Generalized Frobenius Manifold Structures on the Orbit Spaces of Affine Weyl Groups II

GENERALIZED FR OBENIUS MANIF OLD STR UCTURES ON THE ORBIT SP A CES OF AFFINE WEYL GR OUPS I I LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG Abstract. This is a sequel to [14], in whic h an approac h to construct a class of generalized F robenius manifold structures on the orbit spaces of affine W eyl groups is presented. In this paper w e apply this construction to the affine W eyl groups of type A ℓ , B ℓ , C ℓ and D ℓ . Contents 1. In tro duction 1 2. The Case of ( A ℓ , ω ℓ ) 5 2.1. The in v ariant λ -F ourier p olynomial ring 5 2.2. The pencil generators 5 2.3. Flat coordinates of the metric η 7 2.4. The generalized F rob enius manifold sturcture on M D ( A ℓ , ω ℓ ) and mono dromy group 11 2.5. Examples 12 2.6. The Landau-Ginzburg sup erp oten tial 14 3. The Case of ( C ℓ , ω 1 ) 19 3.1. The W a ( R )-inv ariant λ -F ourier polynomial ring 19 3.2. The pencil generators 19 3.3. Flat coordinates of the metric η 22 3.4. The generalized F rob enius manifold structures 25 3.5. Examples 25 3.6. Landau-Ginzburg superp otential 29 4. The Cases of ( B ℓ , ω 1 ) and ( D ℓ , ω 1 ) 35 4.1. The In v ariant λ -F ourier P olynomial Ring 35 4.2. The Relation b etw een the case of ( B ℓ , ω 1 ) , ( D ℓ , ω 1 ) and that of ( C ℓ , ω 1 ) 36 5. Conclusions 36 References 36 1. Introduction In [14], we introduced an approach to construct a class of generalized F rob enius manifold structures on the orbit spaces of affine W eyl groups. As illustrative examples, w e applied this construction Date : F ebruary 26, 2026. Key wor ds and phr ases. Generalized F rob enius manifolds, Affine W eyl groups, Flat co ordinates, Root systems. 1 2 LINGRUI JIANG, SI-QI LIU, YINGCHA O TIAN, YOUJIN ZHANG to the affine W eyl groups of t yp es A 1 , A 2 , A 3 , B 3 , C 3 , D 4 and G 2 . In the presen t pap er, we extend this construction to the affine W eyl groups of t yp es A ℓ , B ℓ , C ℓ and D ℓ , thereby obtaining further examples of generalized F rob enius manifold structures. W e b egin b y recalling the basic framework of this construction. Let R b e an irreducible reduced root system in an ℓ -dimensional Euclidean space with inner product ( · , · ), and α 1 , . . . , α ℓ b e a basis of simple roots. Denote by α ∨ 1 , . . . , α ∨ ℓ and ω 1 , . . . , ω ℓ the corresponding coro ots and fundamental w eights of R . W e fix a w eigh t ω = ℓ X j =1 m j ω j , m j ∈ Z ≥ 0 (1.1) of R , and introduce affine co ordinates ( x 1 , . . . , x ℓ ; c ) on V b y x = cω + x 1 α ∨ 1 + · · · + x ℓ α ∨ ℓ , (1.2) where c ∈ R is a fixed parameter. Denote by W ( R ) and W a ( R ) resp ectively the W eyl group and the affine W eyl group asso ciated with the root system R . The action of W a ( R ) on V given by σ  cω + x 1 α ∨ 1 + · · · + x ℓ α ∨ ℓ  = cω + ˜ x 1 α ∨ 1 + · · · + ˜ x ℓ α ∨ ℓ , σ ∈ W a ( R ) (1.3) yields the change of affine co ordinates σ ( x 1 , . . . , x ℓ ; c ) = ( ˜ x 1 , . . . , ˜ x ℓ ; c ) , σ ∈ W a ( R ) . This c hange of co ordinates induces a righ t action on the F ourier function ring F = span C n e 2 π i ( t 0 c + t 1 x 1 + ··· + t ℓ x ℓ ) | t 0 , t 1 , . . . , t ℓ ∈ R o , whic h has a gradation defined by deg e 2 π ix j = θ j , deg e 2 π ic = − 1 , (1.4) where θ j = ( ω j , ω ). It is shown in [14] that the W a ( R ) action on F preserv es the degrees of monomials. W e denote by F W ( R ) the inv ariant subring of F w.r.t. this action. W e in tro duce a parameter λ = e − 2 π iκc with κ = gcd { ( ω , α r ) | r = 1 , . . . , ℓ } , (1.5) and define the λ -F ourier polynomial ring A as a subring of F by A = C [ λ ] ⊗ C  e 2 π ix j , e − 2 π ix j | j = 1 , . . . , ℓ  ⊂ F . Denote by A W = A ∩ F W the W a ( R )-inv ariant λ -F ourier p olynomial ring, then we kno w from [14] that A W = C [ y 1 , . . . , y ℓ ; λ ], where y j ( x ) = e − 2 π iθ j c Y j ( x ) = 1 N j e − 2 π iθ j c X w ∈ W ( R ) e 2 π i ( ω j ,w ( x )) , j = 1 , . . . , ℓ, (1.6) and N j = # { w ∈ W ( R ) | w ( ω j ) = ω j } . The W a ( R )-inv ariant λ -F ourier p olynomials y 1 , . . . , y ℓ are called basic generators of A W , and they are quasi-homogeneous of deg y j = θ j . AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 3 In order to construct generalized F robenius manifold structures on the orbit space of the affine W eyl group W a ( R ), w e introduce in [14] the notion of prop er generators of A W . Let z 1 , . . . , z ℓ ∈ A W . W e call { z 1 , . . . , z ℓ } a set of prop er generators of A W if z j ∈ A W with deg z j = θ j for j = 1 , . . . , ℓ , and z j   λ =0 = y j   λ =0 , j = 1 , . . . , ℓ. (1.7) In other words, the quasi-homogeneous W a ( R )-inv ariant λ -F ourier p olynomials z 1 , . . . , z ℓ form a set of prop er generators of A W if and only if for any 1 ≤ j ≤ ℓ , there exists a p olynomial s j of λ and elemen ts of { y r | 1 ≤ r ≤ ℓ, θ r < θ j } , suc h that either s j = 0 or deg s j = θ j − κ , and z j = y j + λs j . (1.8) Note that the parameter λ of the W a ( R )-inv ariant λ -F ourier p olynomials