Intertwining Operators of Double Affine Hecke Algebras

a rXiv:q -alg/96514v511J u l1996June (1996),1–36Dedicated to Yuri I.Manin on the occasion of his 60birthdayIntertwining operatorsof double affine Hecke algebras By Ivan Cherednik*Continuing [C3,C4],we study the intertwining operators of double affine Hecke algebras H .They appeared in several papers (especially in [C2,C4,C6]).However for the first time here we apply them systematically to create the non-symmetric [M3,C4]and symmetric [M2]Macdonald polynomials for arbitrary root systems and to start the theory of induced and co-spherical H -modules.The importance of this technique was clearly demonstrated in recent pa-pers by F.Knop and S.Sahi [Kn],[KS],[S].Using the intertwiners of the double affine Hecke algebras in the case of GL (dual to those considered in [C1,C2])they proved the q,t -integrality conjecture by I.Macdonald [M1]and managed to establish the positivity of the coefficients of the Macdonald polynomials in the differential case.As to the integrality,we mention another approach based on the so-called Vinet operators (see [LV]and a recent work by Kirillov,Noumi),and the results by Garsia,Remmel,and Tesler.We do not try in this paper to get the best possible estimates for the denominators of the Macdonald polynomials (generally speaking,the problem looks more complicated than in the stable GL -case).However even rather straightforward analysis of the intertwiners gives a lot.For instance,it is enough to ensure the existence of the restricted Macdonald polynomials at roots of unity from [C3,C4],where we used less convenient methods based directly on the definition or on the recurrence relations.The technique of intertwiners combined with the (projective)action of GL (2,Z )from [C3]gives another proof of the norm and the evaluation formulas (see [C4]).Here the H -embedding of the space of nonsymmetricpolynomialsinto the space of functions on the affine Weyl group ˜W ([C4],Proposition 5.2)plays a key role.The latter representation when restricted to the affine Hecke subalgebra turns into the classical one from [IM]as t is a power of p and q →0(˜W is identified with the set of double cosets of the corresponding p -adic group with respect to the Iwahori subroup).2IV AN CHEREDNIKAnother important application is a calculation of the Fourier transforms of the Macdonald polynomials in the sense of[C3,C4].For instance,it gives a canonical identification of the polynomial representation of the affine Hecke algebra with the representation in functions on the weight lattice(which col-lapses in the p-adic limit).We introduce a proper discretization of theµ-function(the truncated theta-function making Macdonald’s polynomials pairwise orthogonal)and the corresponding discrete inner product on Funct(˜W).It readily gives the pro-portionality of the norms of the Macdonald polynomials[M2,C2,M3,C4]and those defined for the Jackson integral taken instead of the constant term in the inner product.The coefficient of proportionality is described by the Aomoto conjecture(see[A,Ito])recently proved by Macdonald(to calculate it one can also follow[C2],replacing the shift operators by their discretizations).We note that the Macdonald polynomials considered as functions on˜W are square integrable forfinitely many weights only.Here|q|=1and the real partℜ(k)for t=q k is to be negative(otherwise we have none).The program is to describe all integrable and non-integrable eigenfunctions of the discrete Dunkl operators in this representation and to study the corresponding Fourier transform.In contrast to the classical p-adic harmonic analysis(see e.g.