The Satake isomorphism

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

Last update: 27 September 2012

The Satake isomorphism

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

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

\begin{matrix} \tilde{Φ} : & Z (\tilde{H}) = 𝕂 {[P]}^{W} & \overset{Φ_{1}}{⟶} & Z (\tilde{H}) 1_{0} = 1_{0} \tilde{H} 1_{0} & \overset{Φ_{2}}{⟶} & ε_{0} \tilde{H} 1_{0} \\ f & ⟼ & f 1_{0} & ⟼ & A_{ρ} f 1_{0} \\ s_{λ} & ⟼ & s_{λ} 1_{0} & ⟼ & A_{λ + ρ} . \end{matrix}

Proof.

Using the third equality in (2.5),

ε_{0} a_{λ} 1_{0} = ε_{0} (\sum_{w \in W} {(- 1)}^{ℓ (w)} x^{w λ}) 1_{0} = \sum_{w \in W} {(- 1)}^{ℓ (w)} A_{w λ} = ∣ W ∣ A_{λ} .

By Proposition 2.3(c) and Theorem 1.4, $s_{λ} \in 𝕂 {[P]}^{W} = Z (\tilde{H}),$ and so

A_{ρ} s_{λ} 1_{0} = \frac{1}{∣ W ∣} ε_{0} a_{ρ} 1_{0} s_{λ} 1_{0} = \frac{1}{∣ W ∣} ε_{0} a_{ρ} s_{λ} 1_{0}^{2} = \frac{1}{∣ W ∣} ε_{0} a_{λ + ρ} 1_{0} = A_{λ + ρ} .

Since ${s_{λ} ∣ λ \in P^{+}}$ is a basis of $𝕂 {[P]}^{W} = Z (\tilde{H})$ and ${A_{λ + ρ} ∣ λ \in P^{+}}$ is a basis of $ε_{0} \tilde{H} 1_{0},$ the composite map

\begin{matrix} Z (\tilde{H}) & \overset{1_{0}}{⟶} & Z (\tilde{H}) 1_{0} & ↪ & 1_{0} \tilde{H} 1_{0} & \overset{A_{ρ}}{⟶} & ε_{0} \tilde{H} 1_{0} \\ f & ⟼ & f 1_{0} & ⟼ & f 1_{0} & ⟼ & A_{ρ} f 1_{0} \\ s_{λ} & ⟼ & s_{λ} 1_{0} & ⟼ & s_{λ} 1_{0} & ⟼ & A_{λ + ρ} . \end{matrix}

is a vector space isomorphism.

$□$

If $μ \in P$ let

\begin{matrix} W_{μ} = {w \in W ∣ w μ = μ} and W_{μ} (t) = \sum_{w \in W_{μ}} t^{ℓ (w)} . & (2.8) \end{matrix}

In particular, if $μ = 0,$ then $W_{0} = W$ and $W_{0} (t)$ is the polynomial that appears in (2.1).

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

\begin{matrix} P_{μ} (x; t) = \frac{1}{W_{μ} (t)} \sum_{w \in W} w (x^{μ} \prod_{α \in R^{+}} \frac{1 - t x^{- α}}{1 - x^{- α}}), for μ \in P . & (2.9) \end{matrix}

Then $m_{μ} = P_{μ} (x; 1)$ and, using the Weyl denominator formula,

\begin{matrix} P_{μ} (x; 0) & = & \sum_{w \in W} w (\frac{x^{ρ} x^{μ}}{x^{ρ} \prod_{α \in R^{+}} (1 - x^{- α})}) & (2.10) \\ = & \frac{1}{a_{ρ}} \sum_{w \in W} {(- 1)}^{ℓ (w)} w x^{μ + ρ} = \frac{a_{μ + ρ}}{a_{ρ}} = s_{μ}, \end{matrix}

and so, conceptually, the spherical functions $P_{μ} (x; t)$ interpolate between the Schur functions $s_{μ}$ and the monomial symmetric functions $m_{μ} .$

