## The Satake isomorphism

Last update: 27 September 2012

## The Satake isomorphism

The following theorem establishes a $q$-analogue of the isomorphism $\Phi$ from Proposition 2.3(c). The map ${\Phi }_{1}$ in the following theorem is the Satake isomorphism. We shall continue to abuse language and use the term "vector space" in place of "free $𝕂=ℤ\left[q,{q}^{-1}\right]$ module".

The vector space isomorphism $\Phi$ in Proposition 2.3(c) generalizes to a vector space isomorphism

$Φ∼: Z(H∼)= 𝕂[P]W ⟶Φ1 Z(H∼)10= 10H∼10 ⟶Φ2 ε0H∼10 f ⟼ f10 ⟼ Aρf10 sλ ⟼ sλ10 ⟼ Aλ+ρ.$

 Proof. Using the third equality in (2.5), $ε0aλ10= ε0 ( ∑w∈W (-1)ℓ(w) xwλ ) 10=∑w∈W (-1)ℓ(w) Awλ=∣W∣ Aλ.$ By Proposition 2.3(c) and Theorem 1.4, ${s}_{\lambda }\in 𝕂{\left[P\right]}^{W}=Z\left(\stackrel{\sim }{H}\right),$ and so $Aρsλ10= 1∣W∣ε0 aρ10sλ 10=1∣W∣ ε0aρsλ 102= 1∣W∣ε0 aλ+ρ10= Aλ+ρ.$ Since $\left\{{s}_{\lambda }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\lambda \in {P}^{+}\right\}$ is a basis of $𝕂{\left[P\right]}^{W}=Z\left(\stackrel{\sim }{H}\right)$ and $\left\{{A}_{\lambda +\rho }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\lambda \in {P}^{+}\right\}$ is a basis of ${\epsilon }_{0}\stackrel{\sim }{H}{1}_{0},$ the composite map $Z(H∼) ⟶10 Z(H∼)10 ↪ 10H∼10 ⟶Aρ ε0H∼10 f ⟼ f10 ⟼ f10 ⟼ Aρf10 sλ ⟼ sλ10 ⟼ sλ10 ⟼ Aλ+ρ.$ is a vector space isomorphism. $\square$

If $\mu \in P$ let

$Wμ= {w∈W∣wμ=μ} andWμ(t)= ∑w∈Wμtℓ(w). (2.8)$

In particular, if $\mu =0,$ then ${W}_{0}=W$ and ${W}_{0}\left(t\right)$ is the polynomial that appears in (2.1).

The Hall-Littlewood polynomials, or Macdonald spherical functions, are defined by

$Pμ(x;t)= 1Wμ(t) ∑w∈Ww ( xμ∏α∈R+ 1-tx-α 1-x-α ) ,forμ∈P. (2.9)$

Then ${m}_{\mu }={P}_{\mu }\left(x;1\right)$ and, using the Weyl denominator formula,

$Pμ(x;0) = ∑w∈Ww ( xρxμ xρ ∏α∈R+ (1-x-α) ) (2.10) = 1aρ ∑w∈W (-1)ℓ(w) wxμ+ρ= aμ+ρaρ= sμ,$

and so, conceptually, the spherical functions ${P}_{\mu }\left(x;t\right)$ interpolate between the Schur functions ${s}_{\mu }$ and the monomial symmetric functions ${m}_{\mu }\text{.}$

The double cosets in $W\\stackrel{\sim }{W}/W$ are $W{t}_{\lambda }W,$ $\lambda \in {P}^{+}\text{.}$ If $\lambda \in {P}^{+}$ let ${n}_{\lambda }$ and ${m}_{\lambda }$ be the maximal and minimal length elements of $W{t}_{\lambda }W,$ respectively. Theorem 2.9 below will show that under the Satake isomorphism the Weyl characters ${s}_{\lambda }$ correspond to Kazhdan Lusztig basis elements ${C}_{{n}_{\lambda }}^{\prime }$ and the polynomials ${P}_{\mu }\left(x;{q}^{-2}\right)$ correspond to the elements ${M}_{\mu }={1}_{0}{x}^{\mu }{1}_{0}\text{.}$ More precisely, we have the following diagram:

