The Kazhdan-Lusztig basis

Last update: 26 September 2012

The Kazhdan-Lusztig basis

The algebra $\stackrel{\sim }{H}$ has bases

${ xλTw∣ w∈W,λ∈P } and { Twxλ∣ w∈W,λ∈P } .$

The Kazhdan-Lusztig basis $\left\{{C}_{w}^{\prime }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w\in \stackrel{\sim }{W}\right\}$ is another basis of $\stackrel{\sim }{H}$ which plays an important role. It is defined as follows.

The bar involution on $\stackrel{\sim }{H}$ is the $ℤ$-linear automorphism $\stackrel{‾}{\phantom{H}}:\phantom{\rule{0.2em}{0ex}}\stackrel{\sim }{H}\to \stackrel{\sim }{H}$ given by

$q‾=q-1and Tw‾= Tw-1-1, forw∈W∼.$

For $0\le i\le n,$ $\stackrel{‾}{{T}_{i}}={T}_{i}^{-1}={T}_{i}-\left(q-{q}^{-1}\right)$ and the bar involution is a $ℤ$-algebra automorphism of $\stackrel{\sim }{H}\text{.}$ If $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}$ is a reduced word for $w$ then, by the definition of the Bruhat order (defined after Lemma 1.2),

$Tw‾ = Ti1…Tip‾ =Ti1‾… Tip‾= Ti1-1… Tip-1 = (Ti1-(q-q-1))… (Tip-(q-q-1))= Tw+∑v

with ${a}_{vw}\in ℤ\left[\left(q-{q}^{-1}\right)\right]\text{.}$

Setting ${\tau }_{i}=q{T}_{i}$ and $t={q}^{2},$ the second relation in (1.21)

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

Let ${\tau }_{w}={q}^{\ell \left(w\right)}{T}_{w}$ for $w\in \stackrel{\sim }{W}\text{.}$ The Kazhdan-Lusztig basis $\left\{{C}_{w}^{\prime }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\stackrel{\sim }{w}\in \stackrel{\sim }{W}\right\}$ of $\stackrel{\sim }{H}$ is defined [Kal1979] by

$C‾w′= Cw′and Cw′= t-ℓ(w)/2 ( ∑y≤w Pywτy ) , (1.24)$

subject to ${P}_{yw}\in ℤ\left[{t}^{\frac{1}{2}},{t}^{-\frac{1}{2}}\right],$ ${P}_{ww}=1,$ and ${\text{deg}}_{t}\phantom{\rule{0.2em}{0ex}}\left({P}_{yw}\right)\le \frac{1}{2}\left(\ell \left(w\right)-\ell \left(y\right)-1\right)\text{.}$ If

$pyw= q - ( ℓ(w)- ℓ(y) ) Pyw (1.25)$

then

$Cw′= q-ℓ(w) ∑y≤w Pyw qℓ(y)Ty= ∑y≤wPyw q-(ℓ(w)-ℓ(y)) Ty= ∑y≤wpyw Ty, (1.26)$

with

$pyw∈ℤ [q,q-1], pww=1,and pyw∈q-1 ℤ[q-1], (1.27)$

since ${\text{deg}}_{q}\phantom{\rule{0.2em}{0ex}}\left({P}_{yw}\left(q\right){q}^{-\left(\ell \left(w\right)-\ell \left(y\right)\right)}\right)\le \ell \left(w\right)-\ell \left(y\right)-1-\left(\ell \left(w\right)-\ell \left(y\right)\right)=-1\text{.}$ The following proposition establishes the existence and uniqueness of the ${C}_{w}^{\prime }$ and the ${p}_{yw}\text{.}$

Let $\left(\stackrel{\sim }{W},\le \right)$ be a partially ordered set such that for any $u,v\in W$ the interval $\left[u,v\right]=\left\{z\in W\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}u\le z\le v\right\}$ is finite. Let $M$ be a free $ℤ\left[q,{q}^{-1}\right]$-module with basis $\left\{{T}_{w}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w\in \stackrel{\sim }{W}\right\}$ and with a $ℤ$-linear involution $\stackrel{‾}{\phantom{H}}:\phantom{\rule{0.2em}{0ex}}M\to M$ such that

$q‾=q-1and Tw‾=Tw+ ∑v

Then there is a unique basis $\left\{{C}_{w}^{\prime }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w\in \stackrel{\sim }{W}\right\}$ of $M$ such that

1. $\stackrel{‾}{{C}_{w}^{\prime }}={C}_{w}^{\prime },$
2. ${C}_{w}^{\prime }={T}_{w}+\sum _{v

 Proof. The ${p}_{vw}$ are determined by induction as follows. Fix $v,w\in W$ with $v\le w\text{.}$ If $v=w$ then ${p}_{vw}={p}_{ww}=1\text{.}$ For the induction step assume that $v and that ${p}_{zw}$ are known for all $v The matrices $A=\left({a}_{vw}\right)$ and $P=\left({p}_{vw}\right)$ are upper triangular with 1's on the diagonal. The equations $Tw=Tw‾‾ = ∑v avwTv‾= ∑u,vauv avw‾Tuand ∑upuwTu= Cw′ = Cw′‾=∑v pvwTv‾= ∑u,vpvw‾ auvTu,$ imply $A\stackrel{‾}{A}=\text{Id}$ and $P=A\stackrel{‾}{P}\text{.}$ Then $f=∑u is a known element of $ℤ\left[q,{q}^{-1}\right];$ $f=∑k∈ℤfkqk such thatf‾= (puw-p‾uw)‾ =p‾uw-puw=-f.$ Hence ${f}_{k}=-{f}_{-k}$ for all $k\in ℤ$ and ${p}_{uw}$ is given by ${p}_{uw}=\sum _{k\in {ℤ}_{<0}}{f}_{k}{q}^{k}\text{.}$ $\square$

The finite Hecke algebra $H$ and the group algebra of $P$ are the subalgebras of $\stackrel{\sim }{H}$ given, respectively, by

$H = (subalgebra ofH∼ generated byT1,…,Tn ), (1.28) 𝕂[P] = 𝕂-span {xλ∣λ∈P}, where𝕂=ℤ[q,q-1],$

and $𝕂\text{-span}\phantom{\rule{0.2em}{0ex}}\left\{{x}^{\lambda }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\lambda \in P\right\}$ denotes the set of $𝕂$-linear combinations of elements ${x}^{\lambda }$ in $\stackrel{\sim }{H}\text{.}$ The Weyl group $W$ acts on $𝕂\left[P\right]$ by

$wf=∑μ∈P cμxwμ, forw∈Wand f=∑μ∈Pcμ xμ∈𝕂[P]. (1.29)$

Acknowledgements

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).