU = ' H I . Let J be another subset of S, with I J .

Then it follows from (2.4a) that H j C HI. Define to be CIcJ H j , and

let I< = e(u1I2) be the quotient field of A. Then by what has been shown,

it follows that each 3CI/'H; is a "sum" of left cell modules. Precisely,

(sum over left cells, I', with R(F) = I ) .

§3: "R " polynomials

Recall that in [5], polynomials { Rz,y 1 x , y E W } are defined by the formu-

las

Let CY E { -1, u }, and let T, u E W/ WI. Then by (2.4) and (2.6),

EQ = Ea, EU has basis {Tb I u E W/WI ), and E-' has basis {To (

a E W/WI }. Hence we may define elements Rs,o and ET,, in A by the

formulas

- - T o = U;'GT, and PC= u;'&Tr. (3.2)

rEW/W1 TE WI W1

Notice that if I = 0, then the RT,,,'s reduce to the Rt,y's.

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RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS

(3.3) Proposition. Let ~ , a E W/WI, and let y E T. Then,

(a) RT,, = uyl C uyRy,Z, and Z E U

(b) %,u = ~ T C U E ~ U U C C Z U ~ ~ R ~ , ~ . Z E U

Proof. It follows from (3.1) that

Now, comparing the coefficient of Ty in (3.2) and (3.4) proves (a).

To prove (b), we compute

Also,

Comparing coefficients of Ty in (3.5) and (3.6) proves (b). This completes

the proof of (3.3).

It will be a consequence of the following proposition that the RTp's

and the ?Zr,a7~ are the polynomials, R!,,, of [4].

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380 DOUGLASS

(3.7) Proposition. Let a E W/WI with u # WI, and choose T E W/WI

with T 5 a. Let s E S with s o < a . - -

(a) I ~ S T < T, then RT,, = RsT,sQ and RT,, = Rsr,sa -

(b) I f s ~ > T, then RT,, = uRST,Su + (u - l)RT,so and RT,, = -

uRsr,su + (21 - l)?ZT,sa - -

(c) I f s ~ = T, then Rr,, = uRr,sg and Rr,,, = -Rr,sa

Proof. The proof of (3.7) (for both the RTIu9s and the %,,'s) is the ap-

propriate modification of the proof of [4, Lemma 2.8(iv)]. We will omit the

details.

(3.8) Corollary. Let T,U E W/ Wr.

(a) Rr,, = 0 if T $ a , and Rut, = 1. If T 5 a , then RT,, is in

C[u], has degree [(a) - [(T), and has leading coefficient 1.

(b) RT,, = 0 if T $ a, and Ro,o = 1. If T 5 a , then g,g is in

C[u], has degree at most [(a) - [(T), and has constant term

+%. - -1- (c) RT,U = + + U O U ~ RT,,.

Remark. Notice that (c) relates the Rr,,'s and the ET,,'s and generalizes

[5, Lemma 2.l(i)].

Proof. (a) and (b) follow directly from (3.3) and the corresponding result

for the R z , y ' ~ (see [5, $2)).

We now prove (c). Put R:,, = e,eTu,u~~Rr,a. Then by (a), R:,, E

C[u] and R',,, = 1. Because the recursion formulas in (3.7) determine

the %,,'s uniquely, it suffices to show that the R:,, satisfy these recur-

sion formulas also. A straightforward computation using the definition of

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RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS 381

the R:,,'s shows that this is indeed the case. We will omit the details. This

completes the proof of (3.8).

It follows from (3.7), (3.8), and [4, Lemma 2.81 that the polynomials,

R,,,, ( T , U E W/WI), are the polynomials, R!,,, in [4] corresponding to - the choice "u = -1." Also, the polynomials, RT,, (T, a E W/ WI), are the

polynomials, R!,,, in [4] corresponding to the choice "u = q."

Now, as indicated in [4, Proposition 3.11, the proof of [5, Theorem

1.11 can be modified to construct (with the obvious notational conventions)

polynomials, Pr,, and g,, (T, Q E W/WI), which are uniquely determined

by the following properties:

- - (a) PT,,, = P,,, = 0, unless T < u, and P,,, = P,,, = 1.

(b) If 7 I 0, then deg P,,, I ( l ( a ) - l ( ~ ) - 1)/2,

and deg F,,, 5 ([(a) - [(T) - 1)/2. (3.9)

As with the RT,,'s and the Xr,,'s, the PT,,,'s are the P;,,'S of [4]

corresponding to the choice " u = -1," and the FT,,'s are the P;,,'s of [4] -

corresponding to the choice Uu = 9." The Pr,,'s and the P , , ' s are known

as relative Kazhdan-Lusztig polynomials.

Let r , a E W/WI, and let a = xWI and T = yWI, where x,y E X - .

Then it is shown in [4, Proposition 3-41 and [4, Remark 3.81 that P,,, and

PT,, are related to the usual Kazhdan-Lusztig polynomials by

(*) PT,, = PYwI,zwI (recall that wl is the longest element in

WI ), and

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382 DOUGLASS

(**) PT,U = E fwPyw, z . WE w1

Since no proof of [4, Remark 3.81 is given, we mention in passing that (**)

is an easy consequence of [5, Theorem 3.1) and (4 .6 ) .