z 1 , . . . , z ℓ is defined by (1.5) and it lies on the unit circle. How ever, since z j dep end p olynomially on λ , the definition of z j can b e extended naturally to an y λ ∈ C , including λ = 0. Giv en a set { z 1 , . . . , z ℓ } of prop er generators of A W , w e consider the orbit space M := M ( R , ω ) ∼ = C ℓ of the affine W eyl group W a ( R ). F or eac h parameter λ ∈ C , the φ λ -transformation φ λ : ( x 1 , . . . , x ℓ ) 7→ ( z 1 , . . . , z ℓ ) induces a pushing forw ard ( φ λ ) ∗ , whic h transforms the c on tra v ariant metric a (d x i , d x j ) = a ij with ( a ij ) =  ( α ∨ i , α ∨ j )  − 1 , i, j = 1 , . . . , ℓ (1.9) on V ⊗ C to a con tra v ariant metric ( g ij λ ) on M . By definition, g ij λ = 1 4 π 2 ℓ X r,s =1 ∂ z i ∂ x r a rs ∂ z j ∂ x s , i, j = 1 , . . . , ℓ. (1.10) Here a factor 1 / 4 π 2 is in troduced to simplify the expressions of g ij λ . W e say that { z 1 , . . . , z ℓ } is a set of p encil gener ators of A W if the asso ciated metric g λ =  g ij λ  dep ends linearly on λ , i.e., g ij λ can be represen ted in the form g ij λ = g ij + λη ij , where η = ( η ij ) is non-degenerate at generic p oin ts of M , and the Christoffel symbols Γ ij g λ ,k = − g ir λ Γ j g λ ,rk of the Levi-Civita connection of the con trav ariant metric g λ can b e represented in terms of that of g and η as follo ws: Γ ij g λ ,k = Γ ij g ,k + λ Γ ij η ,k . (1.11) W e kno w from [14] that for an y set of p encil generators, the functions g ij λ and Γ ij g λ ,k are quasi- homogeneous polynomials of z 1 , . . . , z ℓ and λ of degree θ i + θ j and θ i + θ j − θ k resp ectiv ely . F or a given set { z 1 , . . . , z ℓ } of p encil generators of A W , w e consider the v ector field E = ℓ X r =1 θ r z r ∂ ∂ z r . (1.12) Denote D η = { z ∈ M   det η ( z ) = 0 } . (1.13) 4 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG Then η = ( η ij ) induces a flat metric ( η ij ) = ( η ij ) − 1 on M \ D η , whic h w e also denote by η . W e assume that we can c hoose a system of flat co ordinates t 1 , . . . , t ℓ of the flat metric η , such that in these co ordinates the vector field E has the form E = ℓ X α,r =1 θ r z r ∂ t α ∂ z r ∂ ∂ t α = ℓ X α =1 d α t α ∂ ∂ t α , (1.14) where d 1 , . . . , d ℓ are some real num b ers. In suc h a case w e call the vector field E is diagonalizable. It is sho wn in [14] that d α m ust be p ositiv e n um bers. Let us denote D 0 = { z ∈ M | ∃ i ∈ S, z i = 0 } , D = D η ∪ D 0 , (1.15) and M D = M \ D , where S = { r ∈ { 1 , . . . , ℓ } | m r = ( ω , α ∨ r ) > 0 } , (1.16) then M D is a dense op en subset of M . Let Γ αβ γ b e the contra v arian t comp onents of the Levi-Civita connection of g in the flat co ordinates t 1 , . . . , t ℓ of η . It is shown in [14] that one can define a F rob enius algebra structure on T ( M D ) with the bilinear form ⟨· , ·⟩ defined by the flat metric η , the multiplication ∂ ∂ t α · ∂ ∂ t β = c γ αβ ∂ ∂ t γ , (1.17) and the unit v ector field e = η ♯ ( ω e ) = − 1 κ grad η ℓ X r =1 m r log z r . (1.18) Here m r are defined in (1.1), c γ αβ = κ d ρ η αν η β ρ η γ ζ Γ ν ρ ζ , ω e = − ℓ X r =1 1 κ m r d log z r , (1.19) and η ♯ is the isomorphism η ♯ : T ∗ M D → T ∗∗ M D = T M D , ω 7→ η ( ω , · ) . (1.20) W e also assume here and in what follo ws summations o ver rep eated upper and low er Greek indices. Theorem 1.1 ( [14]) . Supp ose { z 1 , . . . , z ℓ } b e a set of p encil gener ators asso ciate d with an irr e ducible r e duc e d r o ot system R and a fixe d weight ω , and the ve ctor field E given by (1.12) is diagonalizable, then ther e exists a gener alize d F r ob enius manifold structur e of char ge d = 1 on M D , of which the flat metric is given by η and the multiplic ation is define d by (1.17) ; mor e over, the unit ve ctor field e is define d by (1.18) , the Euler ve ctor field ˜ E = 1 κ E , and the interse ction form c oincides with g . In this pap er, w e are to pro v e the following theorem. Theorem 1.2 (Main Theorem) . F or e ach ( R, ω ) = ( A ℓ , ω ℓ ) , ( B ℓ , ω 1 ) , ( C ℓ , ω 1 ) , ( D ℓ , ω 1 ) , one c an c on- struct a set of p encil gener ators of A W , and a gener alize d F r ob enius manifold structur e on M D ( R, ω ) by using the appr o ach pr op ose d in The or em 1.1. W e organize the pap er as follows. In Sect. 2 and Sect. 3, we prov e the Main Theorem for the cases ( R, ω ) = ( A ℓ , ω ℓ ) and ( R, ω ) = ( C ℓ , ω 1 ) resp ectively . In Sect. 4 we show that that for the cases ( R, ω ) = ( B ℓ , ω 1 ) and ( D ℓ , ω 1 ), the generalized F rob enius manifold structures that are constructed b y using the approach of Theorem 1.1 are isomorphic to the ones obtained for the cases ( R, ω ) = ( C ℓ , ω 1 ). AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 5 2. The Case of ( A ℓ , ω ℓ ) 2.1. The inv ariant λ -F ourier p olynomial ring. Let e 1 , . . . , e ℓ +1 b e an orthonormal basis of R ℓ +1 , and R b e the ro ot system of type A ℓ in the hyperplane V of R ℓ +1 spanned the simple roots α 1 = e 1 − e 2 , . . . , α ℓ = e ℓ − e ℓ +1 . The coroots and the