The Kazhdan-Lusztig basis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 26 September 2012

The Kazhdan-Lusztig basis

The algebra H has bases

{ xλTw wW,λP } and { Twxλ wW,λP } .

The Kazhdan-Lusztig basis {CwwW} is another basis of H which plays an important role. It is defined as follows.

The bar involution on H is the -linear automorphism H:HH given by

q=q-1and Tw= Tw-1-1, forwW.

For 0in, Ti= Ti-1=Ti- (q-q-1) and the bar involution is a -algebra automorphism of H. If w=si1sip is a reduced word for w then, by the definition of the Bruhat order (defined after Lemma 1.2),

Tw = Ti1Tip =Ti1 Tip= Ti1-1 Tip-1 = (Ti1-(q-q-1)) (Tip-(q-q-1))= Tw+v<wavwTv,

with avw [(q-q-1)].

Setting τi=qTi and t=q2, the second relation in (1.21)

Ti2= (q-q-1)Ti +1τi2= (t-1)τi+t. (1.23)

Let τw=q(w)Tw for wW. The Kazhdan-Lusztig basis {CwwW} of H is defined [Kal1979] by

Cw= Cwand Cw= t-(w)/2 ( yw Pywτy ) , (1.24)

subject to Pyw [ t12, t-12 ] , Pww=1, and degt(Pyw) 12((w)-(y)-1). If

pyw= q - ( (w)- (y) ) Pyw (1.25)


Cw= q-(w) yw Pyw q(y)Ty= ywPyw q-((w)-(y)) Ty= ywpyw Ty, (1.26)


pyw [q,q-1], pww=1,and pywq-1 [q-1], (1.27)

since degq ( Pyw(q) q-((w)-(y)) ) (w)-(y) -1- ((w)-(y)) =-1. The following proposition establishes the existence and uniqueness of the Cw and the pyw.

Let (W,) be a partially ordered set such that for any u,vW the interval [u,v]= { zWuzv } is finite. Let M be a free [q,q-1]-module with basis { Tww W } and with a -linear involution H:MM such that

q=q-1and Tw=Tw+ v<wavw Tv.

Then there is a unique basis {CwwW} of M such that

  1. Cw= Cw,
  2. Cw=Tw+ v<w pvw Tv,with pvwq-1 [q-1] forv<w.


The pvw are determined by induction as follows. Fix v,wW with vw. If v=w then pvw=pww=1. For the induction step assume that v<w and that pzw are known for all v<zw.

The matrices A=(avw) and P=(pvw) are upper triangular with 1's on the diagonal. The equations

Tw=Tw = v avwTv= u,vauv avwTuand upuwTu= Cw = Cw=v pvwTv= u,vpvw auvTu,

imply AA=Id and P=AP. Then

f=u<zw auzpzw= ((A-1)P)uw =(AP-P)uw =(P-P)uw= puw-puw,

is a known element of [q,q-1];

f=kfkqk such thatf= (puw-puw) =puw-puw=-f.

Hence fk=-f-k for all k and puw is given by puw= k<0 fkqk.

The finite Hecke algebra H and the group algebra of P are the subalgebras of H given, respectively, by

H = (subalgebra ofH generated byT1,,Tn ), (1.28) 𝕂[P] = 𝕂-span {xλλP}, where𝕂=[q,q-1],

and 𝕂-span{xλλP} denotes the set of 𝕂-linear combinations of elements xλ in H. The Weyl group W acts on 𝕂[P] by

wf=μP cμxwμ, forwWand f=μPcμ xμ𝕂[P]. (1.29)


The research of A. Ram was partially supported by the National Science Foundation (DMS-0097977), the National Security Agency (MDA904-01-1-0032) and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. The research of K. Nelsen was partially supported by the National Science Foundation (DMS-0097977 and a VIGRE grant) and the National Security Agency (MDA904-01-1-0032).

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