The Satake isomorphism

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

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 Φ1 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=[q,q-1] module".

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

Φ: Z(H)= 𝕂[P]W Φ1 Z(H)10= 10H10 Φ2 ε0H10 f f10 Aρf10 sλ sλ10 Aλ+ρ.


Using the third equality in (2.5),

ε0aλ10= ε0 ( wW (-1)(w) xwλ ) 10=wW (-1)(w) Awλ=W Aλ.

By Proposition 2.3(c) and Theorem 1.4, sλ𝕂[P]W =Z(H), and so

Aρsλ10= 1Wε0 aρ10sλ 10=1W ε0aρsλ 102= 1Wε0 aλ+ρ10= Aλ+ρ.

Since {sλλP+} is a basis of 𝕂[P]W=Z(H) and {Aλ+ρλP+} is a basis of ε0H10, the composite map

Z(H) 10 Z(H)10 10H10 Aρ ε0H10 f f10 f10 Aρf10 sλ sλ10 sλ10 Aλ+ρ.

is a vector space isomorphism.

If μP let

Wμ= {wWwμ=μ} andWμ(t)= wWμt(w). (2.8)

In particular, if μ=0, then W0=W and W0(t) is the polynomial that appears in (2.1).

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

Pμ(x;t)= 1Wμ(t) wWw ( xμαR+ 1-tx-α 1-x-α ) ,forμP. (2.9)

Then mμ=Pμ(x;1) and, using the Weyl denominator formula,

Pμ(x;0) = wWw ( xρxμ xρ αR+ (1-x-α) ) (2.10) = 1aρ wW (-1)(w) wxμ+ρ= aμ+ρaρ= sμ,

and so, conceptually, the spherical functions Pμ(x;t) interpolate between the Schur functions sμ and the monomial symmetric functions mμ.

The double cosets in W\W/W are WtλW, λP+. If λP+ let nλ and mλ be the maximal and minimal length elements of WtλW, respectively. Theorem 2.9 below will show that under the Satake isomorphism the Weyl characters sλ correspond to Kazhdan Lusztig basis elements Cnλ and the polynomials Pμ(x;q-2) correspond to the elements Mμ=10xμ10. More precisely, we have the following diagram:

Φ1: Z(H)= 𝕂[P]W Z(H)10 =10H10 ff10 q-(w0) W0(q2)sλ Cnλ Wμ(q-2) W0(q-2) Pμ(x:q-2) Mμ (2.11)

where w0 is the longest element of W. The following three lemmas (of independent interest) are used in the proof of Theorem 2.9.

Let tα, αR+, be commuting variables indexed by the positive roots. For λP+ let Pλ(x:t) be as in (2.9), Wλ as in (2.8), and define

Rλ(x:tα)= wWw ( xλ αR+ 1-tαx-α 1-x-α )


Wλ(tα)= wWλ ( αR(w) tα ) ,

where, as in (1.5), R(w)= { αR+ wα<0 } is the inversion set of w. Then

  1. Rλ(x:tα)= μP+ uλμ sμ, with uλμ[tα], uλμ=0 unless μλ, and uλλ=Wλ(tα).
  2. Pλ(x:t)= μP+ cλμ sμ, with cλμ[t], cλμ=0 unless μλ, and cλλ=1.


(a) If ER+ let

tE=αEtα andαE=αE α,

and let aμ be as defined in (2.4). Using the Weyl denominator formula, Proposition 2.3(a), and the second equality in (2.5),

Rλ = wWw ( xλαR+ 1-tαx-α 1-x-α ) = wWw ( xλ+ρ αR+ (1-tαx-α) xραR+ (1-x-α) ) = 1aρwW (-1)(w)w ( xλ+ρ αR+ (1-tαx-α) ) = 1aρwW (-1)(w)w ( ER+ (-1)E tExλ+ρ-αE ) = 1aρwW (-1)E tEaλ+ρ-αE =ER+ (-1)E tEsλ-αE,

which shows that Rλ is a symmetric function and uλμ[tα].