fundamental weigh ts of R are given by α ∨ i = α i and ω i = 1 ℓ + 1 (( ℓ − i + 1) α 1 + 2( ℓ − i + 1) α 2 + · · · + ( i − 1)( ℓ − i + 1) α i − 1 + i ( ℓ − i + 1) α i + i ( ℓ − i ) α i +1 + · · · + iα ℓ ) , i = 1 , . . . , ℓ. T ake ω = ω ℓ , then we hav e θ j = ( ω j , ω ℓ ) = j ℓ + 1 , j = 1 , . . . , ℓ, and κ = 1. W e define ξ 1 , . . . , ξ ℓ +1 b y the relation cω + x 1 α ∨ 1 + · · · + x ℓ α ∨ ℓ = ξ 1 e 1 + · · · + ξ ℓ +1 e ℓ +1 , then the basic generators of the W a ( R )-inv ariant λ -F ourier p olynomial ring A W can b e represen ted in the form y j ( x ) = λ j ℓ +1 Y j ( x ) = λ j ℓ +1 σ j ( e 2 π iξ 1 , . . . , e 2 π iξ ℓ +1 ) , j = 1 , . . . , ℓ, (2.1) with deg y j = θ j = j ℓ + 1 , deg λ = 1 , j = 1 , . . . , ℓ. Here and in what follows we denote b y σ j ( u 1 , . . . , u ℓ +1 ) the j -th elementary symmetric polynomial of u 1 , . . . , u ℓ +1 defined b y ℓ +1 Y i =1 ( z + u i ) = ℓ +1 X j =0 σ j ( u 1 , . . . , u ℓ +1 ) z ℓ +1 − j . (2.2) W e will also denote by σ j ( u 1 , . . . , b u k , . . . , u ℓ +1 ), or simply b y σ j ( ˆ u k ), the j -th elementary symmet- ric p olynomial of the ℓ v ariables u 1 , . . . , u k − 1 , u k +1 , . . . , u ℓ +1 , and by σ j ( b u k , b u m ) the j -th elementary symmetric polynomial of the ℓ − 1 v ariables u 1 , . . . , u k − 1 , u k +1 , . . . , u m − 1 , u m +1 , . . . , u ℓ +1 . 2.2. The p encil generators. W e are to show in this subsection that { y 1 , . . . , y ℓ } is a set of p encil generators of A W . T o this end, let us consider the comp onents g ij λ of the metric g λ defined b y (1.10) with z i = y i . They can be represen ted in the form g ij λ = 1 4 π 2 ℓ X r,s =1 ∂ y i ∂ x r a rs ∂ y j ∂ x s = 1 4 π 2 ℓ +1 X r,s =1 ∂ y i ∂ ξ r b rs ∂ y j ∂ ξ s , (2.3) where the matrices ( a rs ) and ( b rs ) are defined b y ( a ij ) =  ( α ∨ i , α ∨ j )  − 1 =  i ( ℓ + 1 − j ) ℓ + 1  , ( b rs ) = 1 ℓ + 1      ℓ − 1 · · · − 1 − 1 ℓ · · · − 1 . . . . . . . . . . . . − 1 − 1 · · · ℓ      . 6 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG Lemma 2.1. The quasi-homo gene ous p olynomials g ij λ of y 1 , . . . , y ℓ , λ dep end at most line arly on λ when ℓ + 1 ≤ i + j ≤ 2 ℓ , and do not dep end on λ when 2 ≤ i + j ≤ ℓ . Pr o of. Since deg g ij λ = θ i + θ j = i + j ℓ +1 , we hav e 1 ≤ deg g ij λ < 2 when ℓ + 1 ≤ i + j ≤ 2 ℓ , and deg g ij λ < 1 when 2 ≤ i + j ≤ ℓ . Thus from the fact that deg y j = θ j > 0 , deg λ = 1, it follows that g ij λ dep ends at most linearly on λ when ℓ + 1 ≤ i + j ≤ 2 ℓ , and they do not dep end on λ when 2 ≤ i + j ≤ ℓ . The lemma is prov ed. □ By using this lemma, we can represent g ij λ in the form g ij λ = g ij + λη ij , (2.4) where η ij = 0 when 2 ≤ i + j ≤ ℓ , and the anti-diagonal elements of η i,ℓ +1 − i of η are constan ts. Prop osition 2.2. Al l the anti-diagonal elements of η ar e e qual to ℓ + 1 . Pr o of. Since the anti-diagonal elements of η are constant, we can determine them by calculating g ij λ for special v alues of y 1 , . . . , y ℓ . Let e 2 π iξ 1 0 , . . . , e 2 π iξ ℓ +1 0 b e the ℓ + 1 ro ots of z ℓ +1 + ( − 1) ℓ +1 . Then w e ha v e σ j ( e 2 π iξ 1 0 , . . . , e 2 π iξ ℓ +1 0 ) = 0 , σ ℓ +1 ( e 2 π iξ 1 0 , . . . , e 2 π iξ ℓ +1 0 ) = 1 , j = 1 , . . . , ℓ. By using the iden tit y z ℓ +1 + ( − 1) ℓ +1 z − e 2 π iξ k 0 = z ℓ + e 2 π iξ k 0 z ℓ − 1 + · · · + e 2 π iℓξ k 0 w e also hav e σ j ( e 2 π iξ 1 0 , . . . , \ e 2 π iξ k 0 , . . . , e 2 π iξ ℓ +1 0 ) = ( − 1) j e 2 π ij ξ k 0 , j = 1 , . . . , ℓ + 1 , th us w e obtain ∂ y r ∂ ξ k     ξ = ξ 0 = 2 π iµ r e 2 π iξ k σ r − 1 ( e 2 π iξ 1 , . . . , \ e 2 π iξ k , . . . , e 2 π iξ ℓ +1 )    ξ = ξ 0 = 2 π i ( − 1) r − 1 µ r e 2 π irξ k 0 , where µ = λ 1 ℓ +1 . Now from (2.3) it follows that g rs λ = 1 4 π 2 ( ℓ + 1)   ( ℓ + 1) ℓ +1 X k =1 ∂ y r ∂ ξ k ∂ y s ∂ ξ k − ℓ +1 X j,k =1 ∂ y r ∂ ξ j ∂ y s ∂ ξ k         ξ = ξ 0 = − ( − 1) ℓ +1 µ ℓ +1   ℓ +1 X k =1 e 2 π i ( r + s ) ξ k 0 − 1 ℓ + 1 ℓ +1 X j =1 e 2 π irξ j 0 ℓ +1 X k =1 e 2 π isξ k 0   = ( ℓ + 1) λ when r + s = ℓ + 1. The proposition is prov ed. □ F rom Prop osition 2.2 w e kno w that det( η ij ) = ( − 1) ℓ ( ℓ − 1) 2 ( ℓ + 1) ℓ , (2.5) so η is non-degenerate on M ( A ℓ , ω ℓ ). Theorem 2.3. The b asic gener ators y 1 , . . . , y ℓ form a set of p encil gener ators of A W . AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 7 Pr o of. W e only need to show that the Christoffel symbols Γ ij g λ ,k of the Levi-Civita connection of the con trav ariant metric g λ dep end linearly on λ . Indeed, this follows from the fact that they are quasi- homogeneous polynomials of y 1 , . . . , y ℓ and λ , and deg Γ ij λ,k = θ i + θ j − θ k = i + j − k ℓ + 1 < 2 . The theorem is pro v ed. □ 2.3. Flat co ordinates of the metric η . W e are to sho w in this subsection that one can choose a system of flat coordinates t 1 , . . . , t ℓ of the metric η whic h are quasi-homogeneous p olynomials in y 1 , . . . , y ℓ . T o this end, w e first give the explicit expression of the comp onents of the metric η . Prop osition 2.4. In the c o or dinates y 1 , . . . , y ℓ , the metric η = ( η ij ) is given by η ij = ( 0 , 2 ≤ i + j ≤ ℓ, (2 ℓ + 2 − i − j ) y i + j − 1 − ℓ , ℓ + 1 ≤ i + j ≤ 2 ℓ, (2.6) and the c ontr avariant c omp onents of the L evi-Civita c onne ction of η have the expr essions Γ ij η ,k = ( ℓ + 1 − j ) δ i + j − k,ℓ +1 . (2.7) Let us make some preparations for the proof of this prop osition. Lemma 2.5. The symmetric p olynomials σ a ( u 1 , . . . , u ℓ +1 ) have the fol lowing pr op erties: 1 . ∂ ∂ u i ( u r σ a ( b u r )) = ( u r σ a − 1 ( b u r , b u i ) , r  = i, σ a ( b u r ) , r = i, for a = 1 , . . . , ℓ. 2 . ℓ +1 X r =1 σ a ( b u r ) = ( ℓ + 1 − a ) σ a ( u 1 , . . . , u ℓ +1 ) , a = 1 , . . . , ℓ. 3 . ℓ +1 X r =1 ,r  = i σ a ( b u r , b u i ) = ( ℓ − a ) σ a ( b u i ) , a = 1 , . . . , ℓ − 1 . 