$Φ1: Z(H∼)= 𝕂[P]W ⟶ Z(H∼)10 =10H∼10 f⟼f10 q-ℓ(w0) W0(q2)sλ ⟼ Cnλ′ Wμ(q-2) W0(q-2) Pμ(x:q-2) ⟼ Mμ (2.11)$

where ${w}_{0}$ is the longest element of $W\text{.}$ The following three lemmas (of independent interest) are used in the proof of Theorem 2.9.

Let ${t}_{\alpha },$ $\alpha \in {R}^{+},$ be commuting variables indexed by the positive roots. For $\lambda \in {P}^{+}$ let ${P}_{\lambda }\left(x:t\right)$ be as in (2.9), ${W}_{\lambda }$ as in (2.8), and define

$Rλ(x:tα)= ∑w∈Ww ( xλ ∏α∈R+ 1-tαx-α 1-x-α )$

and

$Wλ(tα)= ∑w∈Wλ ( ∏α∈R(w) tα ) ,$

where, as in (1.5), $R\left(w\right)=\left\{\alpha \in {R}^{+}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w\alpha <0\right\}$ is the inversion set of $w\text{.}$ Then

1. ${R}_{\lambda }\left(x:{t}_{\alpha }\right)=\sum _{\mu \in {P}^{+}}{u}_{\lambda \mu }{s}_{\mu },$ with ${u}_{\lambda \mu }\in ℤ\left[{t}_{\alpha }\right],$ ${u}_{\lambda \mu }=0$ unless $\mu \le \lambda ,$ and ${u}_{\lambda \lambda }={W}_{\lambda }\left({t}_{\alpha }\right)\text{.}$
2. ${P}_{\lambda }\left(x:t\right)=\sum _{\mu \in {P}^{+}}{c}_{\lambda \mu }{s}_{\mu },$ with ${c}_{\lambda \mu }\in ℤ\left[t\right],$ ${c}_{\lambda \mu }=0$ unless $\mu \le \lambda ,$ and ${c}_{\lambda \lambda }=1\text{.}$