By the straightening law for Weyl characters (2.7), sλ-αE=0 or sλ-αE= (-1)(v)sμ with

vWandμP+ such thatμ+ρ= v-1(λ+ρ-αE).

Let Ec denote the complement of E in R+. Since v permutes the elements of R+,

v-1 (λ+ρ-αE) = v-1λ+v-1 ( 12αEc α-12αEα ) = v-1λ+ ( 12αFc α-12αFα ) =v-1λ+ρ-αF,

for some subset FR+ (which could be determined explicitly in terms of E and v). Hence

μ=v-1λ+ρ- αF-ρ=v-1 λ-αFv-1 λλ. (2.12)

This proves that uλμ=0 unless μλ.

In (2.12), μ=λ only if v-1λ=λ and ρ=ρ-αF=v-1(ρ-αE) in which case

ρ-αE=v (12αR+α) =ρ-αR(v)α andE=R(v).


uλλ(tα)= v-1Wλ tR(v).

(b) Set tα=t for all αR+. Applying (a) with λ=0,

R0(x:t)= wWw ( αR+ 1-tx-α 1-x-α ) =W0(t). (2.13)

Let Wλ be a set of minimal length coset representatives of the cosets in W/Wλ. Every element wW can be written uniquely as w=uv with uWλ and vWλ (see [Bou1981, IV §1 Ex. 3]). Let

Z(λ)= { αR+ λ,α=0 } ,

and let Z(λ)c be the complement of Z(λ) in R+. Then vWλ permutes the elements of Z(λ)c among themselves and so

Rλ(x;t) = uWλu ( xλ ( αZ(λ)c 1-tx-α 1-x-α ) vWλv ( αZ(λ) 1-tx-α 1-x-α ) ) = uWλu ( xλ ( αZ(λ)c 1-tx-α 1-x-α ) Wλ(t) ) ,

where the last equality follows from (2.13). Thus there is an element Pλ(x;t)𝔽[P] where 𝔽 is the field of fractions of [tα] such that

Rλ(x;t) Wλ(t) uWλu ( xλ αZ(λ)c 1-tx-α 1-x-α ) = Wλ(t) Pλ(x;t).

Since Rλ is a symmetric polynomial (an element of [t][P]W), Pλ(x;t)𝔽[P]W. Since t only occurs in the numerators of the terms in the sum defining Pλ in fact Pλ is a symmetric polynomial with coefficients in [t]. It follows that all the uλμ appearing in part (a) are divisible by Wλ(t) and

Pλ(x;t)= μPcλμ sμ,where cλμ= 1Wλ(t)uλμ

are polynomials in [t] such that cλλ=1 and cλμ=0 unless μλ.

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

Let ρ and α be as in (1.8) and (1.1), respectively, and let W0(t) be as defined in (2.8).

  1. wWw ( αR+ 1-tx-α 1-x-α ) =W0(t).
  2. αR+ 1-tρ,α+1 1-tρ,α =W0(t).


Part (a) follows from Lemma 2.5 (a) by setting λ=0 and specializing tα=t for all αR+.

(b) Applying the homomorphism

[t±1] [P] [t±1] xλ t-ρ,λ

to both sides of the identity in (a) for the root system R= {ααR} gives

W0(t)= wW αR+ ( 1-tρ,wα+1 1-tρ,wα ) . (2.14)

If wW, w1, and w=si1sip is a reduced word for w then w-1(-αi1)= (si1w)-1 αi1R(w) and so

there is anαR+such that wα=-αi1.


αR+ 1-tρ,wα+1 1-tρ,wα = 1-tρ,-αi1+1 1-tρ,-αi1 αR+ wα-αi1 1-tρ,wα+1 1-tρ,wα = 1-t-1+1 1-t-1 αR+ wα-αi1 1-tρ,wα+1 1-tρ,wα =0.

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

For λP+ let tλW be the translation in λ and let nλ be the maximal length element in the double coset WtλW. Let Mλ=10xλ10, as in (2.4). Then

q-(w0) W0(q2)· W0(q-2) Wλ(q-2) ·Mλ= xWtλW q (x)- (nλ) Tx,

in the affine Hecke algebra H.


