1 Introduction

ITEP/TH-11/17

IITP/TH-5/17

Genus Two Generalization of spherical DAHA

S.Arthamonov111Department of Mathematics, Rutgers, The State University of New Jersey; semeon.artamonov@rutgers.edu
ITEP, Moscow, Russia
and Sh.Shakirov222Harvard Society of Fellows, Harvard University, Cambridge MA; shakirov@fas.harvard.edu
Institute for Information Transmission Problems, Moscow, Russia

ABSTRACT

We consider a system of three commuting difference operators in three variables with two generic complex parameters . This system and its eigenfunctions generalize the trigonometric Ruijsenaars-Schneider model and Macdonald polynomials, respectively. The principal object of study in this paper is the algebra generated by these difference operators together with operators of multiplication by . We represent the Dehn twists by outer automorphisms of this algebra and prove that these automorphisms satisfy all relations of the mapping class group of the closed genus two surface. Therefore we argue from topological perspective this algebra is a genus two generalization of spherical DAHA.

## 1 Introduction

Double affine Hecke algebra [1], introduced and developed by I.Cherednik, plays a crucial role in modern representation theory and mathematical physics from a number of perspectives (see [1] for a comprehensive overview). One perspective which is especially interesting for us is its close relation with topology of the torus and its mapping class group : namely, this mapping class group acts on the DAHA by outer automorphisms. This, in particular, allows to compute DAHA-Jones polynomials of torus [2] and iterated torus [3] links, which generalize and improve WRT invariants [4, 5] and hence are very interesting from the viewpoint of knot theory. DAHA-Jones polynomials are also related to a number of subjects in mathematical physics, such as refined Chern-Simons theory [6] and knot superpolynomials [7, 8].

We concentrate on a particular subalgebra of the double affine Hecke algebra, called spherical DAHA, in the case of the root system. In this case, the spherical DAHA admits a very simple polynomial representation as the algebra generated by two operators,

 ^OB=x+x−1,      ^OA=x−1−txx−1−x ^δ+x−tx−1x−x−1 ^δ−1 (1)

acting on the space of symmetric Laurent polynomials in a single variable ; here is a shift operator. These two operators are associated with the - and -cycles of the torus, as shown in Figure 1, and the mapping class group acts on the algebra generated by these operators by certain explicit outer automorphisms.

In this paper we consider an algebra generated by six operators acting on the space of Laurent polynomials in three variables , associated with the three - and three -cycles of a genus two surface, as shown on Figure 1. We represent the Dehn twists by outer automorphisms of this algebra and prove that these automorphisms satisfy all relations of the mapping class group of a closed genus two surface. This suggests this algebra is a genus two generalization of spherical DAHA. The reason we consider the case of genus two is not because this case is particularly distinguished, but as a first step towards a general construction for arbitrary genus.

## 2 Genus two Macdonald polynomials

We fix the ground field to be the field of rational functions in and .

###### Definition 1.

We call a triple of integers admissible if each of them is non-negative, and is even.

###### Definition 2.

Consider a family of Laurent polynomials in variables which are a common solution to the following system of recursive relations, satisfied for all admissible triples :

 (x12+x−112) Ψj1,j2,j3=∑a,b∈{−1,+1} Ca,b(j1,j2,j3) Ψj1+a,j2+b,j3 (2a) (x13+x−113) Ψj1,j2,j3=∑a,b∈{−1,+1} Ca,b(j1,j3,j2) Ψj1+a,j2,j3+b (2b) (x23+x−123) Ψj1,j2,j3=∑a,b∈{−1,+1} Ca,b(j2,j3,j1) Ψj1,j2+a,j3+b (2c)

with initial conditions: for non-admissible triples , and . Here

 Ca,b(j1,j2,j3)=a b [aj1+bj2+j32,a+b+22]q,t[aj1+bj2−j32,a+b2]q,t[j1−1,2]q,t[j2−1,2]q,t[j1,a+32]q,t[j1−1,a+32]q,t[j2,a+32]q,t[j2−1,a+32]q,t (3)

where

 [n,m]q,t=qn2tm2−q−n2t−m2q12−q−12

We call genus two Macdonald polynomials (of type ), and recursion (2) genus two Pieri rule. Their existence follows from the following Lemma and Proposition.

###### Lemma 3.

For a given admissible triple , the coefficient is non-vanishing if and only if the triple is admissible.

###### Proof.

First note that mod , so that we don’t have to worry about the parity condition. As for the other conditions, there are three cases to consider.

• In this case does not change, and increases by 2, so the triple has to be admissible.

• In this case does not change, but changes by 2. Therefore the triple is admissible unless . This is precisely when the second factor in the numerator of (3) vanishes.

• In this case does not change, and decreases by 2. Therefore the triple is admissible unless . This is precisely when the first factor in the numerator of (3) vanishes.

This technical lemma is important: since each triple that appears on the r.h.s. of (2) with a non-vanishing coefficient is necessarily admissible, one can iterate the genus two Pieri rule.