 Proof. (a) If $E\subseteq {R}^{+}$ let $tE=∏α∈Etα andαE=∑α∈E α,$ and let ${a}_{\mu }$ be as defined in (2.4). Using the Weyl denominator formula, Proposition 2.3(a), and the second equality in (2.5), $Rλ = ∑w∈Ww ( xλ∏α∈R+ 1-tαx-α 1-x-α ) = ∑w∈Ww ( xλ+ρ ∏α∈R+ (1-tαx-α) xρ∏α∈R+ (1-x-α) ) = 1aρ∑w∈W (-1)ℓ(w)w ( xλ+ρ ∏α∈R+ (1-tαx-α) ) = 1aρ∑w∈W (-1)ℓ(w)w ( ∑E⊆R+ (-1)∣E∣ tExλ+ρ-αE ) = 1aρ∑w∈W (-1)∣E∣ tEaλ+ρ-αE =∑E⊆R+ (-1)∣E∣ tEsλ-αE,$ which shows that ${R}_{\lambda }$ is a symmetric function and ${u}_{\lambda \mu }\in ℤ\left[{t}_{\alpha }\right]\text{.}$ By the straightening law for Weyl characters (2.7), ${s}_{\lambda -{\alpha }_{E}}=0$ or ${s}_{\lambda -{\alpha }_{E}}={\left(-1\right)}^{\ell \left(v\right)}{s}_{\mu }$ with $v∈Wandμ∈P+ such thatμ+ρ= v-1(λ+ρ-αE).$ Let ${E}^{c}$ denote the complement of $E$ in ${R}^{+}\text{.}$ Since $v$ permutes the elements of ${R}^{+},$ $v-1 (λ+ρ-αE) = v-1λ+v-1 ( 12∑α∈Ec α-12∑α∈Eα ) = v-1λ+ ( 12∑α∈Fc α-12∑α∈Fα ) =v-1λ+ρ-αF,$ for some subset $F\subseteq {R}^{+}$ (which could be determined explicitly in terms of $E$ and $v\text{).}$ Hence $μ=v-1λ+ρ- αF-ρ=v-1 λ-αF≤v-1 λ≤λ. (2.12)$ This proves that ${u}_{\lambda \mu }=0$ unless $\mu \le \lambda \text{.}$ In (2.12), $\mu =\lambda$ only if ${v}^{-1}\lambda =\lambda$ and $\rho =\rho -{\alpha }_{F}={v}^{-1}\left(\rho -{\alpha }_{E}\right)$ in which case $ρ-αE=v (12∑α∈R+α) =ρ-∑α∈R(v)α andE=R(v).$ Thus $uλλ(tα)= ∑v-1∈Wλ tR(v).$ (b) Set ${t}_{\alpha }=t$ for all $\alpha \in {R}^{+}\text{.}$ Applying (a) with $\lambda =0,$ $R0(x:t)= ∑w∈Ww ( ∏α∈R+ 1-tx-α 1-x-α ) =W0(t). (2.13)$ Let ${W}^{\lambda }$ be a set of minimal length coset representatives of the cosets in $W/{W}_{\lambda }\text{.}$ Every element $w\in W$ can be written uniquely as $w=uv$ with $u\in {W}^{\lambda }$ and $v\in {W}_{\lambda }$ (see [Bou1981, IV §1 Ex. 3]). Let $Z(λ)= { α∈R+∣ ⟨λ,α∨⟩=0 } ,$ and let $Z{\left(\lambda \right)}^{c}$ be the complement of $Z\left(\lambda \right)$ in ${R}^{+}\text{.}$ Then $v\in {W}_{\lambda }$ permutes the elements of $Z{\left(\lambda \right)}^{c}$ among themselves and so $Rλ(x;t) = ∑u∈Wλu ( xλ ( ∏α∈Z(λ)c 1-tx-α 1-x-α ) ∑v∈Wλv ( ∏α∈Z(λ) 1-tx-α 1-x-α ) ) = ∑u∈Wλu ( xλ ( ∏α∈Z(λ)c 1-tx-α 1-x-α ) Wλ(t) ) ,$ where the last equality follows from (2.13). Thus there is an element ${P}_{\lambda }\left(x;t\right)\in 𝔽\left[P\right]$ where $𝔽$ is the field of fractions of $ℤ\left[{t}_{\alpha }\right]$ such that $Rλ(x;t) Wλ(t) ∑u∈Wλu ( xλ ∏α∈Z(λ)c 1-tx-α 1-x-α ) = Wλ(t) Pλ(x;t).$ Since ${R}_{\lambda }$ is a symmetric polynomial (an element of $ℤ\left[t\right]{\left[P\right]}^{W}\text{),}$ ${P}_{\lambda }\left(x;t\right)\in 𝔽{\left[P\right]}^{W}\text{.}$ Since $t$ only occurs in the numerators of the terms in the sum defining ${P}_{\lambda }$ in fact ${P}_{\lambda }$ is a symmetric polynomial with coefficients in $ℤ\left[t\right]\text{.}$ It follows that all the ${u}_{\lambda \mu }$ appearing in part (a) are divisible by ${W}_{\lambda }\left(t\right)$ and $Pλ(x;t)= ∑μ∈Pcλμ sμ,where cλμ= 1Wλ(t)uλμ$ are polynomials in $ℤ\left[t\right]$ such that ${c}_{\lambda \lambda }=1$ and ${c}_{\lambda \mu }=0$ unless $\mu \le \lambda \text{.}$ $\square$

Lemma 2.5 has the following interesting (and useful) corollary, see [Mac1972].

