Double affine Hecke algebras of General Type

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

Last update: 20 September 2012

Double affine Hecke algebras

In the following, for simplicity of exposition we shall assume that we are not in the special case of [15, (2.16)] where the root system is type Cn and 𝔥 is the (co)root lattice. All our results and proofs are valid in this special case but the definition of the double affine Hecke algebra and the formulas in (2.32) and (2.34) may need some slight modification. See Remark 2.3 for details.

Let R=(R)+ (-(R)+) be the set of coroots and dix parameters cβ, indexed by βR+ d, such that for all wW and βR+ d,

cβ= cwβ. Settβ= qcβand ti=tαi . (2.30)

The double affine Hecke algebra H is the group algebra of the double braid group with the additional relations

Ti2= ( ti12- ti-12 ) Ti+1,fori= 0,1,,n. (2.31)

The double affine Hecke algebra H has bases

{ TwXμ wW,μ 𝔥δ } , { YλTw wW, λ𝔥d } ,


{ qkXμTw Yλ wW0,λ 𝔥,μ 𝔥*,k 1e }

(see [6, Prop. 5.4 and Cor. 5.8]).

In the presence of (2.31) the relations (2.29) are equivalent to

TiYλ= YsiλTi + ( ti12- ti-12 ) Yλ- Ysiλ 1-Y-αi ,fori=0,1, ,n. (2.32)


τi=Ti+ ti-12 (1-ti) 1-Y-αi = (Ti)-1+ ti-12 (1-ti) Y-αi 1-Y-αi . (2.34)

Using that the τi satisfy the braid relations and that

gYλ= Ygλ g,write τwYλ= Ywλ τw,for wW.

Let wW and let w= si1 si be a reduced word for w. For k=1,, let

βk= si si-1 sik+1 sikand tβk= tik, (2.35)

so that the sequence β, β-1,, β1 is the sequence of labels of the hyperplanes crossed by the walk w-1= si si-1 si1. For example, in Type A2, with w= s2 s0 s1 s2 s1 s0 s2 s1 the picture is

1 w-1 𝔥β7 𝔥β5 𝔥β8 𝔥β6 𝔥β4 𝔥β2 𝔥β3 𝔥β1

Let vW. An alcove walk of type i1,,i beginning at v is a sequence of steps, where a step of type j is

z zsj - + z zsj - + z zsj - + z zsj - + positivej–crossing negativej–crossing positivej–fold negativej–fold

Let (v,w) be the set of alcove walks of type w=(i1,,i) beginning at v. For a walk p(v,w) let

f+(p) = { kthe kth step ofp is a positive fold } , f-(p) = { kthe kth step ofp is a negative fold } , (2.36)


end(p)= endpoint ofp(an element ofW ). (2.37)

Let v,wW, let w=si1 w=si be a reduced word for w and let β,, β1 be as defined in (2.35). Then, in H,

Xvτw= p(v,w) Xend(p) ( kf+(p) tβk-1/2 (1-tβk) 1- Y-βk ) ( kf-(p) tβk-1/2 (1-tβk) Y-βk 1- Y-βk ) ,

where the sum is over all alcove walks of type w=(i1,,i) beginning at v.


The proof is by induction on the length of w, the base case being the formulas in (2.34). To do the induction step let p(v,w),

F+(p)= ( kf+(p) tβk-1/2 (1-tβk) 1- Y-βk ) , F-(p)= ( kf-(p) tβk-1/2 (1-tβk) Y-βk 1- Y-βk )

and let

p1,p2 (v,wsj) be the two extensions ofp by a step of typej

(by a crossing and a fold, respectively). Let z=end(p). By induction, a term in Xvτwτj is

Xz F+(p) F-(p) τj = Xz τj (sjF+(p)) (sjF-(p)) = { Xz ( Tj+ tj-1/2 (1-tj) 1-Y-αj ) (sjF+(p)) (sjF-(p)) , if Xzsj= XzTj , Xz ( (Tj)-1 + tj-1/2 (1-tj) Y-αj 1-Y-αj ) (sjF+(p)) (sjF-(p)) , if Xzsj= Xz(Tj)-1 , = Xend(p1) F+(p1) F-(p1) + Xend(p2) F+(p2) F-(p2) .

The last step of p2 is

z zsj - + if Xzsj= XzTj, and z zsj - + if Xzsj= Xz (Tj)-1 ,

In some special cases when the affine root system is nonreduced (see [15, (2.1.6)]) the formulas in (2.32) and (2.34) need modification and the definition of the double affine Hecke algebra may need an additional relation. The most involved of these cases is type (Cn,Cn) (see [15, (1.4.3)]) where the double affine Hecke algebra needs additional parameters u01/2 and un1/2 (in the notation of [15], u01/2=τ0 and un1/2=τn) and additional relations

(T0-u012) (T0+u0-12) =0and (T0-un12) (T0+un-12) ,whereT0 =q1-12 Xε1T0-1, (2.38)

and the formulas for τn and τ0 need to be changed to

τn=Tn+ tn-12 (1-tn) + t0-12 (1-t0) Y-εn 1- Y-2εn =Tn-1+ ( tn-12 (1-tn) + t0-12 (1-t0) Yεn ) Y-2εn 1- Y-2εn (2.39)


τ0 = T0+ un-12 (1-un)+ u--12 (1-u0) q12 Yε1 1-q-1 Y-2ε1 (2.40) = (T0)-1 + ( un-12 (1-un)+ u--12 (1-u0) q12 Y-ε1 ) q-1 Y-2ε1 1-q-1 Y-2ε1 . (2.41)

The statement and proof of the analogue of Theorem 2.2 for this case is the same, except with the factors associated to the 0-folds and n-folds replaced by the rational functions in Y which appear in the expressions of τ0 and τn in Equations (2.40), (2.41), and (2.39).

Notes and References

This page is taken from a paper entitled A combinatorial formula for Macdonald Polynomials by Arun Ram and Martha Yip.

page history