Let λP+. Let Wλ=Stab(λ) and let w0 and wλ be the maximal length elements in W and Wλ, respectively. Let mλ and nλ be the minimal and maximal length elements respectively in the double coset WtλW. For each positive root α the hyperplanes Hα,i, 1iλ,α, are between the fundamental alcove A and the alcove tλA and so

(tλ)= αR+ λ,α =2λ,ρ ,whereρ=12 αR+α. (2.15)

Since mλ=tλ(wλw0) and nλ=tw0λw0,

(mλ)= (tλ)- (w0wλ)= (tλ)- ( (w0)- (wλ) ) ,and (nλ)= (tλ)+ (w0)= (mλ)+ (w0)- (wλ)+ (w0). (2.16)

For example, in the setting of Example 1.1, if λ=2ω2 in type C2, then Wλ={1,s1}, wλ=s1, w0=s1s2s1s2, (tλ)=6, (mλ)=3, and (nλ)=10. Labelling the alcove wA by the element w, the 32 alcoves wA with wWtλW 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


10xλ10 = 10Ttλ10 =10 Tmλw0wλ 10=10 TmλTw0wλ 10 = q(w0)-(wλ) 10Tmλ10 = q(w0)-(wλ)-(mλ) W(q2) ( wW q(w)Tw ) q(mλ) Tmλ10.

Let Wλ be a set of minimal length coset representatives of the cosets in W/Wλ. Every element wW has a unique expression w=uv with uWλ and vWλ. If vWλ then

vmλ=vtλwλ w0=tλvwλ w0=mλ (wλw0)-1 vwλw0=mλ ( w0-1 wλ-1 vwλw0 ) .

Since conjugation by wλ and conjugation by w0 are automorphisms of Wλ and W respectively taking simple reflections to simple reflections,

(v)= ( w0-1 wλ-1 vwλw0 ) .


10xλ10 = q (w0)- (wλ)- (mλ) W0(q2) uWλ q(u) Tu vWλ q(v)Tv q(mλ) Tmλ10 = q 2 (w0)-2 (wλ)- (tλ) W0(q2) ( uWλ q(u) Tu q(mλ) Tmλ ) · ( v w0-1 wλ-1 Wλwλw0 q(v)Tv ) 10 = q -2(wλ) -(tλ) W0(q-2) ( uWλ q(u) Tu ) q(mλ) TmλWλ (q2)10 = q -2(wλ) -(tλ) Wλ(q2) W0(q2) W0(q-2) ( uWλ q(u) Tu ) q(mλ) Tmλ ( wW q(w) Tw ) = q-(tλ) Wλ(q-2) W0(q2) W0(q-2) xWtλW q(x)Tx = q-(tλ)+(nλ) Wλ(q-2) W0(q2) W0(q-2) ( xWtλW q(x)-(nλ) Tx ) = q(w0) W0(q2) Wλ(q-2) W0(q-2) ( xWtλW q(x)-(nλ) Tx ) .

Let w0 be the longest element of W and let λP.

  1. xλ= Tw0 xw0λ Tw0-1.
  2. 10= 10and ε0 =ε0.
  3. Ifz [P]Wthen z=z.
  4. q-(w0) Aλ+ρ =q-(w0) Aλ+ρ.


(a) If λP+ then w0tλ= tw0λw0, (w0tλ) =(w0)+ (tλ) and (tw0λw0) =(tw0λ)+ (w0). Thus,

Tw0Ttλ= Tw0tλ= Ttw0λw0 =Ttw0λ Tw0,forλ P+.

Let λP and write λ=μ-ν with μ,νP+. Since -w0μP+ and -w0νP+,

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 1in,

102= 102and Ti10 =Ti-110 =q-110 =q10, ε02= ε02and Tiε0= Ti-1ε0= -qε0= -q-1ε0.

These are the defining properties (before 2.1) of 10 and ε0 and so 10=10 and ε0=ε0.