Let $\rho$ and ${\alpha }^{\vee }$ be as in (1.8) and (1.1), respectively, and let ${W}_{0}\left(t\right)$ be as defined in (2.8).

1. $\sum _{w\in W}w\left(\prod _{\alpha \in {R}^{+}}\frac{1-t{x}^{-\alpha }}{1-{x}^{-\alpha }}\right)={W}_{0}\left(t\right)\text{.}$
2. $\prod _{\alpha \in {R}^{+}}\frac{1-{t}^{⟨\rho ,{\alpha }^{\vee }⟩+1}}{1-{t}^{⟨\rho ,{\alpha }^{\vee }⟩}}={W}_{0}\left(t\right)\text{.}$

 Proof. Part (a) follows from Lemma 2.5 (a) by setting $\lambda =0$ and specializing ${t}_{\alpha }=t$ for all $\alpha \in {R}^{+}\text{.}$ (b) Applying the homomorphism $ℤ[t±1] [P] ⟶ ℤ[t±1] xλ⟼ t⟨-ρ,λ⟩$ to both sides of the identity in (a) for the root system ${R}^{\vee }=\left\{{\alpha }^{\vee }\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}\alpha \in R\right\}$ gives $W0(t)= ∑w∈W ∏α∈R+ ( 1-t⟨ρ,wα∨⟩+1 1-t⟨ρ,wα∨⟩ ) . (2.14)$ If $w\in W,$ $w\ne 1,$ and $w={s}_{{i}_{1}}\dots {s}_{{i}_{p}}$ is a reduced word for $w$ then ${w}^{-1}\left(-{\alpha }_{{i}_{1}}\right)={\left({s}_{{i}_{1}}w\right)}^{-1}{\alpha }_{{i}_{1}}\in R\left(w\right)$ and so $there is anα∈R+such that wα∨=-αi1∨.$ Then $∏α∈R+ 1-t⟨ρ,wα∨⟩+1 1-t⟨ρ,wα∨⟩ = 1-t⟨ρ,-αi1∨⟩+1 1-t⟨ρ,-αi1∨⟩ ∏ α∈R+ wα≠-αi1 1-t⟨ρ,wα∨⟩+1 1-t⟨ρ,wα∨⟩ = 1-t-1+1 1-t-1 ∏ α∈R+ wα≠-αi1 1-t⟨ρ,wα∨⟩+1 1-t⟨ρ,wα∨⟩ =0.$ Thus the only nonzero term on the right hand side of (2.14) occurs for $w=1\text{.}$ $\square$

For $\lambda \in {P}^{+}$ let ${t}_{\lambda }\in \stackrel{\sim }{W}$ be the translation in $\lambda$ and let ${n}_{\lambda }$ be the maximal length element in the double coset $W{t}_{\lambda }W\text{.}$ Let ${M}_{\lambda }={1}_{0}{x}^{\lambda }{1}_{0},$ as in (2.4). Then