[HO])the Plancherel measure coincides with the discretization ofµ(the Fourier transform is self-dual).More generally,we consider the action of the double affine Hecke algebra in the same space Funct(˜W)depending on an arbitrary given weight.Its sub-module generated by the delta-functions is induced(from a character of the standard polynomial subalgebra)and co-spherical.Mainly following[C5],we find out when arbitrary induced representations(in the same sense)are irre-ducible and co-spherical using the technique of intertwiners.The answer is a natural”affinization”of the well-known statements in the p-adic case(see e.g.[KL],[C5]).The classification of co-spherical representations is impor-tant for the harmonic analysis and plays the key role in the theory of affine Knizhnik-Zamolodchikov equations(see[C6,C7,C8]).We also induce up irre-ducible representations of affine Hecke subalgebras([C6]is devoted to appli-cations of such representations).If q is sufficiently general the H-modules we get are irreducible,so one can use the classification of[KL].Thus in this paper we begin a systematic study of the representations of double affine Hecke algebras and related harmonic analysis.The polyno-mial representation considered in the series of papers[C2-4]devoted to the Macdonald conjectures is remarkable,but still just an example.The paper was started during my stay at RIMS(Kyoto University),con-tinued at CRM in Montreal,and completed at the University of Nijmegen.I am grateful to T.Miwa,L.Vinet,G.Heckman and my colleagues at theseINTERTWINERS OF DOUBLE HECKE ALGEBRAS3 institutes for the kind invitations and the hospitality.The author thanks E. Frenkel,G.Heckman,D.Kazhdan,I.Macdonald,and E.Opdam for useful discussions.1.Affine Weyl groupsLet R={α}⊂R n be a root system of type A,B,...,F,G with respect to a euclidean form(z,z′)on R n∋z,z′,normalized by the standard condition that(α,α)=2for longα.Let usfix the set R+of positive roots(R−=−R+),the corresponding simple rootsα1,...,αn,and their dual counterparts a1,...,a n,a i=α∨i,whereα∨=2α/(α,α).The dual fundamental weights b1,...,b n are determined from the relations(b i,αj)=δj i for the Kronecker delta.We will also use the dual root system R∨={α∨,α∈R},R∨+,and the latticesA=⊕n i=1Z a i⊂B=⊕n i=1Z b i,A±,B±for Z±={m∈Z,±m≥0}instead of Z.(In the standard notations, A=Q∨,B=P∨-see[B].)Later on,να=να∨=(α,α),νi=ναi,νR={να,α∈R}⊂{2,1,2/3}.