4 . ℓ +1 X r =1 u r σ a ( b u r ) = ( a + 1) σ a +1 ( u 1 , . . . , u ℓ +1 ) , a = 1 , . . . , ℓ. 5 . n X r =1 ,r  = i u r σ a ( b u r , b u i ) = ( a + 1) σ a +1 ( b u i ) , a = 1 , . . . , ℓ − 1 . Mor e over, supp ose A is a subset of { u 1 , . . . , u ℓ +1 } and u r ∈ A , then we have u r σ a ( b A ) = σ a +1 ( \ A \ u r ) − σ a +1 ( b A ) . No w tak e u j = µe 2 π iξ j for j = 1 , . . . , ℓ + 1, where µ = λ 1 ℓ +1 , then we hav e y j = σ j ( u 1 , . . . , u ℓ +1 ) , j = 1 , . . . , ℓ ; λ = σ ℓ +1 ( u 1 , . . . , u ℓ +1 ) . Th us u 1 , . . . , u ℓ +1 are roots of the p olynomial f ( u ) = u ℓ +1 − y 1 u ℓ + · · · + ( − 1) ℓ y ℓ u + ( − 1) ℓ +1 λ, 8 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG and as functions of y 1 , . . . , y ℓ , λ , they satisfy the relations ∂ u j ∂ λ = ( − 1) ℓ f ′ ( u j ) , j = 1 , . . . , ℓ + 1 . F rom (2.3) we know that g ab λ = − ℓ +1 X r,s =1 ( ℓ + 1) δ rs − 1 ℓ + 1 u r u s σ a − 1 ( b u r ) σ b − 1 ( b u s ) , (2.8) where δ rs is the Kroneck er-Delta function. Th us b y using Lemma 2.5 we obtain η a +1 ,b +1 = ∂ ∂ λ g a +1 ,b +1 λ = ( − 1) ℓ +1 ℓ + 1 ℓ +1 X j =1 1 f ′ ( u j ) ∂ ∂ u j ℓ +1 X r,s =1 (( ℓ + 1) δ rs − 1) u r u s σ a ( b u r ) σ b ( b u s ) ! = ℓ +1 X j =1 ( − 1) ℓ +1 ( F j + G j ) ( ℓ + 1) f ′ ( u j ) , (2.9) where 1 ≤ a, b ≤ ℓ − 1, and F j = − ( ℓ − a )( ℓ − b ) [ σ a +1 ( u 1 , . . . , u ℓ +1 ) σ b ( b u j ) + σ a ( b u j ) σ b +1 ( u 1 , . . . , u ℓ +1 )] , G j = ( ℓ + 1) X r  = j [ σ a ( b u j , b u r ) σ b +1 ( b u r ) + σ a +1 ( b u r ) σ b ( b u j , b u r )] . In order to simplify the expression of (2.9) w e need the follo wing lemma. Lemma 2.6. F or 1 ≤ a, b ≤ ℓ , a + b ≥ ℓ , we have ℓ +1 X j =1 σ a ( b u j ) f ′ ( u j ) = ( − 1) ℓ δ aℓ , ℓ +1 X j =1 σ a ( b u j ) σ b ( b u j ) f ′ ( u j ) = ( − 1) ℓ σ a + b − ℓ ( u 1 , . . . , u ℓ +1 ) . (2.10) Pr o of. The first set of identities follows easily from the fact that σ a ( b u j ) = ( − 1) a Res u =0 f ( u ) ( u − u j ) u ℓ +1 − a , ℓ +1 X j =1 1 f ′ ( u j )( u − u j ) = 1 f ( u ) . AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 9 The second set of identities also hold true, since ℓ +1 X j =1 σ a ( b u j ) σ b ( b u j ) f ′ ( u j ) = ℓ +1 X j =1 1 f ′ ( u j ) Res z =0 Res u =0 ( − 1) a + b f ( z ) f ( u ) ( z − u j )( u − u j ) z ℓ +1 − a u ℓ +1 − b = Res z =0 Res u =0 ( − 1) a + b f ( z ) z ℓ +1 − a f ( u ) u ℓ +1 − b ℓ +1 X j =1 1 f ′ ( u j )  1 z − u j − 1 u − u j  1 u − z = Res z =0 Res u =0 ( − 1) a + b 1 z ℓ +1 − a 1 u ℓ +1 − b 1 u − z [ f ( u ) − f ( z )] = Res z =0 Res u =0 ( − 1) a + b 1 z ℓ +1 − a 1 u ℓ +1 − b ℓ X k =0 ( − 1) k σ k ( u 1 , . . . , u ℓ +1 ) ℓ − k X p =0 u ℓ − k − p z p = ℓ X k =0 ℓ − k X p =0 ( − 1) a + b + k σ k ( u 1 , . . . , u ℓ +1 ) δ ℓ − a,p δ ℓ − b,ℓ − k − p = ( − 1) ℓ σ a + b − ℓ ( u 1 , . . . , u ℓ +1 ) . The lemma is pro v ed. □ Pr o of of Pr op osition 2.4. It follo ws from Lemma 2.6 that ℓ +1 X j =1 ( − 1) ℓ +1 F j f ′ ( u j )( ℓ + 1) = 0 , so w e only need to compute ℓ +1 X j =1 ( − 1) ℓ +1 G j f ′ ( u j )( ℓ + 1) = ℓ +1 X j =1 ( − 1) ℓ +1 f ′ ( u j ) X r  = j ( σ a ( b u j , b u r ) σ b +1 ( b u r ) + σ a +1 ( b u r ) σ b ( b u j , b u r )) for a, b satisfying the conditions 1 ≤ a, b ≤ ℓ − 1 and ℓ ≤ a + b ≤ 2 ℓ − 2. By using the iden tities σ a ( b u j , b u r ) σ b +1 ( b u r ) = ( σ a ( b u j ) − w r σ a − 1 ( b u j , b u r )) σ b +1 ( b u r ) = σ a ( b u j ) σ b +1 ( b u r ) − σ a − 1 ( b u j , b u r ) ( σ b +2 ( u 1 , . . . , u ℓ +1 ) − σ b +2 ( b u r )) = σ a ( b u j ) σ b +1 ( b u r ) + σ a − 1 ( b u j , b u r ) σ b +2 ( b u r ) − σ a − 1 ( b u j , b u r ) σ b +2 ( u 1 , . . . , u ℓ +1 ) , 10 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG and b y using Lemmas 2.5, 2.6 w e ha v e ℓ +1 X j =1 ( − 1) ℓ +1 f ′ ( u j ) X r  = j σ a ( b u j , b u r ) σ b +1 ( b u r ) = ℓ +1 X j =1 ( − 1) ℓ +1 f ′ ( u j ) X r  = j σ a − 1 ( b u j , b u r ) σ b +2 ( b u r ) + ℓ +1 X j =1 ( − 1) ℓ +1 f ′ ( u j ) σ a ( b u j ) (( ℓ − b ) σ b +1 ( u 1 , . . . , u ℓ +1 ) − σ b +1 ( b u j )) − ℓ +1 X j =1 ( − 1) ℓ +1 f ′ ( u j ) ( ℓ − a + 1) σ a − 1 ( b u j ) σ b +2 ( u 1 , . . . , u ℓ +1 ) = ℓ +1 X j =1 ( − 1) ℓ +1 f ′ ( u j ) X r  = j σ a − 1 ( b u j , b u r ) σ b +2 ( b u r ) + σ a + b +1 − ℓ ( u 1 , . . . , u ℓ +1 ) = ℓ +1 X j =1 ( − 1) ℓ +1 f ′ ( u j ) X r  = j σ a + b +1 − ℓ ( b u j , b u r ) σ ℓ ( b u r ) + ( ℓ − b − 1) σ a + b +1 − ℓ ( u 1 , . . . , u ℓ +1 ) = ( ℓ − b ) σ a + b +1 − ℓ ( u 1 , . . . , u ℓ +1 ) . Th us w e arrive at η a +1 ,b +1 = (2 ℓ − a − b ) σ a + b +1 − ℓ ( u 1 , . . . , u ℓ +1 ) = (2 ℓ − a − b ) y a + b +1 − ℓ , whic h yields the form ula (2.6). Finally , The formulae (2.7) for the contra v ariant comp onen ts of η follow from the relations ∂ η ij ∂ y k = Γ ij η ,k + Γ j i η ,k , η is Γ j k η ,s = η j s Γ ik η ,s . The proposition is prov ed. □ Theorem 2.7. Ther e exist quasi-homo gene ous p olynomials t α = t α ( y 1 , . . . , y α ) , α = 1 , . . . , ℓ of de gr e es d α = α ℓ +1 such that t 1 , . . . , t ℓ ar e flat c o or dinates of η . Mor e over, the line ar p art of t α is given by y α . Pr o of. The flat co ordinates of η are solutions to the system of equations η ik ∂ 2 t ∂ y k ∂ y j + Γ ik η ,j ∂ t ∂ y k = 0 . (2.11) Let ψ i = ∂ t ∂ y i , then (2.11) can b e written as the follo wing system of equations for Ψ = ( ψ 1 , . . . , ψ ℓ ): ∂ ψ i ∂ y j − γ k ij ψ k = 0 , i, j = 1 , . . . , ℓ, (2.12) AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 11 where γ k ij = − η ir Γ rk η ,j . F rom (2.6) it follows that γ k ij are quasi-homogeneous p olynomials of y 1 , . . . , y ℓ of degree k − i − j ℓ +1 , so we can find a fundamental system of solutions Ψ α = ( ψ α 1 , . . . , ψ α ℓ ) , α = 1 , . . . ℓ, of (2.12) whic h satisfy the initial condition ψ i j (0) = δ i j , i, j = 1 , . . . , ℓ, and are analytic at ( y 1 , . . . , y ℓ ) = (0 , . . . , 0). Since the system of equations (2.12) is inv ariant w.r.t. the transformation y j → c j ℓ +1 y j , ψ j → c − j ℓ +1 ψ j , j = 1 , . . . , ℓ for any nonzero constant c and deg y i = i ℓ +1 > 0, the functions ψ α j m ust b e quasi-homogeneous p olynomials in y 1 , . . . , y ℓ of degrees α − j ℓ +1 . Thus we can choose the desired system of flat co ordinates t 1 , . . . , t ℓ of the metric η by using the relations ∂ t α ∂ y j = ψ α j . The theorem is prov ed. □ Corollary 2. 8. In the flat c o or dinates t 1 , . . . , t ℓ , the c omp onents of the metric η ar e given by η αβ = ( ℓ + 1) δ β ,ℓ +1 − α ; and the c omp onents of the metric g and the Christoffel symb ols of its L evi-Civita c onne ction ar e quasi- homo gene ous p olynomials in t 1 , . . . , t ℓ with deg g αβ ( t ) = α + β ℓ + 1 , deg Γ αβ γ ( t ) = α + β − γ ℓ + 1 ; mor e over, the ve ctor field define by (1.12) has the expr ession E = ℓ X α =1 d α t α ∂ ∂ t α = ℓ X α =1 α ℓ + 1 t α ∂ ∂ t α . (2.13) Note that if w e define the in v olution ∗ : { 1 , . . . , ℓ } → { 1 , . . . , ℓ } , α 7→ α ∗ := ℓ + 1 − α, then the degrees of t 1 , . . . , t ℓ satisfy the duality relation deg t α + deg t α ∗ = 1 . 2.4. The generalized F rob enius manifold sturcture on M D ( A ℓ , ω ℓ ) and mono drom y group. F rom Theorems 1.1, 2.3 it follows that there is a generalized F rob enius manifold structure of charge d = 1 on M D ( A ℓ , ω ℓ ) with flat metric η and multiplication defined by (1.17). The unit vector field and the Euler v ector field are giv en b y e = η ♯ ( ω e ) = − η ♯ (d log y ℓ ) (2.14) and by (2.13), and the in tersection form is giv en by g . F rom Theorem 2.7 we also know that the con trav ariant comp onen ts Γ αβ γ ( t ) of the Levi-Civita connection of the in tersection form g are quasi- homogeneous p olynomials of the flat co ordinates t 1 , . . . , t ℓ , so the structure constants c αβ γ of the generalized F rob enius algebra defined by (1.19) are also quasi-homogeneous p olynomials of the flat co ordinates. 12 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG The generalized F rob enius manifold structure can also b e characterized b y its p otential F ( t ), which is a quasi-homogeneous polynomial of degree 2 defined b y ∂ 3 F ∂ t α ∂ t β ∂ t γ = η γ ξ c ξ αβ , α, β , γ = 1 , . . . , ℓ. It can also be determined b y using the in tersection form g as follo ws: ∂ 2 F ∂ t α ∂ t β ( t ) = 1 2 − d α − d β η αξ η β ζ g ξζ ( t ) = ℓ + 1 2 ℓ + 2 − α − β η αξ η β ζ g ξζ ( t ) , α, β = 1 , . . . , ℓ. F rom [14] we know that the monodromy group of M D ( A ℓ , ω ℓ ) is given by Stab W ( ω ℓ ) ⋉ Z ℓ , where Stab W ( ω ) = { σ ∈ W ( R ) | σ ( ω ) = ω } (2.15) is a parab olic subgroup of W ( R ) whic h can b e represented by Stab W ( ω ) = ⟨ σ i | i = 1 , . . . , ℓ − 1 ⟩ ∼ = S ℓ . (2.16) 2.5. Examples. Let us giv e some examples to illustrate the abov e construction of generalized F rob e- nius manifold structures on M D ( A ℓ , ω ℓ ). Example 2.1. L et ( R, ω ) = ( A 1 , ω 1 ) . We have the W a ( R ) -invariant λ -F ourier p olynomial y 1 = λ 1 2 Y 1 = e 2 π ix 1 + λe − 2 π ix 1 . The c ontr avariant metric on V ⊗ C is given by (d x 1 , d x 1 ) = ( α ∨ 1 , α ∨ 1 ) − 1 = 1 2 , which induc es the metric g 11 λ = − 1 2 ( y 1 ) 2 + 2 λ on M . In the flat c o or dinate t 1 = y 1 of η , the flat p encil η , g has the form η 11 = 2 , g 11 = − 1 2 ( t 1 ) 2 . Thus we obtain a gener alize d F r ob enius manifold structur e with p otential F = − 1 96 ( t 1 ) 4 . The Euler ve ctor field and the