The double cosets in $W \ \tilde{W} / W$ are $W t_{λ} W,$ $λ \in P^{+} .$ If $λ \in P^{+}$ let $n_{λ}$ and $m_{λ}$ be the maximal and minimal length elements of $W t_{λ} W,$ respectively. Theorem 2.9 below will show that under the Satake isomorphism the Weyl characters $s_{λ}$ correspond to Kazhdan Lusztig basis elements $C_{n_{λ}}^{'}$ and the polynomials $P_{μ} (x; q^{- 2})$ correspond to the elements $M_{μ} = 1_{0} x^{μ} 1_{0} .$ More precisely, we have the following diagram:

\begin{matrix} \begin{matrix} Φ_{1} : & Z (\tilde{H}) = 𝕂 {[P]}^{W} & ⟶ & Z (\tilde{H}) 1_{0} = 1_{0} \tilde{H} 1_{0} \\ f & ⟼ & f 1_{0} \\ q^{- ℓ (w_{0})} W_{0} (q^{2}) s_{λ} & ⟼ & C_{n_{λ}}^{'} \\ \frac{W_{μ} (q^{- 2})}{W_{0} (q^{- 2})} P_{μ} (x : q^{- 2}) & ⟼ & M_{μ} \end{matrix} & (2.11) \end{matrix}

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

Let $t_{α},$ $α \in R^{+},$ be commuting variables indexed by the positive roots. For $λ \in P^{+}$ let $P_{λ} (x : t)$ be as in (2.9), $W_{λ}$ as in (2.8), and define

R_{λ} (x : t_{α}) = \sum_{w \in W} w (x^{λ} \prod_{α \in R^{+}} \frac{1 - t_{α} x^{- α}}{1 - x^{- α}})

and

W_{λ} (t_{α}) = \sum_{w \in W_{λ}} (\prod_{α \in R (w)} t_{α}),

where, as in (1.5), $R (w) = {α \in R^{+} ∣ w α < 0}$ is the inversion set of $w .$ Then

$R_{λ} (x : t_{α}) = \sum_{μ \in P^{+}} u_{λ μ} s_{μ},$ with $u_{λ μ} \in ℤ [t_{α}],$ $u_{λ μ} = 0$ unless $μ \leq λ,$ and $u_{λ λ} = W_{λ} (t_{α}) .$
$P_{λ} (x : t) = \sum_{μ \in P^{+}} c_{λ μ} s_{μ},$ with $c_{λ μ} \in ℤ [t],$ $c_{λ μ} = 0$ unless $μ \leq λ,$ and $c_{λ λ} = 1 .$

Proof.

(a) If $E \subseteq R^{+}$ let

t_{E} = \prod_{α \in E} t_{α} and α_{E} = \sum_{α \in E} α,

and let $a_{μ}$ be as defined in (2.4). Using the Weyl denominator formula, Proposition 2.3(a), and the second equality in (2.5),

\begin{matrix} R_{λ} & = & \sum_{w \in W} w (x^{λ} \prod_{α \in R^{+}} \frac{1 - t_{α} x^{- α}}{1 - x^{- α}}) \\ = & \sum_{w \in W} w (\frac{x^{λ + ρ} \prod_{α \in R^{+}} (1 - t_{α} x^{- α})}{x^{ρ} \prod_{α \in R^{+}} (1 - x^{- α})}) \\ = & \frac{1}{a_{ρ}} \sum_{w \in W} {(- 1)}^{ℓ (w)} w (x^{λ + ρ} \prod_{α \in R^{+}} (1 - t_{α} x^{- α})) \\ = & \frac{1}{a_{ρ}} \sum_{w \in W} {(- 1)}^{ℓ (w)} w (\sum_{E \subseteq R^{+}} {(- 1)}^{∣ E ∣} t_{E} x^{λ + ρ - α_{E}}) \\ = & \frac{1}{a_{ρ}} \sum_{w \in W} {(- 1)}^{∣ E ∣} t_{E} a_{λ + ρ - α_{E}} = \sum_{E \subseteq R^{+}} {(- 1)}^{∣ E ∣} t_{E} s_{λ - α_{E}}, \end{matrix}

which shows that $R_{λ}$ is a symmetric function and $u_{λ μ} \in ℤ [t_{α}] .$

By the straightening law for Weyl characters (2.7), $s_{λ - α_{E}} = 0$ or $s_{λ - α_{E}} = {(- 1)}^{ℓ (v)} s_{μ}$ with

v \in W and μ \in P^{+} such that μ + ρ = v^{- 1} (λ + ρ - α_{E}) .

Let $E^{c}$ denote the complement of $E$ in $R^{+} .$ Since $v$ permutes the elements of $R^{+},$

\begin{matrix} v^{- 1} (λ + ρ - α_{E}) & = & v^{- 1} λ + v^{- 1} (\frac{1}{2} \sum_{α \in E^{c}} α - \frac{1}{2} \sum_{α \in E} α) \\ = & v^{- 1} λ + (\frac{1}{2} \sum_{α \in F^{c}} α - \frac{1}{2} \sum_{α \in F} α) = v^{- 1} λ + ρ - α_{F}, \end{matrix}

for some subset $F \subseteq R^{+}$ (which could be determined explicitly in terms of $E$ and $v).$ Hence

\begin{matrix} μ = v^{- 1} λ + ρ - α_{F} - ρ = v^{- 1} λ - α_{F} \leq v^{- 1} λ \leq λ . & (2.12) \end{matrix}

This proves that $u_{λ μ} = 0$ unless $μ \leq λ .$

In (2.12), $μ = λ$ only if $v^{- 1} λ = λ$ and $ρ = ρ - α_{F} = v^{- 1} (ρ - α_{E})$ in which case

ρ - α_{E} = v (\frac{1}{2} \sum_{α \in R^{+}} α) = ρ - \sum_{α \in R (v)} α and E = R (v) .