(1.1)ρν=(1/2) να=να=(ν/2)νi=νb i,forα∈R+.The vectors˜α=[α,k]∈R n×R⊂R n+1forα∈R,k∈Z form the affine root system R a⊃R(z∈R n are identified with[z,0]).We addα0def=[−θ,1] to the simple roots for the maximal rootθ∈R.The corresponding set R a+of positive roots coincides with R+∪{[α,k],α∈R,k>0}.We denote the Dynkin diagram and its affine completion with{αj,0≤j≤n}as the vertices byΓandΓa(m ij=2,3,4,6ifαi andαj are joined by0,1,2,3laces respectively).The set of the indices of the images ofα0by all the automorphisms ofΓa will be denoted by O(O={0}for E8,F4,G2). Let O∗=r∈O,r=0.The elements b r for r∈O∗are the so-called minuscule weights((b r,α)≤1forα∈R+).Given˜α=[α,k]∈R a,b∈B,lets˜α(˜z)=˜z−(z,α∨)˜α,b′(˜z)=[z,ζ−(z,b)](1.2)for˜z=[z,ζ]∈R n+1.The affine Weyl group W a is generated by all s˜α(simple reflections s j= sαjfor0≤j≤n are enough).It is the semi-direct product W⋉A,where the non-affine Weyl group W is the span of sα,α∈R+.Here and futher we identify b∈B with the corresponding translations.For instance,a=sαs[α,1]=s[−α,1]sαfor a=α∨,α∈R.(1.3)4IV AN CHEREDNIKThe extended Weyl group W b generated by W and B is isomorphic to W⋉B:(wb)([z,ζ])=[w(z),ζ−(z,b)]for w∈W,b∈B.(1.4)Given b+∈B+,letωb+=w0w+0∈W,πb+=b+(ωb+)−1∈W b,ωi=ωbi,πi=πbi,(1.5)where w0(respectively,w+0)is the longest element in W(respectively,in W b+ generated by s i preserving b+)relative to the set of generators{s i}for i>0.The elementsπr def=πbr,r∈O leaveΓa invariant and form a group denoted byΠ,which is isomorphic to B/A by the natural projection{b r→πr}.As to{ωr},they preserve the set{−θ,αi,i>0}.The relationsπr(α0)=αr= (ωr)−1(−θ)distinguish the indices r∈O∗.Moreover(see e.g.[C2]): W b=Π⋉W a,whereπr s iπ−1r=s j ifπr(αi)=αj,0≤j≤n. (1.6)Givenν∈νR,r∈O∗,˜w∈W a,and a reduced decomposition˜w=s jl ...s j2s j1with respect to{s j,0≤j≤n},we call l=l(ˆw)the length ofˆw=πr˜w∈W b.Setting(1.7)λ(ˆw)={˜α1=αj1,˜α2=s j1(αj2),˜α3=s j1s j2(αj3),......,˜αl=˜w−1s jl(αjl)},one can represent(1.8)l=|λ(ˆw)|= νlν,for lν=lν(ˆw)=|λν(ˆw)|,λν(ˆw)={˜αm,ν(˜αm)=ν(˜αjm)=ν},1≤m≤l, where||denotes the number of elements,ν([α,k])def=να.Let us introduce the following affine action of W b on z∈R n:(1.9)(wb) z =w(b+z),w∈W,b∈B,s˜α z =z−((z,α)+k)α∨,˜α=[α,k]∈R a,and the pairing([z,ζ],z′+d)def=(z,z′)+ζ,where we treat d formally(see e.g. [K]).The connection with(1.2,1.3)is as follows:(1.10)(ˆw([z,ζ]),ˆw z′ +d)=([z,ζ],z′+d)forˆw∈W b.Using the affine Weyl chamberC a=nj=0Lαj,L˜α={z∈R n,(z,α)+k>0},INTERTWINERS OF DOUBLE HECKE ALGEBRAS5(1.11)λν(ˆw)={˜α∈R a+, C a ⊂ˆw L˜α ,ν(˜α)=ν} ={˜α∈R a+,lν(ˆws˜α)<lν(ˆw)}.It coincides with(1.8)due to the relations(1.12)λν(ˆwˆu)=λν(ˆu)∪ˆu−1(λν(ˆw)),λν(ˆw−1)=−ˆw(λν(ˆw)) if lν(ˆwˆu)=lν(ˆw)+lν(ˆu).The following proposition is from[C4].Proposition1.1.Given b∈B,the decomposition b=πbωb,ωb∈W can be uniquely determined from the following equivalent conditionsi)l(πb)+l(ωb)=l(b)and l(ωb)is the biggest possible,ii)ωb(b)=b−∈B−and l(ωb)is the smallest possible,iii)πb 0 =b andλ(πb)∩R=∅.We will also use thatλ(b)={˜α,(b,α)>k≥0ifα∈R+,(1.13)(b,α)≥k>0ifα∈R−},λ(πb)={˜α,α∈R−,(b−,α)>k>0if(α,b)<0,(1.14)(b−,α)≥k>0if(α,b)>0},andλ(π−1b)={˜α,−(b,α)>k≥0}for˜α=[α,k]∈R a+.(1.15)Convexity.Let us introduce two orderings on B.Here and further b±are the unique elements from B±which belong to the orbit W(b).Namely, b−=ωbπb=ωb(b),b+=w0(b−)=ω−b(b).So the equality c−=b−(or c+=b+)means that b,c belong to the same orbit.Setb≤c,c≥b for b,c∈B if c−b∈A+,(1.16)b c,c b if b−<c−or b−=c−and b≤c.(1.17)We use<,>,≺,≻respectively if b=c.For instance,c≻b+⇔b+>W(c)>b−,c b−⇔c∈W(b−)or c≻b+.The following sets(1.18)σ(b)def={c∈B,c b},σ∗(b)def={c∈B,c≻b},σ+(b)def={c∈B,c−>b−}=σ∗(b+).are convex.Moreoverσ+is W-invariant.By convex,we mean that if c,d= c+rα∨∈σforα∈R+,r∈Z+,then{c,c+α∨,...,c+(r−1)α∨,d}⊂σ.