unity ar e given by E = 1 2 t 1 ∂ ∂ t 1 , e = − 2 t 1 ∂ ∂ t 1 . Example 2.2. L et ( R, ω ) = ( A 2 , ω 2 ) . We have the W a ( R ) -invariant λ -F ourier p olynomials y 1 = λ 1 3 Y 1 = e 2 π ix 1 + e 2 π i ( x 2 − x 1 ) + λe − 2 π ix 2 , y 2 = λ 2 3 Y 2 = e 2 π ix 2 + λe − 2 π ix 1 + λe 2 π i ( x 1 − x 2 ) . The c ontr avariant metric on V ⊗ C is given by  ( α ∨ i , α ∨ j )  =  2 − 1 − 1 2  ,  d x i , d x j  =  2 − 1 − 1 2  − 1 = 1 3  2 1 1 2  , AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 13 which induc es the metric  g ij λ  = − 2 3 ( y 1 ) 2 + 2 y 2 3 λ − 1 3 y 1 y 2 3 λ − 1 3 y 1 y 2 2 λy 1 − 2 3 ( y 2 ) 2 ! on M . We have the flat p encil  η ij  =  0 3 3 2 y 1  ,  g ij  = − 2 3 ( y 1 ) 2 + 2 y 2 − 1 3 y 1 y 2 − 1 3 y 1 y 2 − 2 3 ( y 2 ) 2 ! . The metric η has flat c o or dinates t 1 = y 1 , t 2 = y 2 − 1 6 ( y 1 ) 2 , in which the metrics η and g have the form  η αβ  =  0 3 3 0  ,  g αβ  = 2 t 2 − 1 3 ( t 1 ) 2 − t 1 t 2 + 1 18 ( t 1 ) 3 − t 1 t 2 + 1 18 ( t 1 ) 3 − 1 54 (6 t 2 − ( t 1 ) 2 ) 2 ! . Thus we obtain a gener alize d F r ob enius manifold M D ( A 2 , ω 2 ) with p otential F = 1 18 ( t 2 ) 3 − 1 36 ( t 1 ) 2 ( t 2 ) 2 + 1 648 ( t 1 ) 4 t 2 − 1 19440 ( t 1 ) 6 . The Euler ve ctor field and the unity ar e given by E = 1 3 t 1 ∂ ∂ t 1 + 2 3 t 2 ∂ ∂ t 2 , e = − 18 ( t 1 ) 2 + 6 t 2 ∂ ∂ t 1 − 6 t 1 ( t 1 ) 2 + 6 t 2 ∂ ∂ t 2 . Example 2.3. L et ( R, ω ) = ( A 3 , ω 3 ) . We have the W a ( R ) -invariant λ -F ourier p olynomials y 1 = e 2 π ix 1 + e − 2 π i ( x 1 − x 2 ) + λe − 2 π ix 3 + e − 2 π i ( x 2 − x 3 ) , y 2 = λ 1 2 Y 2 = e 2 π ix 2 + λe − 2 π ix 2 + λe 2 π i ( x 1 − x 3 ) + λe − 2 π i ( x 1 − x 2 + x 3 ) + e − 2 π i ( x 1 − x 3 ) + e 2 π i ( x 1 − x 2 + x 3 ) , y 3 = e 2 π ix 3 + λe 2 π i ( x 1 − x 2 ) + λe − 2 π ix 1 + λe 2 π i ( x 2 − x 3 ) . The c ontr avariant metric on V ⊗ C is given by (d x i , d x j ) =  ( α ∨ i , α ∨ j )  − 1 = 1 4   3 2 1 2 4 2 1 2 3   , which induc es the metric  g ij λ  =    − 3 4 ( y 1 ) 2 + 2 y 2 − 1 2 y 1 y 2 + 3 y 3 − 1 4 y 1 y 3 + 4 λ − 1 2 y 1 y 2 + 3 y 3 − ( y 2 ) 2 + 2 y 1 y 3 + 4 λ − 1 2 y 2 y 3 + 3 λy 1 − 1 4 y 1 y 3 + 4 λ − 1 2 y 2 y 3 + 3 λy 1 − 3 4 ( y 3 ) 2 + 2 λy 2    . on M . We have the flat p encil  η ij  =   0 0 4 0 4 3 y 1 4 3 y 1 2 y 2   ,  g ij  =    − 3 4 ( y 1 ) 2 + 2 y 2 − 1 2 y 1 y 2 + 3 y 3 − 1 4 y 1 y 3 − 1 2 y 1 y 2 + 3 y 3 − ( y 2 ) 2 + 2 y 1 y 3 − 1 2 y 2 y 3 − 1 4 y 1 y 3 − 1 2 y 2 y 3 − 3 4 ( y 3 ) 2    . 14 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG The metric η has flat c o or dinates t 1 = y 1 , t 2 = y 2 − 1 4 ( y 1 ) 2 , t 3 = y 3 − 1 4 y 1 y 2 + 5 96 ( y 1 ) 3 , in which the metric η has the form  η αβ  =   0 0 4 0 4 0 4 0 0   , and the interse ction form is given by g 11 ( t ) = − 1 4 ( t 1 ) 2 + 2 t 2 , g 21 ( t ) = 1 32 (( t 1 ) 3 − 24 t 1 t 2 + 96 t 3 ) , g 13 ( t ) = − 1 384 ( t 1 ) 4 + 1 8 ( t 1 ) 2 t 2 − 1 2 ( t 2 ) 2 − t 1 t 3 , g 22 ( t ) = − 1 96 ( t 1 ) 4 + 1 4 ( t 1 ) 2 t 2 − ( t 2 ) 2 − t 1 t 3 , g 23 ( t ) = 5(( t 1 ) 2 − 8 t 2 )(( t 1 ) 3 − 24 t 1 t 2 + 96 t 3 ) 3072 , g 33 ( t ) = − ( t 1 ) 6 + 30( t 1 ) 4 t 2 − 288( t 1 ) 2 ( t 2 ) 2 − 48( t 1 ) 3 t 3 + 1152 t 1 t 2 t 3 + 384( t 2 ) 3 − 2304( t 3 ) 2 3072 . We have the p otential F = − ( t 1 ) 8 4128768 + ( t 1 ) 6 t 2 73728 − ( t 1 ) 5 t 3 30720 − ( t 1 ) 4 ( t 2 ) 2 3072 + ( t 1 ) 3 t 2 t 3 384 + ( t 1 ) 2 ( t 2 ) 3 384 − ( t 1 ) 2 ( t 3 ) 2 64 − t 1 ( t 2 ) 2 t 3 32 − ( t 2 ) 4 192 + t 2 ( t 3 ) 2 8 . of the gener alize d F r ob enius manifold. The Euler ve ctor field is given by E = 1 4 t 1 ∂ ∂ t 1 + 1 2 t 2 ∂ ∂ t 2 + 3 4 t 3 ∂ ∂ t 3 , and the unity has the expr ession e = − 12 ( t 1 ) 3 + 24 t 1 t 2 + 96 t 3  32 ∂ ∂ t 1 + 8 t 1 ∂ ∂ t 2 + (( t 1 ) 2 + 8 t 2 ) ∂ ∂ t 3  . 2.6. The Landau-Ginzburg sup erp otential. In this subsection, we will sho w that the abov e gen- eralized F rob enius manifold structures on M D ( A ℓ , ω ℓ ) can b e realized b y using the following LG sup erpotentials: Λ( p ) = p ℓ +1 + a 1 p ℓ + a 2 p ℓ − 1 + · · · + a ℓ p, a 1 , . . . , a ℓ ∈ C , (2.17) follo wing the approach of constructing F rob enius manifold structures on Hurwitz spaces giv en in [6]. Let M ℓ b e the space M ℓ =  ( a 1 , . . . , a ℓ ) ∈ C ℓ  . W e define on M ℓ the follo wing (0 , 2)-type tensor ˜ η ( ∂ ′ , ∂ ′′ ) = X q :Λ ′ ( q )=0 Res p = q ∂ ′ (Λ) ∂ ′′ (Λ) p 2 Λ ′ ( p ) d p, (2.18) AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 15 where the summation runs o v er the critical p oin ts of Λ. W e also define a (0 , 3)-type tensor as follows: ˜ c ( ∂ ′ , ∂ ′′ , ∂ ′′′ ) = X q :Λ ′ ( q )=0 Res p = q ∂ ′ (Λ) ∂ ′′ (Λ) ∂ ′′′ (Λ) p 2 Λ ′ ( p ) d p. (2.19) Lemma 2.9. The tensor ˜ η is a flat metric on M ℓ , and it has flat c o or dinates ˜ t α = − ℓ + 1 α Res p = ∞  Λ α ℓ +1 ( p ) d p p  = a α + f α ( a 1 , . . . , a α − 1 ) , α = 1 , . . . , ℓ. (2.20) Her e f α ar e homo gene ous p olynomials of a 1 , . . . , a α − 1 with deg a α = α ℓ + 1 , α = 1 , . . . , ℓ. Pr o of. Let k b e the following ( ℓ + 1)-th ro ot of Λ( p ): k = Λ 1 ℓ +1 = p + a 1 ℓ + 1 + O  1 p  , p → ∞ . By using the