###### Proposition 4.

Recursive relations (2a), (2b), and (2c) are compatible.

###### Proof.

Let us first prove compatibility of (2a) and (2b). Applying (2a) followed by (2b), we get

 (x13+x−113)(x12+x−112)Ψj1,j2,j3=∑a1,b1∑a2,b2 Ca1,b1(j1,j2,j3) Ca2,b2(j1+a1,j3,j2+b1) Ψj1+a1+a2,j2+b1,j3+b2

In the opposite order we get

 (x12+x−112)(x13+x−113)Ψj1,j2,j3=∑a1,b1∑a2,b2 Ca1,b2(j1,j3,j2) Ca2,b1(j1+a1,j2,j3+b2) Ψj1+a1+a2,j2+b1,j3+b2

Compatibility of (2a) and (2b) follows from the vanishing of their difference

 ∑a1+a2=a(Ca1,b1(j1,j2,j3) Ca2,b2(j1+a1,j3,j2+b1)−Ca1,b2(j1,j3,j2) Ca2,b1(j1+a1,j2,j3+b2))=0

for all , , and , which is a direct computation in . Compatibility of the other two pairs follows by permutation of .

###### Corollary 5.

Genus two Macdonald polynomials are symmetric w.r.t. sumultaneous permutations:

 Ψj1,j2,j3(x12,x13,x23)=Ψjσ1,jσ2,jσ3(xσ1σ2,xσ1σ3,xσ2σ3),   ∀σ∈S3 (4)
###### Corollary 6.

Usual Macdonald polynomials [9] are a particular case of genus two Macdonald polynomials,

 Ψℓ,ℓ,0(x12,x13,x23)=cℓPℓ(x12),   Ψℓ,0,ℓ(x12,x13,x23)=cℓPℓ(x13),   Ψ0,ℓ,ℓ(x12,x13,x23)=cℓPℓ(x23)

where is a normalization constant, which is nothing but the well-known principal specialization of :

 cℓ:=tℓ2 ℓ−1∏i=01−qit1−qit2=Pℓ(t12)
###### Proof.

By corollary 5, it suffices to prove one of the three specializations, the other two will follow. Consider the case of the genus two Pieri rule (2a). It is easy to see that it takes form

 (x12+x−112)Ψℓ,ℓ,0=t12 1−qℓt1−qℓt2 Ψℓ+1,ℓ+1,0+t−12 (1−qℓ)(1−qℓ−1t2)2(1−qℓt)(1−qℓ−1t)2 Ψℓ−1,ℓ−1,0

After renormalization this recursion takes form

 (x12+x−112)Ψ′ℓ,ℓ,0=Ψ′ℓ+1,ℓ+1,0+(1−qℓ)(1−qℓ−1t2)(1−qℓt)(1−qℓ−1t) Ψ′ℓ−1,ℓ−1,0

This is precisely the Pieri rule for usual Macdonald polynomials, with the same initial conditions for and . Consequently, . ∎

###### Lemma 7.

Genus two Macdonald polynomials have a form

 Ψj1,j2,j3=xd312xd213xd123 ∑n1,n2,n3≥0n1+n2+n3≤j1+j2+j3 Kn1,n2,n3(j1,j2,j3) (x23x12x13)n1 (x13x12x23)n2 (x12x13x23)n3 (5)

where

 d1=−j1+j2+j32,   d2=j1−j2+j32,   d3=j1+j2−j32

and for all admissible triples .

###### Proof.

The proof is by induction in . The base case is obvious. Assume that the lemma holds for all admissible triples such that . Pick any admissible triple with . Applying the Pieri rule (2a) we get as a -linear combination of and . Consider first . By the inductive assumption, it has the form (5) with the leading term

 xd3+212xd2−213xd1−223=xd312xd213xd123⋅(x12x13x23)2

therefore, every monomial in the expansion of has a form

 xd312xd213xd123⋅ (x23x12x13)n1 (x13x12x23)n2 (x12x13x23)n3 (6)

where the sum cannot be bigger than . The same holds for and . As for , by the inductive assumption, it has the form (5) with the leading term

 xd312xd213xd123⋅ x−112

therefore every monomial in the expansion of has the form (6) where the sum cannot be bigger than , and the coefficient in front of is non-vanishing. We therefore proved that has this form too. ∎

Let be the space of all Laurent polynomials in (as everywhere in this paper, with coefficients over ) symmetric under the group of Weyl inversions

 (x12,x13,x23)   ↦   (xu12,xv13,xw23),      u,v,w∈{−1,+1}
###### Proposition 8.

Genus two Macdonald polynomials for all admissible form a basis of .