$q-ℓ(w0) W0(q2)· W0(q-2) Wλ(q-2) ·Mλ= ∑x∈WtλW q ℓ(x)- ℓ(nλ) Tx,$

in the affine Hecke algebra $\stackrel{\sim }{H}\text{.}$

 Proof. Let $\lambda \in {P}^{+}\text{.}$ Let ${W}_{\lambda }=\text{Stab}\phantom{\rule{0.2em}{0ex}}\left(\lambda \right)$ and let ${w}_{0}$ and ${w}_{\lambda }$ be the maximal length elements in $W$ and ${W}_{\lambda },$ respectively. Let ${m}_{\lambda }$ and ${n}_{\lambda }$ be the minimal and maximal length elements respectively in the double coset $W{t}_{\lambda }W\text{.}$ For each positive root $\alpha$ the hyperplanes ${H}_{\alpha ,i},$ $1\le i\le ⟨\lambda ,{\alpha }^{\vee }⟩,$ are between the fundamental alcove $A$ and the alcove ${t}_{\lambda }A$ and so $ℓ(tλ)= ∑α∈R+ ⟨λ,α∨⟩ =2⟨λ,ρ∨⟩ ,whereρ∨=12 ∑α∈R+α∨. (2.15)$ Since ${m}_{\lambda }={t}_{\lambda }\left({w}_{\lambda }{w}_{0}\right)$ and ${n}_{\lambda }={t}_{{w}_{0}\lambda }{w}_{0},$ $ℓ(mλ)= ℓ(tλ)- ℓ(w0wλ)= ℓ(tλ)- ( ℓ(w0)- ℓ(wλ) ) ,and ℓ(nλ)= ℓ(tλ)+ ℓ(w0)= ℓ(mλ)+ ℓ(w0)- ℓ(wλ)+ ℓ(w0). (2.16)$ For example, in the setting of Example 1.1, if $\lambda =2{\omega }_{2}$ in type ${C}_{2},$ then ${W}_{\lambda }=\left\{1,{s}_{1}\right\},$ ${w}_{\lambda }={s}_{1},$ ${w}_{0}={s}_{1}{s}_{2}{s}_{1}{s}_{2},$ $\ell \left({t}_{\lambda }\right)=6,$ $\ell \left({m}_{\lambda }\right)=3,$ and $\ell \left({n}_{\lambda }\right)=10\text{.}$ Labelling the alcove $wA$ by the element $w,$ the 32 alcoves $wA$ with $w\in W{t}_{\lambda }W$ make up the four shaded diamonds $Hα1 Hφ = Hα1+α2 Hα2 H α1+2α2 H α1+α2, 1 = Hφ,1 = Hα0 tλ mλ wλ w0 nλ The double cosetWtλW$ Then $10xλ10 = 10Ttλ10 =10 Tmλw0wλ 10=10 TmλTw0wλ 10 = qℓ(w0)-ℓ(wλ) 10Tmλ10 = qℓ(w0)-ℓ(wλ)-ℓ(mλ) W(q2) ( ∑w∈W qℓ(w)Tw ) qℓ(mλ) Tmλ10.$ Let ${W}^{\lambda }$ be a set of minimal length coset representatives of the cosets in $W/{W}_{\lambda }\text{.}$ Every element $w\in W$ has a unique expression $w=uv$ with $u\in {W}^{\lambda }$ and $v\in {W}_{\lambda }\text{.}$ If $v\in {W}_{\lambda }$ then $vmλ=vtλwλ w0=tλvwλ w0=mλ (wλw0)-1 vwλw0=mλ ( w0-1 wλ-1 vwλw0 ) .$ Since conjugation by ${w}_{\lambda }$ and conjugation by ${w}_{0}$ are automorphisms of ${W}_{\lambda }$ and $W$ respectively taking simple reflections to simple reflections, $ℓ(v)=ℓ ( w0-1 wλ-1 vwλw0 ) .$ Then $10xλ10 = q ℓ(w0)- ℓ(wλ)- ℓ(mλ) W0(q2) ∑u∈Wλ qℓ(u) Tu ∑v∈Wλ qℓ(v)Tv qℓ(mλ) Tmλ10 = q 2 ℓ(w0)-2 ℓ(wλ)- ℓ(tλ) W0(q2) ( ∑u∈Wλ qℓ(u) Tu qℓ(mλ) Tmλ ) · ( ∑ v∈ w0-1 wλ-1 Wλwλw0 qℓ(v)Tv ) 10 = q -2ℓ(wλ) -ℓ(tλ) W0(q-2) ( ∑u∈Wλ qℓ(u) Tu ) qℓ(mλ) TmλWλ (q2)10 = q -2ℓ(wλ) -ℓ(tλ) Wλ(q2) W0(q2) W0(q-2) ( ∑u∈Wλ qℓ(u) Tu ) qℓ(mλ) Tmλ ( ∑w∈W qℓ(w) Tw ) = q-ℓ(tλ) Wλ(q-2) W0(q2) W0(q-2) ∑ x∈WtλW qℓ(x)Tx = q-ℓ(tλ)+ℓ(nλ) Wλ(q-2) W0(q2) W0(q-2) ( ∑ x∈WtλW qℓ(x)-ℓ(nλ) Tx ) = qℓ(w0) W0(q2) Wλ(q-2) W0(q-2) ( ∑ x∈WtλW qℓ(x)-ℓ(nλ) Tx ) .$ $\square$