54: The inversion formula

Throughout this section assume that W is finite. We will prove an inver-

sion formula (Theorem [4 .6 ] ) , which states roughly that the matrix of the - PT,,'s ( 7 , a E W/WI) is the inverse of the matrix of the P,,,'s.

Let wo be the longest element in W . In order to state our results,

we need a relative version of the poset anti-automorphism, x I+ wox, of W .

Let a and r be in W/WI. Then woa is also in W/WI. It follows

from (2.2) that o 5 r if and only if WOT 5 WOO. Hence a I+ WOO defines

a poset anti-automorphism of order two of W/WI. Let o = xWI, where

x E X - . It is easily shown that w o x w ~ E X- n woo.

(4 .1 ) Proposition. Let a E W/WI, let s E S, and put so = woswo.

(a) If so < a, then sowoo > woa.

(b) If sa > a , then sowoo < woo.

(c) If so = u , then sowoa = woo.

Proof. These results follow directly from the definitions. We will omit fur-

ther details.

(4.2) Lemma. Let a , ~ E W/WI, and let s E S with so = a and ss > 7. Then RT,, = uR,,,,.

Proof. By (2.3), TsT, = uTu, SO

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RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS 383

It follows from (3.2) that the coeffecient of T, on the right hand side of

(4.3) is u - l u ~ ~ R r , a . Since T;' = (u-I - l)T1 + U - ~ T ~ , it follows from

(3.2) and (2.3) that the coeffecient of T, on the left hand side of (4.3) is

By assumption, ST > T. Let y E T n X-. Then as noted in the proof of

(2.3), sy > y and sy E ST n X-. Hence us, = UUT, and (4.2) follows by

comparing the coeffecient of TT in both sides of (4.3).

We can now prove the "relativen version of [5, Lemma 2.l(iv)], for

the Rs,a 's.

(4.4) Proposition. Let T,U E W/WI. Then RT,# = Rwoa,wor.

Proof. It suffices to prove the result when T 5 a. We will do this using

induction on l (u) .

If [(a) = l(wI), then a = T = WI, and so (4.4) follows from (3.8).

Assume that t ( a ) > t (wr) , and choose s E S with s o < a. Then by

(2.2), [(so) < [(a). Let T E W/WI with T < u, and put so = woswo. There

are three cases:

Case 1. Suppose that ST < T. Then

Rr,u = Rss,sa (by [3.7aI)

= ~ w o s a , w o s ~ (by induction)

= RsoWO~,SOwo~

= Rwoo,wos

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Case 2. Suppose that ST > T . Then

R , o = uRsr,so + (u - l)Rr,so (by [3.7bl)

= u R w o ~ o , w O ~ ~ + (u - l ) R ~ O ~ o , W O ~ (by induction)

= ~ R ~ O W O o , ~ O ~ O ~ + ( u - ~ ) R S ~ W ~ ~ , W ~ T

= Rwou,wor

Case 3. Suppose that ST = T . Then

Rr,u = uRr,so

= URW,,SU,W~~

= uRsOwOo,wO~

= Rw0u,w0r

This completes the proof of (4.4).

(by 13.7~1)

(by induction)

(4.5) Lemma. Let p , r E W/ WI with p < T . Then

Proof. This result follows directly from (3.9~) and (3.8~). We will omit fur-

ther details.

We can now state the promised inversion formula.

(4.6) Theorem. Let p , a E W/WI with p 5 o. Then

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RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS 385

Proof. The proof we give is the appropriate modification of the proof of [5,

Theorem 3.11.

Since p 5 T 5 u if and only if woo 5 wor 5 wop, it suffices to prove

6p,u = C t p t r ~ p , r ~ w ~ a , ~ r . (4.7) PITSO

We will prove (4.7) using induction on B(a) - [(p).

If t (u) = l (p), then (4.7) follows from (3.9a).

Assume that t(p) < [(a), and let a, p E W/WI with a < /3. Define

Ma,p to be the right hand side of (4.7) with (p, a ) replaced by ( a , P) . By

induction, we may assume that ma,^ = ba,B whenever

We will show that Mp,a = 0, and hence that (4.7) holds.

Since u&uwoT = uUu;l, we may conclude from (4.8), (4.5), and the defi-

nition of that

Hence, by induction,

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By [4, Lemma 2.8(iv)],

DOUGLASS

- Therefore, M p p = U ~ ' U ~ M ~ , ~ . Hence,

It follows from the definition of Mp,, and (3.9) that u ~ ~ 2 u ; 1 1 2 ~ p , u

is in c [ u 1 i 2 ] and has no constant term. Therefore, (4.9) implies that

Thus M p , U = 0, as claimed. This completes the proof of (4.6).

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C. CURTIS, On Lusztig's isomorphism theorem for Hecke algebras,

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RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS 387

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Ann. of Math. Studies, No. 107, Princeton University Press, Prince-

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Received: April 1988 Revised: September 1989

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