Thus

u_{λ λ} (t_{α}) = \sum_{v^{- 1} \in W_{λ}} t_{R (v)} .

(b) Set $t_{α} = t$ for all $α \in R^{+} .$ Applying (a) with $λ = 0,$

\begin{matrix} R_{0} (x : t) = \sum_{w \in W} w (\prod_{α \in R^{+}} \frac{1 - t x^{- α}}{1 - x^{- α}}) = W_{0} (t) . & (2.13) \end{matrix}

Let $W^{λ}$ be a set of minimal length coset representatives of the cosets in $W / W_{λ} .$ Every element $w \in W$ can be written uniquely as $w = u v$ with $u \in W^{λ}$ and $v \in W_{λ}$ (see [Bou1981, IV §1 Ex. 3]). Let

Z (λ) = {α \in R^{+} ∣ ⟨ λ, α^{\lor} ⟩ = 0},

and let $Z {(λ)}^{c}$ be the complement of $Z (λ)$ in $R^{+} .$ Then $v \in W_{λ}$ permutes the elements of $Z {(λ)}^{c}$ among themselves and so

\begin{matrix} R_{λ} (x; t) & = & \sum_{u \in W^{λ}} u (x^{λ} (\prod_{α \in Z {(λ)}^{c}} \frac{1 - t x^{- α}}{1 - x^{- α}}) \sum_{v \in W_{λ}} v (\prod_{α \in Z (λ)} \frac{1 - t x^{- α}}{1 - x^{- α}})) \\ = & \sum_{u \in W^{λ}} u (x^{λ} (\prod_{α \in Z {(λ)}^{c}} \frac{1 - t x^{- α}}{1 - x^{- α}}) W_{λ} (t)), \end{matrix}

where the last equality follows from (2.13). Thus there is an element $P_{λ} (x; t) \in 𝔽 [P]$ where $𝔽$ is the field of fractions of $ℤ [t_{α}]$ such that

R_{λ} (x; t) W_{λ} (t) \sum_{u \in W^{λ}} u (x^{λ} \prod_{α \in Z {(λ)}^{c}} \frac{1 - t x^{- α}}{1 - x^{- α}}) = W_{λ} (t) P_{λ} (x; t) .

Since $R_{λ}$ is a symmetric polynomial (an element of $ℤ [t] {[P]}^{W}),$ $P_{λ} (x; t) \in 𝔽 {[P]}^{W} .$ Since $t$ only occurs in the numerators of the terms in the sum defining $P_{λ}$ in fact $P_{λ}$ is a symmetric polynomial with coefficients in $ℤ [t] .$ It follows that all the $u_{λ μ}$ appearing in part (a) are divisible by $W_{λ} (t)$ and

P_{λ} (x; t) = \sum_{μ \in P} c_{λ μ} s_{μ}, where c_{λ μ} = \frac{1}{W_{λ} (t)} u_{λ μ}

are polynomials in $ℤ [t]$ such that $c_{λ λ} = 1$ and $c_{λ μ} = 0$ unless $μ \leq λ .$

$□$

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

Let $ρ$ and $α^{\lor}$ be as in (1.8) and (1.1), respectively, and let $W_{0} (t)$ be as defined in (2.8).

$\sum_{w \in W} w (\prod_{α \in R^{+}} \frac{1 - t x^{- α}}{1 - x^{- α}}) = W_{0} (t) .$
$\prod_{α \in R^{+}} \frac{1 - t^{⟨ ρ, α^{\lor} ⟩ + 1}}{1 - t^{⟨ ρ, α^{\lor} ⟩}} = W_{0} (t) .$

Proof.