(1.19)6IV AN CHEREDNIKThe elements fromσ(b)strictly between c and d(i.e.c+qα,0<q<r) belong toσ+(b).πb,where i p are from any se-Proposition 1.2.a)Letˆu=s˜αi m...s˜αi1quence1≤i1<i2<...<i m≤l=l(b)in a reduced decomposition of ˆw=π−1b(see(1.7)).In other words,ˆu is obtained by crossing out any number of{s j}from a reduced decomposition ofπb.Then c def=ˆu 0 ∈σ∗(b).More-over,c∈σ+(b)if and only if at least one of˜αi p=[α,k]for1≤p≤m has k>0.b)If c,b belong to the same W-orbit then the converse is ly, settingωbc def=πbπ−1c,the following relations are equivalent:(i)c≻b(which means that c>b),(ii)(α,c)>0for allα∈λ(ωbc),(iii)l(πb)=l(ωbc)+l(πc),It also results from(i)thatωbc is the smallest possible element w∈W such that b=w(c).Proof.Assertion a)is a variant of Proposition1.2from[C4].For the sake of completeness we will outline the proof of b).Taking u(c)≤b<c,we will check(ii),(iii)by induction supposing that{u′(c)≤b′<c}⇒{(ii),(iii)}for all b′,u′such that l(u′)<l(u),which is obvious when l(u′)=0.Settingβ=u(α)forα∈λ(u),u(sα(c))=u(c)−(α,c)β∨andβ∈R−(see the definition ofλ(α)).One can assume that(α,c)>0for all suchα. Otherwise usα(c)≤u(c)≤c and we can argue by induction.Applying(1.12) and(1.13),we see that l(uc)=l(u)+l(c).Indeed,the intersection ofλ(c)andc−1(λ(u))={[α,(c,α)],α∈λ(u)}is empty.Hence the product uπc is reduced(i.e.l(uπc)=l(u)+l(πc))and λ(uπc)=ωc c−1(λ(u))∪λ(πc)contains no roots from R+.Finally,Proposition 1.1leads to(iii)(and the uniqueness of u of the minimal possible length).This reasoning gives the equivalence of(ii)and(iii)as well.Assertion(i)readily results from(ii).We will also use(cf.Proposition5.2,[C4])the relationsπb=πrπc for b=πr c and any c∈B,r∈O and the equivalence of the following three conditions:(1.20)(αj,c+d)>0⇔αj∈λ(π−1c)⇔{s jπc=πb,c≻b}for0≤j≤n.When j>0it is a particular case of Proposition1.2b). Assuming that(α0,c+d)=1−(θ,c)>0,b=s0 c =c+(α0,c+d)θ>c>c−θ=sθ(b).INTERTWINERS OF DOUBLE HECKE ALGEBRAS7 Hence c∈σ+(b).If the product s0πc is reducible then we can apply statement a)to come to a contradiction.Therefore s0πc=πb,since s0is simple.The remaining implications are obvious.2.Intertwining operatorsWe put m=2for D2k and C2k+1,m=1for C2k,B k,otherwise m=|Π|.Let us sett˜α=tν(˜α),t j=tαj,where˜α∈R a,0≤j≤n,X˜b =ni=1X k i i q k if˜b=[b,k],(2.1)for b=ni=1k i b i∈B,k∈18IV AN CHEREDNIKdoes not depend on the choice of the reduced decomposition(because{T} satisfy the same“braid”relations as{s}do).Moreover,Tˆv Tˆw=Tˆvˆw whenever l(ˆvˆw)=l(ˆv)+l(ˆw)forˆv,ˆw∈W b.(2.3)In particular,we arrive at the pairwise commutative elementsY b=ni=1Y k i i if b=ni=1k i b i∈B,where Y i def=T bi,(2.4)satisfying the relations(2.5)T−1iY b T−1i=Y b Y−1a iif(b,αi)=1,T i Y b=Y b T i if(b,αi)=0,1≤i≤n.The following maps can be extended to involutions of H(see[C1,C3]):ε:X i→Y i,Y i→X i,T i→T−1i ,(2.6)tν→t−1ν,q→q−1,τ:X b→X b,Y r→X r Y r q−(b r,b r)/2,Yθ→X−10T−20Yθ,(2.7)T i→T i,tν→tν,q→q,1≤i≤n,r∈O∗,X0=qX−1θ.Let us give some explicit formulas:(2.8)ε(T0)=XθT−1Yθ=XθT sθ,ε(πr)=X r Tω−1r,τ(T0)=X−10T−1,τ(πr)=q−(b r,b r)/2X rπr=q(b r,b r)/2πr X−1r∗,πr X r∗π−1r=q(b r,b r)X−1r,X r∗TωrX r=T−1ωr∗.Theorem2.3from[C3]says that the map(2.9) 0−1−10 →ε, 1101→τcan be extended to a homomorphism of GL2(Z)up to conjugations