Differential Geometry of Orbit space of Extended Affine Jacobi Group $A_n$: Part I

I define a certain extension of Jacobi group $A_n$, prove an analogue of Chevalley Theorem for its invariants.


Introduction
Dubrovin-Frobenius manifold is a geometric object that encodes the same information of WDVV equations [5]. In [5] the author associated to each Dubrovin-Frobenius manifold a Fuchsian system and consequently a monodromy group. Therefore monodromy group may contribute to classify solutions of WDVV equations. It was shown previously that any finite Coxeter group can serve as a monodromy group of a polynomial Frobenius manifold, see [4]. Moreover, in [4] it was proved that the orbit space of finite Coxeter groups have the structure of Dubrovin-Frobenius manifold. Afterwards, in [6] it was obtained Dubrovin-Frobenius structure on orbit spaces of extended affine Weyl groups, and in [2], [3] the same was done for Jacobi groups.
In the present paper I introduce a new class of groups that can be realized as monodromy groups of Dubrovin-Frobenius manifolds. This groups are higher dimension analogue of the group introduced in [1] and is denoted by J (Ã n ). In the part I, we will given a notion of invariant ring for this group. This invariant ring will be a suitable ring of meromorphic Jacobi forms. The main result of this part I is prove that this invariant ring is finitely generated ring. In the part II the structure of Dubrovin-Frobenius manifold will be constructed on the orbit space of this new group, and we will prove that this structute is isomorphic to that on Hurwitz-spaceH 1,n−1,0 . Acknowledgements I am grateful to Professor Boris Dubrovin for proposing this problem, for his remarkable advises and guidance, for the always fruitful discussions. I would like also to thanks Prof. Davide Guzzetti, and Prof. Marco Bertola for the helpful conversations, and guidance of this paper.

Ordinary Jacobi group J (A n )
Let me recall some definitions about ordinary Jacobi group, for details [2]: Let A n be a finite Coxeter group that acts on a vector space (L An , <, > An ) with a bilinear form <, > An , where L An is defined below The bilinear for L An are Recall that A n acts on L An by permutations: Moreover, A n also acts on the complexfication of L An ⊗ C. Let us consider the following group L An × L An × Z with the following group operation Note that <, > An is invariant under A n group, then A n acts on L An × L An × Z. Hence, we can take the semidirect product A n ⋉ (L An × L An × Z) given by the following product.
(w, λ, µ, k) • (w,λ,μ,k) = (ww, wλ +λ, wµ +μ, k +k+ < λ,λ > An ) Denoting W (A n ) := A n ⋉ (L An × L An × Z), we can define Definition 2.1. The Jacobi group J (Ã n ) is defined as a semidirect product W (A n ) ⋊ SL 2 (Z). The group action of SL 2 (Z) on W (A n ) is defined as for (w, t = (λ, µ, k)) ∈ W (Ã n ), γ ∈ SL 2 (Z). Then the multiplication rule is given as follows Let us use the following identification Z n+1 ∼ = L An , C n+1 ∼ = L An ⊗ C that is possible due to maps Then the action of Jacobi group J (A n ) on Ω := C ⊕ C n+1 ⊕ H is given as follows For the purpose of this paper, I will consider the A n in the following extended space The action of A n on LÃ n is given by permutations in the first n + 1 variables. Moreover, A n also acts on the complexfication of LÃ n ⊗ C. Let the quadratic form <, >Ã n given by Consider the following group LÃ n × LÃ n × Z with the following group operation ∀(λ, µ, k), (λ,μ,k) ∈ LÃ n × LÃ n × Z (λ, µ, k) • (λ,μ,k) = (λ +λ, µ +μ, k +k+ < λ,λ >Ã n ) Note that <, >Ã n is invariant under A n group, then A n acts onLÃ n × LÃ n × Z. Hence, we can take the semidirect product A n ⋉ (LÃ n × LÃ n × Z) given by the following product.
Then the multiplication rule is given as follows Let us use the following identification Z n+2 ∼ = LÃ n , C n+2 ∼ = LÃ n ⊗ C that is possible due to maps Then the action of Jacobi group J (Ã n ) on Ω := C ⊕ C n+2 ⊕ H is given as follows The proof is straightforward, but rather long, then it is left to the reader.