Part (a) follows from Lemma 2.5 (a) by setting $λ = 0$ and specializing $t_{α} = t$ for all $α \in R^{+} .$

(b) Applying the homomorphism

\begin{matrix} ℤ [t^{\pm 1}] [P] & ⟶ & ℤ [t^{\pm 1}] \\ x^{λ} & ⟼ & t^{⟨ - ρ, λ ⟩} \end{matrix}

to both sides of the identity in (a) for the root system $R^{\lor} = {α^{\lor} ∣ α \in R}$ gives

\begin{matrix} W_{0} (t) = \sum_{w \in W} \prod_{α \in R^{+}} (\frac{1 - t^{⟨ ρ, w α^{\lor} ⟩ + 1}}{1 - t^{⟨ ρ, w α^{\lor} ⟩}}) . & (2.14) \end{matrix}

If $w \in W,$ $w \neq 1,$ and $w = s_{i_{1}} \dots s_{i_{p}}$ is a reduced word for $w$ then $w^{- 1} (- α_{i_{1}}) = {(s_{i_{1}} w)}^{- 1} α_{i_{1}} \in R (w)$ and so

there is an α \in R^{+} such that w α^{\lor} = - α_{i_{1}}^{\lor} .

Then

\begin{matrix} \prod_{α \in R^{+}} \frac{1 - t^{⟨ ρ, w α^{\lor} ⟩ + 1}}{1 - t^{⟨ ρ, w α^{\lor} ⟩}} & = & \frac{1 - t^{⟨ ρ, - α_{i_{1}}^{\lor} ⟩ + 1}}{1 - t^{⟨ ρ, - α_{i_{1}}^{\lor} ⟩}} \prod_{\underset{w α \neq - α_{i_{1}}}{α \in R^{+}}} \frac{1 - t^{⟨ ρ, w α^{\lor} ⟩ + 1}}{1 - t^{⟨ ρ, w α^{\lor} ⟩}} \\ = & \frac{1 - t^{- 1 + 1}}{1 - t^{- 1}} \prod_{\underset{w α \neq - α_{i_{1}}}{α \in R^{+}}} \frac{1 - t^{⟨ ρ, w α^{\lor} ⟩ + 1}}{1 - t^{⟨ ρ, w α^{\lor} ⟩}} = 0 . \end{matrix}

Thus the only nonzero term on the right hand side of (2.14) occurs for $w = 1 .$

$□$

For $λ \in P^{+}$ let $t_{λ} \in \tilde{W}$ be the translation in $λ$ and let $n_{λ}$ be the maximal length element in the double coset $W t_{λ} W .$ Let $M_{λ} = 1_{0} x^{λ} 1_{0},$ as in (2.4). Then

q^{- ℓ (w_{0})} W_{0} (q^{2}) \cdot \frac{W_{0} (q^{- 2})}{W_{λ} (q^{- 2})} \cdot M_{λ} = \sum_{x \in W t_{λ} W} q^{ℓ (x) - ℓ (n_{λ})} T_{x},

in the affine Hecke algebra $\tilde{H} .$

Proof.

Let $λ \in P^{+} .$ Let $W_{λ} = Stab (λ)$ and let $w_{0}$ and $w_{λ}$ be the maximal length elements in $W$ and $W_{λ},$ respectively. Let $m_{λ}$ and $n_{λ}$ be the minimal and maximal length elements respectively in the double coset $W t_{λ} W .$ For each positive root $α$ the hyperplanes $H_{α, i},$ $1 \leq i \leq ⟨ λ, α^{\lor} ⟩,$ are between the fundamental alcove $A$ and the alcove $t_{λ} A$ and so

\begin{matrix} ℓ (t_{λ}) = \sum_{α \in R^{+}} ⟨ λ, α^{\lor} ⟩ = 2 ⟨ λ, ρ^{\lor} ⟩, where ρ^{\lor} = \frac{1}{2} \sum_{α \in R^{+}} α^{\lor} . & (2.15) \end{matrix}

Since $m_{λ} = t_{λ} (w_{λ} w_{0})$ and $n_{λ} = t_{w_{0} λ} w_{0},$

\begin{matrix} \begin{matrix} ℓ (m_{λ}) = ℓ (t_{λ}) - ℓ (w_{0} w_{λ}) = ℓ (t_{λ}) - (ℓ (w_{0}) - ℓ (w_{λ})), and \\ ℓ (n_{λ}) = ℓ (t_{λ}) + ℓ (w_{0}) = ℓ (m_{λ}) + ℓ (w_{0}) - ℓ (w_{λ}) + ℓ (w_{0}) . \end{matrix} & (2.16) \end{matrix}