by the central elements from the group generated by T1,...,T n.The involutionη=τ−1ετcorresponding to the matrix −1011 will play an important role in the paper:η:X r→q(b r,b r)/2X−1r Y r=πr X r∗Tωr ,(2.10)Y r→q(b r,b r)/2X−1r Y r X r=πr T−1ωr∗,Yθ→T−10T−1sθ,T j→T−1j(0≤j≤n),πr→πr(r∈O∗),tν→t−1ν,q→q−1.INTERTWINERS OF DOUBLE HECKE ALGEBRAS9 We note thatεandηcommute with the main anti-involution∗from[C2]:(2.11)X∗i=X−1i,Y∗i=Y−1i,T∗i=T−1i,tν→t−1ν,q→q−1,0≤i≤n,(AB)∗=B∗A∗.The X-intertwiners(see e.g.[C2,C5,C6])are introduced as follows:Φj=T j+(t1/2j −t−1/2j)(X aj−1)−1,G j=Φj(φj)−1,˜G j=(φj)−1Φj,(2.12)φj=t1/2j +(t1/2j−t−1/2j)(X aj−1)−1,for0≤j≤n.They belong to the extension of H by thefield C q,t(X)of rational functions in{X}.The elements G j and G′j satisfy the same relations as{s j,πr}do,{Φj}satisfy the relations for{T j}(i.e.the homogeneous Coxeter relations and those withπr).Hence the elements(2.13)Gˆw=πr G jl ···G j1,whereˆw=πr s jl···s j1∈W b,are well-defined and G is a homomorphism of W b.The same holds for˜G.AstoΦ,the decomposition ofˆw should be reduced.The simplest way to see this is to use the following property of{Φ}which fixes them uniquely up to left or right multiplications by functions of X:Φˆw X b=Xˆw(b)Φˆw,ˆw∈W b.(2.14)Onefirst checks(2.14)for s j andπr,then observes thatΦfrom(2.13)satisfy (2.14)for any choice of the reduced decomposition,and uses the normalizing conditions to see that they are uniquely determined from the intertwining relations(2.14).We note thatΦj,φj are self-adjoint with respect to the anti-involution (2.11).HenceΦ∗ˆw=Φˆw−1,G∗ˆw=˜Gˆw−1,ˆw∈W b.(2.15)It follows from the quadratic relations for T.To define the Y-intertwiners we apply the involutionεtoΦˆw and to G,˜G. The formulas can be easily calculated using(2.8).In the case of GL n one getsthe intertwiners from[Kn].For w∈W,we just need to replace X b by Y−1b and conjugate q,t(cf.[C4]).However it will be more convenient to consider η(Φ)instead ofε(Φ)to create the Macdonald polynomials.Both constructions gives the intertwiners satisfying the∗-relations from(2.15).3.Standard representationsIt was observed in[C4],Section5that there is a natural passage from the representation of H in polynomials to a representation in functions on10IV AN CHEREDNIKW b.We will continue this line,beginning with the construction of the basic representaions of level0,1.Settingx˜b =ni=1x k i i q k if˜b=[b,k],b=ni=1k i b i∈B,k∈1representation V0is induced from the character{T j→t j,πr→1}.Namely, the imageˆH is uniquely determined from the following condition:(3.7)ˆH(f(x))=g(x)for H∈H if Hf(X)−g(X)∈ni=0H(T i−t i)+ r∈O∗H(πr−1).To make the statement about V1quite obvious let us introduce the Gauss-ianγ=Const qΣn i=1z i zαi/2,where formallyx b=q z b,z a+b=z a+z b,z i=z bi,(wa)(z b)=z w(b)−(a,b),a,b∈R n. More exactly,it is a W-invariant solution of the following difference equations:(3.8)b j(γ)=Const q(1/2)Σn i=1(z i−(b j,b i))(zαi−δj i)=q−z j+(b j,b j)/2γ=q(b j,b j)/2x−1jγfor1≤j≤n.The Gaussian commutes with T j for1≤j≤n because it is W-invariant.A straightforward calculation gives that(3.9)γ(X)T0γ(X)−1=X−10T−1=τ(T0),γ(X)Y rγ(X)−1=q−(b r,b r)/2X r Y r=τ(Y r),r∈O.Hence the conjugation byγinducesτ.We can put in the following way.There is a formal