definition (2.20), one can obtain log p k = − 1 ℓ + 1 ℓ X α =1 ˜ t α k α + O  1 k ℓ +1  , k → ∞ , (2.21) whic h implies that − 1 p ∂ p ( k , ˜ t ) ∂ ˜ t α = 1 ℓ + 1 1 k α + O  1 k ℓ +1  , k → ∞ . According to the implicit function theorem, w e hav e ∂ ˜ t α (Λ( p ( k , ˜ t )) = − Λ ′ ( p ) ∂ p ( k , ˜ t ) ∂ ˜ t α , d p = ( ℓ + 1) k ℓ d k . So w e hav e ˜ η ( ∂ ˜ t α , ∂ ˜ t β ) = − Res k = ∞ 1 p 2 ∂ p ∂ ˜ t α ∂ p ∂ ˜ t β ( ℓ + 1) k ℓ d k = 1 ℓ + 1 δ α + β ,ℓ +1 . (2.22) The lemma is pro v ed. □ Let us define an op eration of multiplication on the tangent spaces of M ℓ b y the following relation: ˜ η ( ∂ ′ · ∂ ′′ , ∂ ′′′ ) = ˜ c ( ∂ ′ , ∂ ′′ , ∂ ′′′ ) . (2.23) W e are to show that this op eration yields a F rob enius manifold structure on M ℓ . Supp ose q 1 , . . . , q ℓ are the distinct critical p oints of Λ( p ), then we hav e Λ ′ ( p ) = ( ℓ + 1) ℓ Y i =1 ( p − q i ) . Let u i = Λ( q i ) b e the critical v alues of Λ( p ). Note that ∂ u i Λ( p ) | p = q j = δ ij , by using the Lagrange in terp olation formula, we hav e ∂ u i Λ( p ) = p Λ ′ ( p ) ( p − q i ) q i Λ ′′ ( q i ) , (2.24) whic h implies that ˜ η ( ∂ u i , ∂ u j ) = δ ij q 2 i Λ ′′ ( q i ) , ˜ c ( ∂ u i , ∂ u j , ∂ u k ) = δ ij δ ik q 2 i Λ ′′ ( q i ) . (2.25) 16 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG F rom (2.23) it follows that ∂ u i · ∂ u j = δ ij ∂ u i , th us u 1 , . . . , u ℓ are canonical co ordinates of the multiplication. The unity vector field and the Euler v ector field are giv en b y ˜ e = ℓ X i =1 ∂ u i , ˜ E = ℓ X i =1 u i ∂ u i . Lemma 2.10. The unity ve ctor field and the Euler ve ctor field c an also b e r epr esente d in the form ˜ e = − grad ˜ η log a ℓ , ˜ E = ℓ X α =1 α ℓ + 1 a α ∂ a α = ℓ X α =1 α ℓ + 1 ˜ t α ∂ ˜ t α . (2.26) Pr o of. Note that ∂ u i Λ( p ) = ℓ X α =1 ( ∂ u i a α ) p ℓ +1 − α , one can obtain from (2.24) that ∂ u i a ℓ = ( − 1) ℓ − 1 ( ℓ + 1) σ ℓ ( q i ) q 2 i Λ ′′ ( q i ) = − a ℓ q 2 i Λ ′′ ( q i ) . By using (2.25), we hav e ˜ e = ℓ X i =1 ∂ u i = − grad ˜ η log a ℓ . The second formula comes from the homogeneity of u i and ˜ t α . □ Lemma 2.11. If ther e lo c al ly exists a function φ , such that the unity c an b e r epr esente d as ˜ e = grad η φ, (2.27) then the (0 , 4) -typ e tensor ˜ c αβ γ ξ := ∂ ˜ t ξ ˜ c ( ∂ ˜ t α , ∂ ˜ t β , ∂ ˜ t γ ) is symmetric. Pr o of. Denote f i = 1 / ( q 2 i Λ ′′ ( q i )), then we hav e ˜ η ( ∂ u i , ∂ u j ) = f i δ ij , ˜ c ( ∂ u i , ∂ u j , ∂ u k ) = f i δ ij δ ik . F rom the condition (2.27) w e see that f i = ∂ φ ∂ u i , i = 1 , . . . , ℓ. (2.28) W e denote the Lam´ e coefficients and the rotation co efficien ts respectively by h i = p f i , γ ij = 1 2 p f i f j ∂ f i ∂ u j , i  = j . It follo ws from (2.28) that γ ij = γ j i , i.e., ˜ η is an Egoroff metric. Let e i = 1 h i ∂ u i , i = 1 , . . . , ℓ be the orthogonal frame. Supp ose the flat frame has the following expansion w.r.t. the orthogonal frame: ∂ ˜ t α = ℓ X i =1 ψ iα e i , α = 1 , . . . , ℓ. (2.29) AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 17 By calculating the co v ariant deriv ativ es ∇ u j ( ∂ ˜ t α ), one can obtain ∂ ψ j α ∂ u i = γ j i ψ iα , i  = j , (2.30) ∂ ψ iα ∂ u i = − X k  = i γ ki ψ kα . (2.31) On the other hand, ˜ c αβ γ = ˜ c ( ∂ ˜ t α , ∂ ˜ t β , ∂ ˜ t γ ) = ℓ X j =1 ψ j α ψ j β ψ j γ h j . (2.32) By using (2.29)-(2.32), we can sho w that ˜ c αβ γ ξ − ˜ c ξβ γ α = 1 2 X i  = j γ ij − γ j i h i h j ( ψ iα ψ j ξ − ψ j α ψ iξ )( ψ iβ ψ j γ + ψ j γ ψ iβ ) . Since γ ij = γ j i , w e hav e ˜ c αβ γ ξ = ˜ c ξβ γ α , the lemma is prov ed. □ F rom Lemma 2.11, we see that ( M ℓ , ˜ η , · ) gives a generalized F rob enius manifold structure. It is easy to verify that the Euler vector field ˜ E satisfies the follo wing relations: L ˜ E ( ∂ ′ · ∂ ′′ ) = L ˜ E ∂ ′ · ∂ ′′ + ∂ ′ · L ˜ E ∂ ′′ + ∂ ′ · ∂ ′′ , (2.33) L ˜ E ˜ η ( ∂ ′ , ∂ ′′ ) = ˜ η ( L ˜ E ∂ ′ , ∂ ′′ ) + ˜ η ( ∂ ′ , L ˜ E ∂ ′′ ) + ˜ η ( ∂ ′ , ∂ ′′ ) , (2.34) so M ℓ is a conformal generalized F rob enius manifold with charge d = 1. The in tersection form of M ℓ can be represen ted b y the follo wing residue formula: ˜ g ( ∂ ′ , ∂ ′′ ) = X q :Λ ′ ( q )=0 Res p = q ∂ ′ (Λ) ∂ ′′ (Λ) p 2 Λ( p )Λ ′ ( p ) d p. (2.35) It is defined on M ℓ \ Σ 0 , where Σ 0 := { p | Λ( p ) = 0 , Λ ′ ( p ) = 0 } . In terms of the canonical co ordinates, w e hav e ˜ g (d u i , d u j ) = u i q 2 i Λ ′′ ( q i ) δ ij . (2.36) Let us pro ceed to sho w that the generalized F robenius manifold structures on M ℓ and M D ( A ℓ , ω ℓ ) are isomorphic. W e factorize the p olynomial Λ( p ) as Λ( p ) = p ℓ Y α =1 ( p − e iϕ α ) , (2.37) and let h : M D ( A ℓ , ω ℓ ) → M ℓ b e induced by the map ( x 1 , . . . , x ℓ ) 7→ ( ϕ 1 , . . . , ϕ ℓ ) , (2.38) where ϕ 1 = 2 π x 1 , ϕ α = 2 π ( x α − x α − 1 ) , α = 2 , . . . , ℓ. (2.39) F rom (1.5),(2.1),(2.37) and (2.39), it follows that the map h : ( y 1 , . . . , y ℓ ) 7→ ( a 1 , . . . , a ℓ ) is given by a α = ( − 1) α y α | λ =0 , α = 1 , . . . , ℓ, (2.40) 18 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG so we can know that h is a diffeomorphism, and