For example, in the setting of Example 1.1, if $λ = 2 ω_{2}$ in type $C_{2},$ then $W_{λ} = {1, s_{1}},$ $w_{λ} = s_{1},$ $w_{0} = s_{1} s_{2} s_{1} s_{2},$ $ℓ (t_{λ}) = 6,$ $ℓ (m_{λ}) = 3,$ and $ℓ (n_{λ}) = 10 .$ Labelling the alcove $w A$ by the element $w,$ the 32 alcoves $w A$ with $w \in W t_{λ} W$ make up the four shaded diamonds

\begin{matrix} The double coset W t_{λ} W \end{matrix}

Then

\begin{matrix} 1_{0} x^{λ} 1_{0} & = & 1_{0} T_{t_{λ}} 1_{0} = 1_{0} T_{m_{λ} w_{0} w_{λ}} 1_{0} = 1_{0} T_{m_{λ}} T_{w_{0} w_{λ}} 1_{0} \\ = & q^{ℓ (w_{0}) - ℓ (w_{λ})} 1_{0} T_{m_{λ}} 1_{0} \\ = & \frac{q^{ℓ (w_{0}) - ℓ (w_{λ}) - ℓ (m_{λ})}}{W (q^{2})} (\sum_{w \in W} q^{ℓ (w)} T_{w}) q^{ℓ (m_{λ})} T_{m_{λ}} 1_{0} . \end{matrix}

Let $W^{λ}$ be a set of minimal length coset representatives of the cosets in $W / W_{λ} .$ Every element $w \in W$ has a unique expression $w = u v$ with $u \in W^{λ}$ and $v \in W_{λ} .$ If $v \in W_{λ}$ then

v m_{λ} = v t_{λ} w_{λ} w_{0} = t_{λ} v w_{λ} w_{0} = m_{λ} {(w_{λ} w_{0})}^{- 1} v w_{λ} w_{0} = m_{λ} (w_{0}^{- 1} w_{λ}^{- 1} v w_{λ} w_{0}) .

Since conjugation by $w_{λ}$ and conjugation by $w_{0}$ are automorphisms of $W_{λ}$ and $W$ respectively taking simple reflections to simple reflections,

ℓ (v) = ℓ (w_{0}^{- 1} w_{λ}^{- 1} v w_{λ} w_{0}) .

Then

\begin{matrix} 1_{0} x^{λ} 1_{0} & = & \frac{q^{ℓ (w_{0}) - ℓ (w_{λ}) - ℓ (m_{λ})}}{W_{0} (q^{2})} \sum_{u \in W^{λ}} q^{ℓ (u)} T_{u} \sum_{v \in W_{λ}} q^{ℓ (v)} T_{v} q^{ℓ (m_{λ})} T_{m_{λ}} 1_{0} \\ = & \frac{q^{2 ℓ (w_{0}) - 2 ℓ (w_{λ}) - ℓ (t_{λ})}}{W_{0} (q^{2})} (\sum_{u \in W^{λ}} q^{ℓ (u)} T_{u} q^{ℓ (m_{λ})} T_{m_{λ}}) \cdot (\sum_{v \in w_{0}^{- 1} w_{λ}^{- 1} W_{λ} w_{λ} w_{0}} q^{ℓ (v)} T_{v}) 1_{0} \\ = & \frac{q^{- 2 ℓ (w_{λ}) - ℓ (t_{λ})}}{W_{0} (q^{- 2})} (\sum_{u \in W^{λ}} q^{ℓ (u)} T_{u}) q^{ℓ (m_{λ})} T_{m_{λ}} W_{λ} (q^{2}) 1_{0} \\ = & \frac{q^{- 2 ℓ (w_{λ}) - ℓ (t_{λ})} W_{λ} (q^{2})}{W_{0} (q^{2}) W_{0} (q^{- 2})} (\sum_{u \in W^{λ}} q^{ℓ (u)} T_{u}) q^{ℓ (m_{λ})} T_{m_{λ}} (\sum_{w \in W} q^{ℓ (w)} T_{w}) \\ = & \frac{q^{- ℓ (t_{λ})} W_{λ} (q^{- 2})}{W_{0} (q^{2}) W_{0} (q^{- 2})} \sum_{x \in W t_{λ} W} q^{ℓ (x)} T_{x} \\ = & \frac{q^{- ℓ (t_{λ}) + ℓ (n_{λ})} W_{λ} (q^{- 2})}{W_{0} (q^{2}) W_{0} (q^{- 2})} (\sum_{x \in W t_{λ} W} q^{ℓ (x) - ℓ (n_{λ})} T_{x}) \\ = & \frac{q^{ℓ (w_{0})}}{W_{0} (q^{2})} \frac{W_{λ} (q^{- 2})}{W_{0} (q^{- 2})} (\sum_{x \in W t_{λ} W} q^{ℓ (x) - ℓ (n_{λ})} T_{x}) . \end{matrix}

$□$

Let $w_{0}$ be the longest element of $W$ and let $λ \in P .$

$\overline{x^{λ}} = T_{w_{0}} x^{w_{0} λ} T_{w_{0}}^{- 1} .$
$\overline{1_{0}} = 1_{0} and \overline{ε_{0}} = ε_{0} .$
$If z \in ℤ {[P]}^{W} then \overline{z} = z .$
$\overline{q^{- ℓ (w_{0})} A_{λ + ρ}} = q^{- ℓ (w_{0})} A_{λ + ρ} .$

Proof.