H-homomorphism:(3.10)V0∋v→ˆv def=vγ−1∈V1.One has to complete V0,1to make this map well-defined(see the discrete rep-resentations below).We will later need an extended version of Proposition3.6from[C2].Proposition3.2.a)The operators{Y i,1≤i≤n}acting in V0preserve Σ(b)def=⊕c∈σ(b)C q,t x c andΣ∗(b)(defined forσ∗(b))for arbitrary b∈B.b)The operators{T j,0≤j≤n}acting in V0preserveΣ+(b)=Σ∗(b+):(3.11)ˆTj(x b)modΣ+(b)=t1/2jx b if(b,αj)=0,=t1/2js j(x b)+(t1/2j−t−1/2j)x b if(b,αj)<0,=t−1/2js j(x b)if(b,αj)>0.c)Coming to V1,if(αj,b+d)>0(0≤j≤n)then(3.12)ˆT j x b modΣ+(s j b )=t−1/2js j x b .Otherwise,(αj ,b +d )≤0and(3.13)ˆTj x b ∈Σ(b )for (αj ,b +d )≤0,ˆT j x b =t 1/2jx b if (αj ,b +d )=0.Proof.Due to Proposition 3.3from [C4]it suffices to check c)for j =0.The first inequality,the definition of ˆT 0 x b = c ∈B u bc x c ,and (1.20)readily give that (for nonzero u )c =b +rθ(r ∈Z )and(3.14)s θ(b ′)=b −θ<b <c ≤b +(α0,b +d )θ=s 0 b def =b ′.Hence c ∈σ+(b ′)if c =b ′.The coefficient u bb ′equals t −1/20.Let (α0,b +d )≤0.Then(3.15)s θ(b )=b −(b,θ)θ<c ≤b and c ∈{Σ+(b )∪b }∈Σ(b )(cf.Proposition 1.2,a)).Discretization.We go to the lattice version of the functions and opera-tors.Let ξbe a ”generic”character of C [x ]:x a (ξ)def =n i =1ξk i i ,a =n i =1k i b i ∈B,for independent parameters ξi .The discretizations of functions g (x )in x ∈C n and the operators from the algebra A def =⊕ˆu ∈W b C q,t (X )ˆu ,are described by the formulas:δx a (bw )=x a (q b w (ξ))=q (a,b )x w −1(a )(ξ),(δˆu (δg ))(bw )=δg (ˆu −1bw ).(3.16)For instance,(δX a (δg ))(bw )=x a (bw )g (bw )(we will sometimes omitδand put g (ˆw )instead of δg (ˆw )).The image of g ∈C q,t (x )belongs to the space F ξdef=Funct(W b ,C ξ)of C ξ-valued functions on W b ,where C ξdef =C q,t (ξ1,...,ξn ).Considering the discretizations of operators ˆHfor H ∈H we come to the functional representation of H in F ξ.Similarly,introducing the group algebra C ξ[W b ]=⊕ˆw ∈W b C ξδˆwfor (for-mal)delta-functions ,we can consider the dual anti-action on the indices:(3.17)δ(g (x )ˆu )( ˆw ∈W b c ˆw δˆw )= ˆw ∈W bc ˆw g (ˆw )δˆu −1ˆw ,c ˆw ∈C ξ.Composing it with the anti-involution of H(3.18)T ⋄j =T j (0≤j ≤n ),π⋄r =π−1r (r ∈O ),X ⋄i =X i (0≤i ≤n ),sending q,t to q,t(and AB to B⋄A⋄),we get the delta-representation∆ξof H in Cξ[W b]:(3.19)H→δ(ˆH⋄)def=δ(H)for H∈H.Explicitly,δπr=πr=δ(πr),r∈O,and forˆw=bwδ(Ti(g))(ˆw))=t1/2ix ai(w(ξ))q(a i,b)−t−1/2ix ai(w(ξ))q(a i,b)−1g(ˆw)for0≤i≤n,(3.20)δ(T i)(δˆw)=t1/2ix ai(w(ξ))q(a i,b)−t−1/2x ai(w(ξ))q(a i,b)−1δˆw for0≤i≤n.(3.21)There is a natural Cξ-linear pairing between Fξand∆ξ.Given g∈Funct(W b,Cξ),ˆw∈W b,{g,δˆw}def=g(ˆw),{H(g),δˆw}={g,H⋄(δˆw)},H∈H.(3.22)It also gives a nondegenerate pairing between V0and∆ξ.For arbitrary ope-rators A∈A,the relation is as follows:{δA(g),δˆw}={g,δA(δˆw)}.Let us extend the discretization map and the pairing with∆ξto V1.We use the map from(3.10)for theδ-Gaussian:(3.23)δγ(bw)def=q(b,b)/2xb(w(ξ)),which satisfies(3.8)and is a discretization ofγfor a proper constant(cf.[C4], (6.20)).The representations Fξand∆ξcan be introduced when q,t,{ξi}are con-sidered as complex numbers ensuring that x˜a(ξ)=1for all˜a∈(R a)∨.Follow-ing Proposition5.2from[C4],let us specialize the definition of∆forξ=t−ρ. In this casex a(bw)=x a(q b t−w(ρ))=q(a,b) νt−(w(ρν),a)ν.