Let ${w}_{0}$ be the longest element of $W$ and let $\lambda \in P\text{.}$

1. $\stackrel{‾}{{x}^{\lambda }}={T}_{{w}_{0}}{x}^{{w}_{0}\lambda }{T}_{{w}_{0}}^{-1}\text{.}$
2. $\stackrel{‾}{{1}_{0}}={1}_{0}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\stackrel{‾}{{\epsilon }_{0}}={\epsilon }_{0}\text{.}$
3. $\text{If}\phantom{\rule{0.2em}{0ex}}z\in ℤ{\left[P\right]}^{W}\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}\stackrel{‾}{z}=z\text{.}$
4. $\stackrel{‾}{{q}^{-\ell \left({w}_{0}\right)}{A}_{\lambda +\rho }}={q}^{-\ell \left({w}_{0}\right)}{A}_{\lambda +\rho }\text{.}$

 Proof. (a) If $\lambda \in {P}^{+}$ then ${w}_{0}{t}_{\lambda }={t}_{{w}_{0}\lambda }{w}_{0},$ $\ell \left({w}_{0}{t}_{\lambda }\right)=\ell \left({w}_{0}\right)+\ell \left({t}_{\lambda }\right)$ and $\ell \left({t}_{{w}_{0}\lambda }{w}_{0}\right)=\ell \left({t}_{{w}_{0}\lambda }\right)+\ell \left({w}_{0}\right)\text{.}$ Thus, $Tw0Ttλ= Tw0tλ= Ttw0λw0 =Ttw0λ Tw0,forλ ∈P+.$ Let $\lambda \in P$ and write $\lambda =\mu -\nu$ with $\mu ,\nu \in {P}^{+}\text{.}$ Since $-{w}_{0}\mu \in {P}^{+}$ and $-{w}_{0}\nu \in {P}^{+},$ $xλ‾ = TtμTtν-1‾ =Tt-μ-1 Tt-ν=Tw0 Tt-w0μ-1 Tt-w0ν Tw0-1 = Tw0 (x-w0λ)-1 Tw0-1= Tw0xw0λ Tw0-1.$ (b) For $1\le i\le n,$ $102‾= 10‾2and Ti10‾ =Ti-110‾ =q-110‾ =q10‾, ε02‾= ε0‾2and Tiε0‾= Ti-1ε0‾= -qε0‾= -q-1ε0‾.$ These are the defining properties (before 2.1) of ${1}_{0}$ and ${\epsilon }_{0}$ and so $\stackrel{‾}{{1}_{0}}={1}_{0}$ and $\stackrel{‾}{{\epsilon }_{0}}={\epsilon }_{0}\text{.}$ (c) If $z=\sum _{\mu \in P}{c}_{\mu }{x}^{\mu }\in ℤ{\left[P\right]}^{W},$ then, since ${c}_{\mu }\in ℤ,$ $\stackrel{‾}{{c}_{\mu }}={c}_{\mu }$ and, by (a), $z‾=∑μ∈P cμ‾xμ‾= ∑μ∈Pcμ Tw0xw0μ Tw0-1= Tw0 ( ∑μ∈P cμxw0μ ) Tw0-1= Tw0z Tw0-1,$ since $z\in ℤ{\left[P\right]}^{W}$ is $W$-invariant. Finally, since $ℤ{\left[P\right]}^{W}\subseteq Z\left(\stackrel{\sim }{H}\right),$ $z$ is central, and $\stackrel{‾}{z}={T}_{{w}_{0}}z{T}_{{w}_{0}}^{-1}=z\text{.}$ (d) By (a), (b) and the third equality in (2.5), $q-ℓ(w0) Aλ+ρ ‾ = qℓ(w0) ε0xλ+ρ 10 ‾ =qℓ(w0) ε0Tw0 xw0(λ+ρ) Tw0-110 = qℓ(w0) (-q-1)ℓ(w0) ε0xw0(λ+ρ) 10q-ℓ(w0)= (-q-1)ℓ(w0) Aw0(λ+ρ) = q-ℓ(w0) Aλ+ρ.$ $\square$