(c) If z=μP cμxμ [P]W, then, since cμ, cμ=cμ and, by (a),

z=μP cμxμ= μPcμ Tw0xw0μ Tw0-1= Tw0 ( μP cμxw0μ ) Tw0-1= Tw0z Tw0-1,

since z[P]W is W-invariant. Finally, since [P]W Z(H), z is central, and z=Tw0z Tw0-1=z.

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

q-(w0) Aλ+ρ = q(w0) ε0xλ+ρ 10 =q(w0) ε0Tw0 xw0(λ+ρ) Tw0-110 = q(w0) (-q-1)(w0) ε0xw0(λ+ρ) 10q-(w0)= (-q-1)(w0) Aw0(λ+ρ) = q-(w0) Aλ+ρ.

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

If μP let Wμ be the stabilizer of μ and let Wμ(t) be as in (2.8).

  1. Let μP. Let Pμ(x;t) be the Macdonald spherical function defined in (2.9) and define Mμ=10xμ10 as in (2.4). In the affine Hecke algebra H, Wμ(q-2) W0(q-2) ·Pμ (x;q-2)10 =Mμ.
  2. For λP+ let tλW be the translation in λ and let nλ be the maximal length element in the double coset WtλW. Let sλ be the Weyl character and let Cnλ be the Kazhdan-Lusztig basis element as defined in (2.6) and (1.26), respectively. In the affine Hecke algebra H, q-(w0) W0(q2)·sλ 10=Cnλ.


(a) By Theorem 2.4 there is an element Pλ𝕂[P]W such that Pλ10=10xλ10. To find Pλ 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 10 is a linear combination of products of Ti it can also be written as a linear combination of products of q-1+Ti. Thus 10xλ10 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.


10xλ10= Pλ10, wherePλ= wW xwλwcw,

and the cw are some linear combinations of products of terms of the form (q-q-1xα)/ (1-xα) for roots αR. Since Pλ is an element of 𝕂[P]W,

Pλ=wW w ( xw0λw0 cw0 ) ,

where w0 is the longest element of W. The coefficient w0cw0 comes from the highest term in the expansion of

10= 1W0(q2) ( q2(w0) Tw0 +lower terms )

in terms of linear combination of products of the (q-1+Ti). If w0=si1sip is a reduced word for w0 then

w0cw0 = q(w0) W0(q2) si1 ( q-q-1x-αi1 1-x-αi1 ) sip ( q-q-1x-αip 1-x-αip ) = q(w0) W0(q2) si1sip ( q-q-1 x-sipsi2αi1 1- x -sip si2αi1 ) ( q-q-1 x-sipsi3αi2 1- x -sip si3αi2 ) ( q-q-1x-αip 1-x-αip ) = q(w0) W0(q2) w0 αR+ q-q-1x-α 1-x-α = q2(w0) W0(q2) w0 αR+ q-q-2x-α 1-x-α ,

by Lemma 1.2 and the fact that (w0)=Card(R+). Thus, since q-2(w0) W0(q2)=W0 (q-2),

Pλ= 1W0(q-2) wWw ( xλαR+ 1-q-2x-α 1-x-α ) .

(b) Since W0(q-2)= q-2(w0) W0(q2), Lemma 2.8 gives

q-(w0) W0(q2)sλ 10 = q(w0)W0 (q-2)sλ 10=q-(w0) W0(q2)sλ10.

By Lemma 2.5(b),

sλ=μP+ Kλμ(t) Pμ(x;t),

where Kλμ(t) [t], Kλμ(t)=0 unless μλ and Kλλ(t)=1. Thus, by part (a) and Lemma 2.7

q-(w0)W0 (q2)sλ10 = μP+ q-(w0) W0(q2) Kλμ (q-2)Pμ (x;q-2)10 = μP+ xWtμW q(x)-(nμ) Kλμ(q-2) Tx,

where the polynomials Kλμ(q-2) [q-2] are 0 unless μλ and Kλλ(q-2)=1. Hence q-(w0)W (q2)sλ10 is a bar invariant element of H such that its expansion in terms of the basis {TwwW} is triangular with coefficient of Tnλ equal to 1 and all other coefficients in q-1 [q-1]. These are the defining properties (1.26)-(1.27) of Cnλ.


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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