(3.24)Proposition3.3.The H-module∆(−ρ)def=∆t−ρcontains the H-sub-module∆#def=⊕b∈B Cδπb.This also holds for any q∈C∗and generic t. Moreover,∆#is irreducible if and only if q is not a root of unity.When q→0and t is a power of prime p the action of the algebra H a generated by{T j,0≤j≤n}in∆(−ρ)coincides with the standard action ofthe p-adic Hecke algebra H(G//I)∼=H a on the(linear span of)delta-functions on I\G/I∼=W a.Here I is the Iwahori subgroup of the split semisimple p-adic group G(see[IM]).However∆#does not remain a submodule in this limit.Multiplying the delta-functions on the right by the operator of t-symmet-rization we can get an H a-submodule isomorphic to∆#(upon the restriction to H a).Its limit readily exists and coincides with the space of delta-functions on I\G/K for the maximal parahoric subgroup K.However the latter space can be identified with neither spaces of delta-functions for smaller subsets of W a(as in Proposition3.3).It is possible only for the q-deformation under consideration.Practically,when calculating with right K-invariant functions in the p-adic case one needs to consider their values on the whole W a(that is an obviousflaw since much fewer number of points is enough to reconstruct them uniquely).4.OrthogonalityThe coefficient of x0=1(the constant term)of a polynomilal f∈C q,t[x] will be denoted by f 0.Let(4.1)µ= a∈R∨+∞ i=0(1−x a q i a)(1−x−1a q i+1a).(1−x a(tρ)t a q i a)(1−x a(tρ)t−1a q i a)Here x b(t±ρq c)=q(b,c) νt±(b,ρν)ν.We note thatµ∗0=µ0with respect to the involutionx∗b=x−b,t∗=t−1,q∗=q−1.Setting(4.3)f,g 0= µ0f g∗ 0= g,f ∗0for f,g∈C(q,t)[x],we introduce the non-symmetric Macdonald polynomials e b(x),b∈B−,by means of the conditions(4.4)e b−x b∈Σ∗(b), e b,x c =0for c∈σ∗={c∈B,c≻b}in the setup of Section1.They can be determined by the Gram-Schmidt process because the pairing is non-degenerate and form a basis in C(q,t)[x].This definition is due to Macdonald[M3](for tν=q k,k∈Z+)who extended Opdam’s nonsymmetric polynomials introduced in the degenerate (differential)case in[O2].He also established the connection with the Y-operators.The general case was considered in[C4].The notations are from Proposition1.1and(1.1).We use the involution ¯x a=x−1a,¯q=q,¯t=t,a∈B.Proposition4.1.a)For any H∈H and the anti-involution∗from (2.11), ˆH(f),g 0= f,ˆH∗(g) 0.Here f,g are either from V0or from V1.All products of{X b,Y b,T j,πr,q,tν}are unitary operators.b)The polynomials{e b,b∈B}are eigenvectors of the operators{L f def= f(Y1,···,Y n),f∈C[x]}:L¯f(e b)=f(#b)e b,where#b def=πb=bω−1b ,(4.5)x a(#b)def=x a(q b t−ω−1b(ρ))=q(a,b) νt−(ω−1b(ρν),a)ν,w∈W.(4.6)Proof.Assertion a)for V0is from[C2].Using(3.10)we come to V1(a formal proof is equally simple).Since operators{Y b}are unitary relative to , 0and leave allΣ(a),Σ∗(a)invariant(Proposition3.2),their eigenvectors in C q,t[x]are exactly{e}.See[M3,C4].The theorem results immediately in the orthogonality of{e b}for pairwise distinct b.Macdonald also gives the formula for the squares of e b(for tν= q k,k∈Z+)and writes that he deduced it from the corresponding formula in the W-symmetric case(proved in[C2]).The general case was considered in [C4]where we used the recurrence relations.A direct simple proof(based on the