The following theorem is due to Lusztig [Lus1983]. Part (a) was originally proved in a different formulation by Macdonald [Mac1971, (4.1.2)].

If $\mu \in P$ let ${W}_{\mu }$ be the stabilizer of $\mu$ and let ${W}_{\mu }\left(t\right)$ be as in (2.8).

1. Let $\mu \in P\text{.}$ Let ${P}_{\mu }\left(x;t\right)$ be the Macdonald spherical function defined in (2.9) and define ${M}_{\mu }={1}_{0}{x}^{\mu }{1}_{0}$ as in (2.4). In the affine Hecke algebra $\stackrel{\sim }{H},$ $Wμ(q-2) W0(q-2) ·Pμ (x;q-2)10 =Mμ.$
2. For $\lambda \in {P}^{+}$ let ${t}_{\lambda }\in \stackrel{\sim }{W}$ be the translation in $\lambda$ and let ${n}_{\lambda }$ be the maximal length element in the double coset $W{t}_{\lambda }W\text{.}$ Let ${s}_{\lambda }$ be the Weyl character and let ${C}_{{n}_{\lambda }}^{\prime }$ be the Kazhdan-Lusztig basis element as defined in (2.6) and (1.26), respectively. In the affine Hecke algebra $\stackrel{\sim }{H},$ $q-ℓ(w0) W0(q2)·sλ 10=Cnλ′.$