Jacobi forms of J (Ã n )
Since we want to study the geometric structure of the orbit space J (Ã n ), it will be necessary to study the algebra of the invariant functions. Therefore, the main goal of this paper will be to prove a version of Chevalley theorem for the group J (Ã n ). Before stating the main theorem, it will be necessary to define the notion of ring of invariants. Hence, motivated by the following definition of Jacobi forms of group A n defined in [8], and used in the context of Dubrovin-Frobenius manifold in [2], [3]: Moreover, The space of Invariant functions of J (A n ) of weight k, and index m is denoted by J An k,m .
Motivated by the ordinary definition of Jacobi forms of A n , we can make the following extended definition. Moreover, The space ofÃ n -invariant Jacobi forms of weight k, order l, and index m is denoted by JÃ n k,l,m , and J J (Ãn) •,•,• = k,l,m JÃ n k,l,m is the space of Jacobi formsÃ n invariant.
and the function f (v ′ , v n+1 , τ ) has the following transformation law The functions f (v ′ , v n+1 , τ ) are more closely related with the definition of Jacobi form of Eichler-Zagier type [7]. The coordinate φ works as kind of automorphic correction in this functions f (v ′ , v n+1 , τ ). Further, the coordinate φ will be crucial to construct an equivariant metric on the orbit space of J (Ã 1 ) (See section 4).
The main result of section is the following.
The ring ofÃ n invariant Jacobi forms is polynomial over a suitable ring Proof. Using the Remark 3.1, we know that functions ϕ ) are holomorphic, elliptic function for any i = n + 1. Therefore, by Liouville theorem, these function are constant in v ′ . Similar argument shows that these function do not depend on v n+1 , because l + 2m = 0, i.e there is no pole. Then, ϕ = ϕ(τ ) are standard holomorphic modular form. Now, I am able to state the following lemma for fixed τ the function τ → ϕ(v n+1 , τ ) is a elliptic function with poles of order at most l v n+1 = j n + lτ n , 0 ≤ l, j ≤ n − 1mod Z ⊕ τ Z.
Proof. The proof follows essentially in the same way of the lemma (4.1) proof, the only difference is that now we have poles on v n+1 = j n + lτ n , 0 ≤ l, j ≤ n − 1mod Z ⊕ τ Z. Then, we have depedence in v n+1 .
As a consequence of lemma 4.2, the function ϕ ∈ E k,l = J J (Ãn) k,l,0 has the following form where f (τ ) is holomorphic modular form of weight k, and for fixed τ , the function v n+1 → g(v n+1 , τ ) is an elliptic function of order at most l on the poles v n+1 = j n + lτ n , 0 ≤ l, j ≤ n − 1 mod Z ⊕ τ Z.
Before defining ϕÃ n 0 , ϕÃ n 1 , ϕÃ n 2 , .., ϕÃ n n , some auxiliaries lemmas are needed. Lemma 4.3. There is an one-to-one correspondence between Ω/J (Ã n ) and H 1,n−1,0 , i.e the space of elliptic functions with 1 pole of order n, and one simple pole.
Proof. The correspondence is realized by the map: Note that this map is well defined and one to one. Indeed:

Well defined
Note that proof that the map does not depend on the choice of the representant of [(φ, v 0 , v 1 , .., v n−1 , v n+1 , τ )] is equivalent to prove that the function (5) is invariant under the action of J (Ã n ). Indeed

A n invariant
The A n group acts on (5) by permuting its roots, thus (5) remais invariant under this operation.

Injectivity
Two elliptic functions are equal if they have the same zeros and poles with multiplicity.

Surjectivity
Any elliptic function can be written as rational functions of Weierstrass sigma function up to a multiplication factor [9]. By using the formula to relate Weierstrass sigma function and Jacobi theta function where E 2 (τ ) is Eisenstein 2. Hence, we get the desire result.
Proof. Let us prove each item separated.

A n invariant, translation invariant
The l.h.s of (9) are A n invariant, and translation invariant by the lemma (4.3). Then, by the uniqueness pf Laurent expansion of λÃ n , we have that ϕÃ n i are A n invariant, and translation invariant.

SL 2 (Z) equivariant
The l.h.s of (9) are SL 2 (Z) invariant, but the Weierstrass functions of the r.h.s have the following transformation law Then, ϕÃ n k must have the following transformation law
At this stage, the principal theorem can be state in precise way as follows.

Proof of the main theorem
Before proving this lemma, an auxialiary lemma will be necessary.
Let me recall an useful theorem for that proof.
Proof. Note the following relation λ An+1 Then, the desired result is obtained by doing a Laurent expansion in the variable z in both side of the equality.
As a consequence of the previous lemma, we have 2 ) m h m (ϕÃ n 0 , ϕÃ n 1 , ϕÃ n 2 , .., ϕÃ n n ). Let us expand the functions ϕÃ n i in the variables v i , thenh m vanishes iff its vanishes in each order of this expansion.