Double affine Hecke algebras of General Type

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 ${C}_{n}$ and ${𝔥}_{ℤ}$ 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 ${R}^{\vee }={\left({R}^{\vee }\right)}^{+}\cup \left(-{\left({R}^{\vee }\right)}^{+}\right)$ be the set of coroots and dix parameters ${c}_{{\beta }^{\vee }},$ indexed by ${\beta }^{\vee }\in {R}^{\vee }+ℤd,$ such that for all $w\in W$ and ${\beta }^{\vee }\in {R}^{\vee }+ℤd,$

$cβ∨= cwβ∨. Settβ∨= qcβ∨and ti=tαi∨ . (2.30)$

The double affine Hecke algebra $\stackrel{\sim }{H}$ is the group algebra $ℂ\stackrel{\sim }{ℬ}$ 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 $\stackrel{\sim }{H}$ has bases

${ TwXμ∣ w∈W,μ∈ 𝔥ℤ∨⊕ℤδ } , { Yλ∨Tw∨ ∣w∈W∨, λ∨∈𝔥ℤ⊕ℤd } ,$

and

${ qkXμTw Yλ∨∣ w∈W0,λ∨ ∈𝔥ℤ,μ∈ 𝔥ℤ*,k∈ 1eℤ }$

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

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

$Ti∨Yλ∨= Ysiλ∨Ti∨ + ( ti12- ti-12 ) Yλ∨- Ysiλ∨ 1-Y-αi∨ ,fori=0,1, …,n. (2.32)$

where

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

Using that the ${\tau }_{i}^{\vee }$ satisfy the braid relations and that

$g∨Yλ∨= Yg∨λ∨ g∨,write τw∨Yλ∨= Ywλ∨ τw∨,for w∈W.$

Let $w\in W$ and let $w={s}_{{i}_{1}}^{\vee }\dots {s}_{{i}_{\ell }}^{\vee }$ be a reduced word for $w.$ For $k=1,\dots ,\ell$ let

$βk∨= siℓ∨ siℓ-1∨ … sik+1∨ sik∨and tβk∨= tik, (2.35)$

so that the sequence ${\beta }_{\ell }^{\vee },{\beta }_{\ell -1}^{\vee },\dots ,{\beta }_{1}^{\vee }$ is the sequence of labels of the hyperplanes crossed by the walk ${w}^{-1}={s}_{{i}_{\ell }}^{\vee }{s}_{{i}_{\ell -1}}^{\vee }\dots {s}_{{i}_{1}}^{\vee }\text{.}$ For example, in Type ${A}_{2},$ with $w={s}_{2}^{\vee }{s}_{0}^{\vee }{s}_{1}^{\vee }{s}_{2}^{\vee }{s}_{1}^{\vee }{s}_{0}^{\vee }{s}_{2}^{\vee }{s}_{1}^{\vee }$ the picture is

$1 w-1 𝔥β7∨ 𝔥β5∨ 𝔥β8∨ 𝔥β6∨ 𝔥β4∨ 𝔥β2∨ 𝔥β3∨ 𝔥β1∨$

Let $v\in {W}^{\vee }\text{.}$ An alcove walk of type ${i}_{1},\dots ,{i}_{\ell }$ 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 $ℬ\left(v,\stackrel{\to }{w}\right)$ be the set of alcove walks of type $\stackrel{\to }{w}=\left({i}_{1},\dots ,{i}_{\ell }\right)$ beginning at $v.$ For a walk $p\in ℬ\left(v,\stackrel{\to }{w}\right)$ let

$f+(p) = { k∣the kth step ofp is a positive fold } , f-(p) = { k∣the kth step ofp is a negative fold } , (2.36)$

and

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

Let $v,w\in W,$ let $w={s}_{{i}_{1}}^{\vee }\dots w={s}_{{i}_{\ell }}^{\vee }$ be a reduced word for $w$ and let ${\beta }_{\ell }^{\vee },\dots ,{\beta }_{1}^{\vee }$ be as defined in (2.35). Then, in $\stackrel{\sim }{H},$

$Xvτw∨= ∑p∈ℬ(v,w→) Xend(p) ( ∏k∈f+(p) tβk∨-1/2 (1-tβk∨) 1- Y-βk∨ ) ( ∏k∈f-(p) tβk∨-1/2 (1-tβk∨) Y-βk∨ 1- Y-βk∨ ) ,$

where the sum is over all alcove walks of type $\stackrel{\to }{w}=\left({i}_{1},\dots ,{i}_{\ell }\right)$ beginning at $v\text{.}$

 Proof. 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 $p\in ℬ\left(v,\stackrel{\to }{w}\right),$ $F+(p)= ( ∏k∈f+(p) tβk∨-1/2 (1-tβk∨) 1- Y-βk∨ ) , F-(p)= ( ∏k∈f-(p) tβk∨-1/2 (1-tβk∨) Y-βk∨ 1- Y-βk∨ )$ and let $p1,p2∈ℬ (v,w→sj) be the two extensions ofp by a step of typej$ (by a crossing and a fold, respectively). Let $z=\text{end}\phantom{\rule{0.2em}{0ex}}\left(p\right)\text{.}$ By induction, a term in ${X}^{v}{\tau }_{w}^{\vee }{\tau }_{j}^{\vee }$ 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 ${p}_{2}$ is $z zsj - + if Xzsj= XzTj∨, and z zsj - + if Xzsj= Xz (Tj∨)-1 ,$ $\square$

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 $\left({C}_{n}^{\vee },{C}_{n}\right)$ (see [15, (1.4.3)]) where the double affine Hecke algebra needs additional parameters ${u}_{0}^{1/2}$ and ${u}_{n}^{1/2}$ (in the notation of [15], ${u}_{0}^{1/2}={\tau }_{0}^{\prime }$ and ${u}_{n}^{1/2}={\tau }_{n}^{\prime }$) 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 ${\tau }_{n}^{\vee }$ and ${\tau }_{0}^{\vee }$ need to be changed to

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

and

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

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 ${\tau }_{0}^{\vee }$ and ${\tau }_{n}^{\vee }$ 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.