 Proof. (a) By Theorem 2.4 there is an element ${\stackrel{\sim }{P}}_{\lambda }\in 𝕂{\left[P\right]}^{W}$ such that ${\stackrel{\sim }{P}}_{\lambda }{1}_{0}={1}_{0}{x}^{\lambda }{1}_{0}\text{.}$ To find ${\stackrel{\sim }{P}}_{\lambda }$ first do a rank 1 calculation, $(q-1+Ti) xλ10 = ( q-1xλ+ xsiλTi+ (q-q-1) ( xλ-xsiλ 1-x-αi ) ) 10 = 11-x-αi ( q-1xλ (1-x-αi) +qxsiλ (1-x-αi) +qxλ-qxsiλ -q-1xλ+q-1 xsiλ ) 10 = (1-x-αi)-1 ( -q-1xλ-αi -qxsiλ-αi+ qxλ+q-1xsiλ ) 10 = (1-x-αi)-1 ( xλ (q-q-1x-αi) +xsiλ (q-1-qx-αi) ) 10 = ( q-q-1x-αi 1-x-αi ·xλ+ x-αix-αi · q-1xαi-q xαi-1 ·xsiλ ) 10 = (1+si) ( q-q-1x-αi 1-x-αi xλ ) 10.$ Since ${1}_{0}$ is a linear combination of products of ${T}_{i}$ it can also be written as a linear combination of products of ${q}^{-1}+{T}_{i}\text{.}$ Thus ${1}_{0}{x}^{\lambda }{1}_{0}$ can be written as a linear combination of terms of the form $(1+si1) ( q-q-1x-αi1 1-x-αi1 ) …(1+sip) ( q-q-1x-αip 1-x-αip ) xλ10.$ Thus $10xλ10= P∼λ10, whereP∼λ= ∑w∈W xwλwcw,$ and the ${c}_{w}$ are some linear combinations of products of terms of the form $\left(q-{q}^{-1}{x}^{\alpha }\right)/\left(1-{x}^{\alpha }\right)$ for roots $\alpha \in R\text{.}$ Since ${\stackrel{\sim }{P}}_{\lambda }$ is an element of $𝕂{\left[P\right]}^{W},$ $P∼λ=∑w∈W w ( xw0λw0 cw0 ) ,$ where ${w}_{0}$ is the longest element of $W\text{.}$ The coefficient ${w}_{0}{c}_{{w}_{0}}$ comes from the highest term in the expansion of $10= 1W0(q2) ( q2ℓ(w0) Tw0 +lower terms )$ in terms of linear combination of products of the $\left({q}^{-1}+{T}_{i}\right)\text{.}$ If ${w}_{0}={s}_{{i}_{1}}\dots {s}_{{i}_{p}}$ is a reduced word for ${w}_{0}$ then $w0cw0 = qℓ(w0) W0(q2) si1 ( q-q-1x-αi1 1-x-αi1 ) … sip ( q-q-1x-αip 1-x-αip ) = qℓ(w0) W0(q2) si1…sip ( q-q-1 x-sip…si2αi1 1- x -sip… si2αi1 ) ( q-q-1 x-sip…si3αi2 1- x -sip… si3αi2 ) … ( q-q-1x-αip 1-x-αip ) = qℓ(w0) W0(q2) w0 ∏α∈R+ q-q-1x-α 1-x-α = q2ℓ(w0) W0(q2) w0 ∏α∈R+ q-q-2x-α 1-x-α ,$ by Lemma 1.2 and the fact that $\ell \left({w}_{0}\right)=\text{Card}\phantom{\rule{0.2em}{0ex}}\left({R}^{+}\right)\text{.}$ Thus, since ${q}^{-2\ell \left({w}_{0}\right)}{W}_{0}\left({q}^{2}\right)={W}_{0}\left({q}^{-2}\right),$ $P∼λ= 1W0(q-2) ∑w∈Ww ( xλ∏α∈R+ 1-q-2x-α 1-x-α ) .$ (b) Since ${W}_{0}\left({q}^{-2}\right)={q}^{-2\ell \left({w}_{0}\right)}{W}_{0}\left({q}^{2}\right),$ Lemma 2.8 gives $q-ℓ(w0) W0(q2)sλ 10 ‾ = qℓ(w0)W0 (q-2)sλ‾ 10=q-ℓ(w0) W0(q2)sλ10.$ By Lemma 2.5(b), $sλ=∑μ∈P+ Kλμ(t) Pμ(x;t),$ where ${K}_{\lambda \mu }\left(t\right)\in ℤ\left[t\right],$ ${K}_{\lambda \mu }\left(t\right)=0$ unless $\mu \le \lambda$ and ${K}_{\lambda \lambda }\left(t\right)=1\text{.}$ Thus, by part (a) and Lemma 2.7 $q-ℓ(w0)W0 (q2)sλ10 = ∑μ∈P+ q-ℓ(w0) W0(q2) Kλμ (q-2)Pμ (x;q-2)10 = ∑μ∈P+ ∑x∈WtμW qℓ(x)-ℓ(nμ) Kλμ(q-2) Tx,$ where the polynomials ${K}_{\lambda \mu }\left({q}^{-2}\right)\in ℤ\left[{q}^{-2}\right]$ are 0 unless $\mu \le \lambda$ and ${K}_{\lambda \lambda }\left({q}^{-2}\right)=1\text{.}$ Hence ${q}^{-\ell \left({w}_{0}\right)}W\left({q}^{2}\right){s}_{\lambda }{1}_{0}$ is a bar invariant element of $\stackrel{\sim }{H}$ such that its expansion in terms of the basis $\left\{{T}_{w}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w\in \stackrel{\sim }{W}\right\}$ is triangular with coefficient of ${T}_{{n}_{\lambda }}$ equal to 1 and all other coefficients in ${q}^{-1}ℤ\left[{q}^{-1}\right]\text{.}$ These are the defining properties (1.26)-(1.27) of ${C}_{{n}_{\lambda }}^{\prime }\text{.}$ $\square$

## 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).