(a) If $λ \in P^{+}$ then $w_{0} t_{λ} = t_{w_{0} λ} w_{0},$ $ℓ (w_{0} t_{λ}) = ℓ (w_{0}) + ℓ (t_{λ})$ and $ℓ (t_{w_{0} λ} w_{0}) = ℓ (t_{w_{0} λ}) + ℓ (w_{0}) .$ Thus,

T_{w_{0}} T_{t_{λ}} = T_{w_{0} t_{λ}} = T_{t_{w_{0} λ} w_{0}} = T_{t_{w_{0} λ}} T_{w_{0}}, for λ \in P^{+} .

Let $λ \in P$ and write $λ = μ - ν$ with $μ, ν \in P^{+} .$ Since $- w_{0} μ \in P^{+}$ and $- w_{0} ν \in P^{+},$

\begin{matrix} \overline{x^{λ}} & = & \overline{T_{t_{μ}} T_{t_{ν}}^{- 1}} = T_{t_{- μ}}^{- 1} T_{t_{- ν}} = T_{w_{0}} T_{t_{- w_{0} μ}}^{- 1} T_{t_{- w_{0} ν}} T_{w_{0}}^{- 1} \\ = & T_{w_{0}} {(x^{- w_{0} λ})}^{- 1} T_{w_{0}}^{- 1} = T_{w_{0}} x^{w_{0} λ} T_{w_{0}}^{- 1} . \end{matrix}

(b) For $1 \leq i \leq n,$

\begin{matrix} \overline{1_{0}^{2}} = {\overline{1_{0}}}^{2} and T_{i} \overline{1_{0}} = \overline{T_{i}^{- 1} 1_{0}} = \overline{q^{- 1} 1_{0}} = q \overline{1_{0}}, \\ \overline{ε_{0}^{2}} = {\overline{ε_{0}}}^{2} and T_{i} \overline{ε_{0}} = \overline{T_{i}^{- 1} ε_{0}} = \overline{- q ε_{0}} = - q^{- 1} \overline{ε_{0}} . \end{matrix}

These are the defining properties (before 2.1) of $1_{0}$ and $ε_{0}$ and so $\overline{1_{0}} = 1_{0}$ and $\overline{ε_{0}} = ε_{0} .$

(c) If $z = \sum_{μ \in P} c_{μ} x^{μ} \in ℤ {[P]}^{W},$ then, since $c_{μ} \in ℤ,$ $\overline{c_{μ}} = c_{μ}$ and, by (a),

\overline{z} = \sum_{μ \in P} \overline{c_{μ}} \overline{x^{μ}} = \sum_{μ \in P} c_{μ} T_{w_{0}} x^{w_{0} μ} T_{w_{0}}^{- 1} = T_{w_{0}} (\sum_{μ \in P} c_{μ} x^{w_{0} μ}) T_{w_{0}}^{- 1} = T_{w_{0}} z T_{w_{0}}^{- 1},

since $z \in ℤ {[P]}^{W}$ is $W$ -invariant. Finally, since $ℤ {[P]}^{W} \subseteq Z (\tilde{H}),$ $z$ is central, and $\overline{z} = T_{w_{0}} z T_{w_{0}}^{- 1} = z .$

(d) By (a), (b) and the third equality in (2.5),

\begin{matrix} \overline{q^{- ℓ (w_{0})} A_{λ + ρ}} & = & q^{ℓ (w_{0})} \overline{ε_{0} x^{λ + ρ} 1_{0}} = q^{ℓ (w_{0})} ε_{0} T_{w_{0}} x^{w_{0} (λ + ρ)} T_{w_{0}}^{- 1} 1_{0} \\ = & q^{ℓ (w_{0})} {(- q^{- 1})}^{ℓ (w_{0})} ε_{0} x^{w_{0} (λ + ρ)} 1_{0} q^{- ℓ (w_{0})} = {(- q^{- 1})}^{ℓ (w_{0})} A_{w_{0} (λ + ρ)} \\ = & q^{- ℓ (w_{0})} A_{λ + ρ} . \end{matrix}

$□$

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

If $μ \in P$ let $W_{μ}$ be the stabilizer of $μ$ and let $W_{μ} (t)$ be as in (2.8).