intertwiners)will be given below.The symmetric Macdonald polynomials form a basis in the space C q,t[x]W of all W-invariant polynomials and can be expressed as follows:(4.7)p b=P t b e b=P1b e b,b=b+∈B+,P t def= c∈W(b) νt lν(w c)/2νˆTw c,w cdef=ω−1c w0,P1b=P t=1b.This presentation is from[M3,C4](from[O2]in the differential case). Here one can take the complete symmetrizations(with proper coefficients) since e b is W b-invariant for the stabilizer W b of b.Macdonald introduced these polynomials in[M1,M2]by the conditionsp b−m b∈Σ+(b), p b,m c 0=0,b∈B+,c≻b,(4.8)for the monomial symmetric functions m b= c∈W(b)x c.One can also define {p}as eigenvectors for the(W-invariant)operators L f,f∈C q,t[x]W:L f(p b)=f(q b o tρ)p b,b∈B+,b o=−w0(b),(4.9)normalized as above.Applying any elements from H Y=<T j,Y b>to e c(c∈W(b+))we get solutions of(4.9),because symmetric Y-polynomials are central in H Y(due to I.Bernstein).It readily gives the coincidence of(4.7)and(4.9).Functional representations.The representations Fξ,∆ξalso have in-variant skew-symmetric forms.Letµ1(bw)=µ(bw)/µ(1)def=(4.10)a∈R∨+∞i=0(1−x a(bw)q i a)(1−x−1a(bw)q i+1a)(1−x a(1)t a q i a)(1−x−1a(1)t a q i+1a)t1/2α−qjαt−1/2αx a(ξ),(4.11)where a=α∨,and we extend the conjugation∗from C q,t to Cξsettingξ∗i=ξ−1i.b)The following Cξ-valued scalar product is well-defined for f,g from the H-submodule offinitely supported functions Fξ⊂Fξ=Funct(W b,Cξ):f,g 1= ˆw∈W bµ1(ˆw)f(ˆw)g(ˆw)∗= g,f ∗1.(4.12)c)Assertion a)from Proposition4.1holds for Fξand∆ξ,where the latter module is endowed with the scalar productf,g −1= ˆw∈W b(µ1(ˆw))−1uˆw v∗ˆw,f= uˆwδˆw,g= vˆwδˆw.(4.13)Namely, H(f),g ±1= f,H∗(g) ±1.Proof.Since x˜a(ˆw)=x˜a′(1)for˜a=˜α∨∈(R a+)∨,where˜a′def=ˆw−1(˜a), one has forˆw=bw:(4.14)µ1(ˆw)= ˜α∈R a+(1−x˜a(ˆw))(1−t a x˜a(1))(1−x˜a(1))(1−t a x˜a′(1)) = ˜α∈λ(ˆw)(1−x−1˜a(1))(1−t a x˜a(1))Here we use thatˆw−1(R a+)={−λ(ˆw)}∪{R a+\λ(ˆw)}.The invariance of µ1(ˆw)∈Cξwith respect to the conjugation∗is obvious.Other statements are completely analogous to those forµ0(and follow from them).The key relation(4.15)ˆHµ(X)=µ(X)(ˆH∗)+,H∈H,readily holds after the discretization.Here by+we mean the anti-involutionˆw+=ˆw−1∈W b,x+b=x−1b,b∈B,q,t→q−1,t−1.Its discretization conjugates the values of functions from Fξand the coefficients ofδˆw in∆ξ(fixingδˆw).The characteristic functions fˆw∈Fξ(ˆw∈W b)are defined from the rela-tions fˆw(ˆu)=δˆw,ˆu for the Kronecker delta.The action of the operators X b on them is the same as for{δˆw}:X b(fˆw)=x b(ˆw)fˆw,X b(δˆw)=x b(ˆw)δˆw,b∈B,ˆw∈W b.Moreover the mapfˆw→µ1(ˆw)δˆw,ˆw∈W b,(4.16)establishes an H-isomorphism between Fξand∆ξ,taking , 1to , −1.It readily results from the formulas:δTi(fˆw)=t1/2ix−1ai(w(ξ))q−(a i,b)−t−1/2x ai(w(ξ))q(a i,b)−1fˆw for0≤i≤n,(4.17)and the formulas for the action of{πr}.Let us consider the special caseξ=t−ρ(see(3.24)).Using the pairing(3.22),we see that the subspace(4.18)F#=⊕ˆw∈#B C q,t fˆw⊂F(−ρ)=F t−ρ,where#B={#b=πb∈W b,b∈B},is an H-submodule.It is exactly the radical of the form , 1,which is well-defined for suchξ.Indeed,anyˆw can be uniquely represented in the form(see[C2])ˆw=πb w,where b=ˆw b ,w∈W,l(ˆw)=l(πb)+l(w). Hence,{ˆw∈#B}⇒{αi∈λ(ˆw)for some i>0}⇒{µ1(ˆw)=0}.On the other hand,µ1(#b)= a∈R∨+ t−1/2α−q jαt1/2αx a(tρ)。