The Kazhdan-Lusztig basis

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 26 September 2012

The Kazhdan-Lusztig basis

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

$$\{x^{\lambda }{T}_w\mid w\in W,\lambda \in P\}\text{and}\{T_w{x}^{\lambda }\mid w\in W,\lambda \in P\}\text{.}$$

The Kazhdan-Lusztig basis $\{C_w^{\prime }\mid w\in \stackrel{\sim }{W}\}$ 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 $\mathbb{Z}$-linear automorphism $\bar{\phantom{H}}:\stackrel{\sim }{H}\to \stackrel{\sim }{H}$ given by

$$\bar{q}=q^{-1}\text{and}\overline{T_w}=T_{w^{-1}}^{-1},\text{for}w\in \stackrel{\sim }{W}\text{.}$$

For $0\le i\le n,$ $\overline{T_i}=T_i^{-1}=T_i-(q-q^{-1})$ and the bar involution is a $\mathbb{Z}$-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),

$$\begin{array}{ccc}\overline{T_w}& =& \overline{T_{i_1}\dots T_{i_p}}=\overline{T_{i_1}}\dots \overline{T_{i_p}}=T_{i_1}^{-1}\dots T_{i_p}^{-1}\\ & =& (T_{i_1}-(q-q^{-1}))\dots (T_{i_p}-(q-q^{-1}))=T_w+\sum _{v<w}{a}_{vw}{T}_v,\end{array}$$

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

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

$$\begin{array}{cc}{T}_i^2=(q-q^{-1})T_i+1{\tau }_i^2=(t-1){\tau }_i+t\text{.}& \text{(1.23)}\end{array}$$

Let ${\tau }_w=q^{\ell \left(w\right)}{T}_w$ for $w\in \stackrel{\sim }{W}\text{.}$ The Kazhdan-Lusztig basis $\{C_w^{\prime }\mid \stackrel{\sim }{w}\in \stackrel{\sim }{W}\}$ of $\stackrel{\sim }{H}$ is defined [Kal1979] by

$$\begin{array}{cc}{\bar{C}}_w^{\prime }=C_w^{\prime }\text{and}{C}_w^{\prime }=t^{-\ell \left(w\right)/2}\left(\sum _{y\le w}{P}_{yw}{\tau }_y\right),& \text{(1.24)}\end{array}$$

subject to $P_{yw}\in \mathbb{Z}[t^{\frac{1}{2}},t^{-\frac{1}{2}}],$ $P_{ww}=1,$ and ${\text{deg}}_t\left(P_{yw}\right)\le \frac{1}{2}(\ell \left(w\right)-\ell \left(y\right)-1)\text{.}$ If

$$\begin{array}{cc}{p}_{yw}=q^{-(\ell \left(w\right)-\ell \left(y\right))}{P}_{yw}& \text{(1.25)}\end{array}$$

then

$$\begin{array}{cc}{C}_w^{\prime }=q^{-\ell \left(w\right)}\sum _{y\le w}{P}_{yw}{q}^{\ell \left(y\right)}{T}_y=\sum _{y\le w}{P}_{yw}{q}^{-(\ell \left(w\right)-\ell \left(y\right))}{T}_y=\sum _{y\le w}{p}_{yw}{T}_y,& \text{(1.26)}\end{array}$$

with

$$\begin{array}{cc}{p}_{yw}\in \mathbb{Z}[q,q^{-1}],p_{ww}=1,\text{and}{p}_{yw}\in q^{-1}\mathbb{Z}\left[q^{-1}\right],& \text{(1.27)}\end{array}$$

since ${\text{deg}}_q\left(P_{yw}\left(q\right)q^{-(\ell \left(w\right)-\ell \left(y\right))}\right)\le \ell \left(w\right)-\ell \left(y\right)-1-(\ell \left(w\right)-\ell \left(y\right))=-1\text{.}$ The following proposition establishes the existence and uniqueness of the $C_w^{\prime }$ and the $p_{yw}\text{.}$

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

$$\bar{q}=q^{-1}\text{and}\overline{T_w}=T_w+\sum _{v<w}{a}_{vw}{T}_v\text{.}$$

Then there is a unique basis $\{C_w^{\prime }\mid w\in \stackrel{\sim }{W}\}$ of $M$ such that

1. $\overline{C_w^{\prime }}=C_w^{\prime },$
2. $C_w^{\prime }=T_w+{\displaystyle \sum _{v<w}}{p}_{vw}{T}_v,\text{with}{p}_{vw}\in q^{-1}\mathbb{Z}\left[q^{-1}\right]\text{for}v<w\text{.}$

		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 $v<w$ and that $p_{zw}$ are known for all $v<z\le w\text{.}$ The matrices $A=\left(a_{vw}\right)$ and $P=\left(p_{vw}\right)$ are upper triangular with 1's on the diagonal. The equations $$\begin{array}{ccc}{T}_w=\stackrel{\overline{‾}}{T_w}& =& \sum _v\overline{a_{vw}{T}_v}=\sum _{u,v}{a}_{uv}\overline{a_{vw}}{T}_u\text{and}\\ \sum _u{p}_{uw}{T}_u=C_w^{\prime }& =& \overline{C_w^{\prime }}=\sum _v\overline{p_{vw}{T}_v}=\sum _{u,v}\overline{p_{vw}}{a}_{uv}{T}_u,\end{array}$$ imply $A\bar{A}=\text{Id}$ and $P=A\bar{P}\text{.}$ Then $$f=\sum _{u<z\le w}{a}_{uz}{\bar{p}}_{zw}={\left((A-1)\bar{P}\right)}_{uw}={(A\bar{P}-\bar{P})}_{uw}={(P-\bar{P})}_{uw}=p_{uw}-{\bar{p}}_{uw},$$ is a known element of $\mathbb{Z}[q,q^{-1}];$ $$f=\sum _{k\in \mathbb{Z}}{f}_k{q}^k\text{such that}\bar{f}=\overline{(p_{uw}-{\stackrel{‾}{p}}_{uw})}={\bar{p}}_{uw}-p_{uw}=-f\text{.}$$ Hence $f_k=-f_{-k}$ for all $k\in \mathbb{Z}$ and $p_{uw}$ is given by $p_{uw}={\displaystyle \sum _{k\in {\mathbb{Z}}_{<0}}}{f}_k{q}^k\text{.}$ $\square$

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

$$\begin{array}{cccc}H& =& \text{(subalgebra of}\stackrel{\sim }{H}\text{generated by}{T}_1,\dots ,T_n\text{),}& \text{(1.28)}\\ 𝕂\left[P\right]& =& 𝕂\text{-span}\{x^{\lambda }\mid \lambda \in P\},\text{where}𝕂=\mathbb{Z}[q,q^{-1}],\end{array}$$

and $𝕂\text{-span}\{x^{\lambda }\mid \lambda \in P\}$ 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

$$\begin{array}{cc}wf=\sum _{\mu \in P}{c}_{\mu }{x}^{w\mu },\text{for}w\in W\text{and}f=\sum _{\mu \in P}{c}_{\mu }{x}^{\mu }\in 𝕂\left[P\right]\text{.}& \text{(1.29)}\end{array}$$

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).

page history