Let $μ \in P .$ Let $P_{μ} (x; t)$ be the Macdonald spherical function defined in (2.9) and define $M_{μ} = 1_{0} x^{μ} 1_{0}$ as in (2.4). In the affine Hecke algebra $\tilde{H},$ $\frac{W_{μ} (q^{- 2})}{W_{0} (q^{- 2})} \cdot P_{μ} (x; q^{- 2}) 1_{0} = M_{μ} .$
For $λ \in P^{+}$ let $t_{λ} \in \tilde{W}$ be the translation in $λ$ and let $n_{λ}$ be the maximal length element in the double coset $W t_{λ} W .$ Let $s_{λ}$ be the Weyl character and let $C_{n_{λ}}^{'}$ be the Kazhdan-Lusztig basis element as defined in (2.6) and (1.26), respectively. In the affine Hecke algebra $\tilde{H},$ $q^{- ℓ (w_{0})} W_{0} (q^{2}) \cdot s_{λ} 1_{0} = C_{n_{λ}}^{'} .$

Proof.

(a) By Theorem 2.4 there is an element ${\tilde{P}}_{λ} \in 𝕂 {[P]}^{W}$ such that ${\tilde{P}}_{λ} 1_{0} = 1_{0} x^{λ} 1_{0} .$ To find ${\tilde{P}}_{λ}$ first do a rank 1 calculation,

\begin{matrix} (q^{- 1} + T_{i}) x^{λ} 1_{0} & = & (q^{- 1} x^{λ} + x^{s_{i} λ} T_{i} + (q - q^{- 1}) (\frac{x^{λ} - x^{s_{i} λ}}{1 - x^{- α_{i}}})) 1_{0} \\ = & \frac{1}{1 - x^{- α_{i}}} (\binom{q^{- 1} x^{λ} (1 - x^{- α_{i}}) + q x^{s_{i} λ} (1 - x^{- α_{i}})}{+ q x^{λ} - q x^{s_{i} λ} - q^{- 1} x^{λ} + q^{- 1} x^{s_{i} λ}}) 1_{0} \\ = & {(1 - x^{- α_{i}})}^{- 1} (- q^{- 1} x^{λ - α_{i}} - q x^{s_{i} λ - α_{i}} + q x^{λ} + q^{- 1} x^{s_{i} λ}) 1_{0} \\ = & {(1 - x^{- α_{i}})}^{- 1} (x^{λ} (q - q^{- 1} x^{- α_{i}}) + x^{s_{i} λ} (q^{- 1} - q x^{- α_{i}})) 1_{0} \\ = & (\frac{q - q^{- 1} x^{- α_{i}}}{1 - x^{- α_{i}}} \cdot x^{λ} + \frac{x^{- α_{i}}}{x^{- α_{i}}} \cdot \frac{q^{- 1} x^{α_{i}} - q}{x^{α_{i}} - 1} \cdot x^{s_{i} λ}) 1_{0} \\ = & (1 + s_{i}) (\frac{q - q^{- 1} x^{- α_{i}}}{1 - x^{- α_{i}}} x^{λ}) 1_{0} . \end{matrix}

Since $1_{0}$ is a linear combination of products of $T_{i}$ it can also be written as a linear combination of products of $q^{- 1} + T_{i} .$ Thus $1_{0} x^{λ} 1_{0}$ can be written as a linear combination of terms of the form

(1 + s_{i_{1}}) (\frac{q - q^{- 1} x^{- α_{i_{1}}}}{1 - x^{- α_{i_{1}}}}) \dots (1 + s_{i_{p}}) (\frac{q - q^{- 1} x^{- α_{i_{p}}}}{1 - x^{- α_{i_{p}}}}) x^{λ} 1_{0} .

Thus

1_{0} x^{λ} 1_{0} = {\tilde{P}}_{λ} 1_{0}, where {\tilde{P}}_{λ} = \sum_{w \in W} x^{w λ} w c_{w},

and the $c_{w}$ are some linear combinations of products of terms of the form $(q - q^{- 1} x^{α}) / (1 - x^{α})$ for roots $α \in R .$ Since ${\tilde{P}}_{λ}$ is an element of $𝕂 {[P]}^{W},$

{\tilde{P}}_{λ} = \sum_{w \in W} w (x^{w_{0} λ} w_{0} c_{w_{0}}),

where $w_{0}$ is the longest element of $W .$ The coefficient $w_{0} c_{w_{0}}$ comes from the highest term in the expansion of