###### Proof.

Let us first prove that is spanned by genus two Macdonald polynomials. For this, note that is spanned by the set of monomials where . This set contains the generators . Since span() is closed under multiplication by (see (2)) genus two Macdonald polynomials span . By Lemma 7 their leading terms in total degree are all different, thus they are linearly independent and form a basis of . ∎

## 3 Algebra of Knot operators

Denote by the algebra of all -difference operators with coefficients to be rational functions in over . Here

 ^δ12f(x12,x23,x13) =f(q12x12,x23,x13)% for allf∈k(x12,x23,x13)

and the other two: and are obtained by permutation of indexes.

###### Definition 9.

Let be the following -difference operators acting on

 ^OA1 = ∑a,b∈{±1} ab (1−t12x+123xa12xb13)(1−t12x−123xa12xb13)t12xa12xb13(x+112−x−112)(x+113−x−113) ^δa12^δb13 (7a) ^OA2 = ∑a,b∈{±1} ab (1−t12x+113xa12xb23)(1−t12x−113xa12xb23)t12xa12xb23(x+112−x−112)(x+123−x−123) ^δa12^δb23 (7b) ^OA3 = ∑a,b∈{±1} ab (1−t12x+112xa13xb23)(1−t12x−112xa13xb23)t12xa13xb23(x+113−x−113)(x+123−x−123) ^δa13^δb23 (7c)

and be the following multiplication operators:

 ^OB12 = x12+x−112 (8a) ^OB13 = x13+x−113 (8b) ^OB23 = x23+x−123 (8c)
###### Definition 10.

Let be the algebra generated by the above six operators. We will call the algebra of genus two knot operators.

###### Remark 11.

The action of the operator on functions that depend only on is identical to the action of the Macdonald operator of the spherical DAHA (1). Therefore the action of the subalgebra on functions that depend only on is isomorphic to the action of the spherical DAHA. By symmetry, a similar statement holds for and .

###### Lemma 12.

The algebra is not free: among others, the following relations hold for all

 [^OA1,^OA2]=[^OA2,^OA3]=[^OA1,^OA3]=0 (9a) [^OB12,^OB23]=[^OB12,^OB13]=[^OB23,^OB13]=0 (9b) [^OA1,^OB23]=[^OA2,^OB13]=[^OA3,^OB12]=0 (9c) (q12−q−12)2 ^OBij=−^OAi^OAi^OBij+(q12+q−12) ^OAi^OBij^OAi−^OBij^OAi^OAi (9d) (q12−q−12)2 ^OAi=−^OBij^OBij^OAi+(q12+q−12) ^OBij^OAi^OBij−^OAi^OBij^OBij (9e) ^OB13^OA1^OB12−^OB12^OA1^OB13=^OA2^OB23^OA3−^OA3^OB23^OA2 (9f) ^OA1^OB12^OB13− (q12+q−12)^OB12^OA1^OB13+^OB13^OB12^OA1= ^OA3^OA2^OB23−(q12+q−12)^OA3^OB23^OA2+^OB23^OA2^OA3 (9g)
###### Proof.

For each relation apply both sides to an arbitrary the result will follow by a direct computation in . ∎

###### Proposition 13.

Genus two Macdonald polynomials are common eigenfunctions of

 ^OAkΨj1,j2,j3=(qjk2t12+q−jk2t−12)Ψj1,j2,j3 (10)
###### Proof.

See Appendix A. ∎

###### Corollary 14.

Algebra of knot operators is acting on the space of genus two symmetric polynomials .

###### Proof.

Indeed, is acting on , so the only thing we have to prove is that this action preserves . Combining (8) with (10) we conclude that generators of leave invariant and thus so does . ∎

This allows one to define a polynomial representation of algebra . In what follows we denote the natural image of generators by the same letters.

## 4 Mapping Class Group Action

###### Definition 15.

Define to be the following automorphisms of

 bij(xkl)=xkl,bij(^δkl)={q14xij^δij(ij)=(kl),^δkl(ij)≠(kl)for all1≤k

Note that all relations in are preserved by (11).

###### Lemma 16.

Automorphisms act on the generators of in the following way

 b±1ij(^OAk)= ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩±q±14q12−q12^OAk^OBij∓q∓14q12−q−12^OBij^OAkk∈{i,j},^OAkk∉{i,j}, (12a) b±1ij(^OBkl)= ^OBklfor all1≤k
###### Proof.

Identity (12b) is immediate. To prove (12a) note that (11) imply . Applying to (7a)–(7c) and collecting coefficients in front of on both sides of (12a) we get the result. ∎

###### Corollary 17.

Algebra is invariant under , as a result the above automorphisms act on as well.

###### Definition 18.

Let be defined on basis elements as

 Akψj1,j2,j3=qj2k4tjk2ψj1,j2,j3. (13)

Denote the corresponding adjoint action of on as

 ak(h)=A−1khAk,for allh∈H.
###### Lemma 19.

Automorphisms act on generators of in the following way

 a±1k(^OBij)= ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩∓q∓14q12−q−12^OAk^OBij±q±14q12−q−12^OBij^OAkk∈{i,j},^OBijk∉{i,j}, (14a) a±k(^OAj)= ^O