1_{0} = \frac{1}{W_{0} (q^{2})} (q^{2 ℓ (w_{0})} T_{w_{0}} + lower terms)

in terms of linear combination of products of the $(q^{- 1} + T_{i}) .$ If $w_{0} = s_{i_{1}} \dots s_{i_{p}}$ is a reduced word for $w_{0}$ then

\begin{matrix} w_{0} c_{w_{0}} & = & \frac{q^{ℓ (w_{0})}}{W_{0} (q^{2})} s_{i_{1}} (\frac{q - q^{- 1} x^{- α_{i_{1}}}}{1 - x^{- α_{i_{1}}}}) \dots s_{i_{p}} (\frac{q - q^{- 1} x^{- α_{i_{p}}}}{1 - x^{- α_{i_{p}}}}) \\ = & \frac{q^{ℓ (w_{0})}}{W_{0} (q^{2})} s_{i_{1}} \dots s_{i_{p}} (\frac{q - q^{- 1} x^{- s_{i_{p}} \dots s_{i_{2}} α_{i_{1}}}}{1 - x^{- s_{i_{p}} \dots s_{i_{2}} α_{i_{1}}}}) (\frac{q - q^{- 1} x^{- s_{i_{p}} \dots s_{i_{3}} α_{i_{2}}}}{1 - x^{- s_{i_{p}} \dots s_{i_{3}} α_{i_{2}}}}) \dots (\frac{q - q^{- 1} x^{- α_{i_{p}}}}{1 - x^{- α_{i_{p}}}}) \\ = & \frac{q^{ℓ (w_{0})}}{W_{0} (q^{2})} w_{0} \prod_{α \in R^{+}} \frac{q - q^{- 1} x^{- α}}{1 - x^{- α}} = \frac{q^{2 ℓ (w_{0})}}{W_{0} (q^{2})} w_{0} \prod_{α \in R^{+}} \frac{q - q^{- 2} x^{- α}}{1 - x^{- α}}, \end{matrix}

by Lemma 1.2 and the fact that $ℓ (w_{0}) = Card (R^{+}) .$ Thus, since $q^{- 2 ℓ (w_{0})} W_{0} (q^{2}) = W_{0} (q^{- 2}),$

{\tilde{P}}_{λ} = \frac{1}{W_{0} (q^{- 2})} \sum_{w \in W} w (x^{λ} \prod_{α \in R^{+}} \frac{1 - q^{- 2} x^{- α}}{1 - x^{- α}}) .

(b) Since $W_{0} (q^{- 2}) = q^{- 2 ℓ (w_{0})} W_{0} (q^{2}),$ Lemma 2.8 gives

\overline{q^{- ℓ (w_{0})} W_{0} (q^{2}) s_{λ} 1_{0}} = q^{ℓ (w_{0})} W_{0} (q^{- 2}) \overline{s_{λ}} 1_{0} = q^{- ℓ (w_{0})} W_{0} (q^{2}) s_{λ} 1_{0} .

By Lemma 2.5(b),

s_{λ} = \sum_{μ \in P^{+}} K_{λ μ} (t) P_{μ} (x; t),

where $K_{λ μ} (t) \in ℤ [t],$ $K_{λ μ} (t) = 0$ unless $μ \leq λ$ and $K_{λ λ} (t) = 1 .$ Thus, by part (a) and Lemma 2.7

\begin{matrix} q^{- ℓ (w_{0})} W_{0} (q^{2}) s_{λ} 1_{0} & = & \sum_{μ \in P^{+}} q^{- ℓ (w_{0})} W_{0} (q^{2}) K_{λ μ} (q^{- 2}) P_{μ} (x; q^{- 2}) 1_{0} \\ = & \sum_{μ \in P^{+}} \sum_{x \in W t_{μ} W} q^{ℓ (x) - ℓ (n_{μ})} K_{λ μ} (q^{- 2}) T_{x}, \end{matrix}

where the polynomials $K_{λ μ} (q^{- 2}) \in ℤ [q^{- 2}]$ are 0 unless $μ \leq λ$ and $K_{λ λ} (q^{- 2}) = 1 .$ Hence $q^{- ℓ (w_{0})} W (q^{2}) s_{λ} 1_{0}$ is a bar invariant element of $\tilde{H}$ such that its expansion in terms of the basis ${T_{w} ∣ w \in \tilde{W}}$ is triangular with coefficient of $T_{n_{λ}}$ equal to 1 and all other coefficients in $q^{- 1} ℤ [q^{- 1}] .$ These are the defining properties (1.26)-(1.27) of $C_{n_{λ}}^{'} .$

$□$

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).

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