here the p olynomial Λ( p ) coincides with f 0 ( z ) := f ( z ) | λ =0 in (2.3). Theorem 2.12. The map h is an isomorphism b etwe en gener alize d F r ob enius manifolds. Pr o of. F rom (2.26) and (2.40), it follows that h ∗ e = ˜ e, h ∗ E = ˜ E . Since deg ˜ t α = d α > 0, we only need to prov e that h ∗ ˜ η = η, h ∗ ˜ g = g . According to the definition of the canonical co ordinates, w e ha v e d u i = ℓ X α =1 q ℓ +1 − α i d a α . By using the in v erse of the V andermonde matrix, we obtain ˜ η (d a α , d a β ) = ( ℓ + 1) 2 ℓ X s =1 σ α − 1 ( b q s ) σ β − 1 ( b q s ) Λ ′′ ( q s ) , here the notations σ α − 1 ( ˆ q s ) , σ β − 1 ( b q s ) are introduced in Section 2.1. By using the same metho d that is employ ed in the pro of of Lemma 2.6, w e can show that ˜ η (d a α , d a β ) =      ( − 1) α + β − ℓ − 1 (2 ℓ + 2 − α − β ) a α + β − ℓ − 1 , α + β ≥ ℓ + 2 , ℓ + 1 , α + β = ℓ + 1 , 0 , α + β ≤ ℓ, so w e hav e h ∗ ˜ η = η . F rom (2.24) and (2.37), it follows that ∂ u i Λ( p ) = − ℓ X α =1 ie iϕ α Λ( p ) p − e iϕ α ∂ u i ϕ α , ∂ ϕ α ∂ u i = i ( e iϕ α − q i ) q i Λ ′′ ( q i ) . Then w e hav e ˜ g (d ϕ α , d ϕ β ) = ℓ X j,k =1 ˜ g (d u j , d u k ) ∂ ϕ α ∂ u j ∂ ϕ β ∂ u k = − ℓ X j =1 u j ( e iϕ α − q j )( e iϕ β − q j )Λ ′′ ( q j ) = − ℓ X j =1 Res p = q j Λ( p ) ( p − e iϕ α )( p − e iϕ β )Λ ′ ( p ) d p =  Res p = e iϕ α + Res p = e iϕ β + Res p = ∞  Λ( p ) ( p − e iϕ α )( p − e iϕ β )Λ ′ ( p ) d p = δ αβ − 1 ℓ + 1 , whic h implies that h ∗ ˜ g = g . The theorem is pro ved. □ AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 19 Remark 2.1. The intr o duction of the gener alize d F r ob enius manifold structur e on M ℓ in terms of the r esidue formulae (2.18) and (2.19) is also given by Zhonglun Cao in his Ph.D. thesis [5]. In his thesis he attempte d to c onstruct this gener a lize d F r ob enius manifold structur e by using the ge ometry of the orbit sp ac e of a c ertain extension of the affine Weyl gr oup of typ e A ℓ , however, he did not pr ovide a rigor ous c onstruction. He also showe d in [5] that this gener alize d F r ob enius manifold c an b e obtaine d fr om the disp ersionless limit of the bihamiltonian structur e of the q -deforme d Gelfand–Dickey hier ar chy [11]. 3. The Case of ( C ℓ , ω 1 ) 3.1. The W a ( R ) -inv ariant λ -F ourier p olynomial ring. Let R b e the ro ot system of type C ℓ in an ℓ -dimensional Euclidean space V with orthonormal basis e 1 , . . . , e ℓ . W e tak e the simple roots α 1 = e 1 − e 2 , . . . , α ℓ − 1 = e ℓ − 1 − e ℓ , α ℓ = 2 e ℓ . The coroots and the fundamental weigh ts are giv en by α ∨ j = α j , α ∨ ℓ = 1 2 α ℓ , j = 1 , . . . , ℓ − 1 . ω j = α 1 + 2 α 2 + · · · + ( j − 1) α j − 1 + j  α j + · · · + α ℓ − 1 + 1 2 α ℓ  , j = 1 , . . . , ℓ. T ake ω = ω 1 , then we hav e θ j = ( ω j , ω 1 ) = 1 , j = 1 , . . . , ℓ, and κ = 1. W e define ξ 1 , . . . , ξ ℓ b y the relation cω 1 + x 1 α ∨ 1 + · · · + x ℓ α ∨ ℓ = cω 1 + ξ 1 e 1 + · · · + ξ ℓ e ℓ , and denote ζ j = e 2 π iξ j + e − 2 π iξ j , j = 1 , . . . , ℓ. The basic generators of the W a ( R )-inv ariant λ -F ourier p olynomial ring A W can b e represented in the form [4] y j := y j ( x ) = λσ j ( ζ 1 , . . . , ζ ℓ ) , j = 1 , . . . , ℓ, (3.1) here λ = e − 2 π ic . Let y 0 = λ , then we hav e the follo wing generating function for y 1 , . . . , y ℓ : P ( u ) = ℓ X j =0 y j u ℓ − j = λ ℓ Y k =1 ( u + ζ k ) . (3.2) 3.2. The p encil generators. Unlike the ( R, ω ) = ( A ℓ , ω ℓ ) case that w e studied in the last section, the basic y 1 , . . . , y ℓ are not p encil generators of A W . In order to find a set of p encil generators of A W , w e need to compute explicitly the metric g λ defined by (1.10) and the contra v arian t comp onen ts of its Levi-Civita connection. 20 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG Lemma 3.1. The fol lowing formulae hold true for the gener ating functions of the metric g λ and the c ontr avariant c omp onents of its L evi-Civita c onne ction in the c o or dinates y 1 , . . . , y ℓ : ℓ X i,j =1 g ij λ ( y ) u ℓ − i v ℓ − j = − ℓP ( u ) P ( v ) + u 2 − 4 u − v P ′ ( u ) P ( v ) − v 2 − 4 u − v P ( u ) P ′ ( v ) , (3.3) ℓ X i,j,k =1 Γ ij λ,k ( y )d y k u ℓ − i v ℓ − j = − ℓP ( u )d P ( v ) + u 2 − 4 u − v P ′ ( u )d P ( v ) − v 2 − 4 u − v P ( u )d P ′ ( v ) + uv − 4 ( u − v ) 2 ( P ( v )d P ( u ) − P ( u )d P ( v )) , (3.4) wher e P ′ ( u ) = ∂ P ∂ u ( u ) , P ′ ( v ) = ∂ P ∂ v ( v ) . Pr o of. (cf. [10]) F rom (1.9) we kno w that the contra v arian t metric a defined on V ⊗ C has the prop erty a (d ξ i , d ξ j ) = δ ij , i, j = 1 , . . . , ℓ. Th us b y using the iden tities ∂ P ∂ ξ k ( u ) = 2 π iP ( u )( e 2 π iξ k − e − 2 π iξ k ) u + ζ k , 1 ≤ k ≤ ℓ, (3.5) P ′ ( u ) = P ( u ) ℓ X k =1 1 u + ζ k , (3.6) w e obtain ℓ X j,k =0  ( φ λ ) ∗ a  (d y j , d y k ) u ℓ − j v ℓ − k = 1 4 π 2 ℓ X a =1 ∂ P ( u ) ∂ ξ a ∂ P ( v ) ∂ ξ a = − ℓ X a =1 P ( u ) P ( v ) ( ζ a ) 2 − 4 ( u + ζ a )( v + ζ a ) = − ℓ X a =1 P ( u ) P ( v )(1 − u 2 − 4 u − v 1 u + ζ a + v 2 − 4 u − v 1 v + ζ a ) = − ℓP ( u ) P ( v ) + u 2 − 4 u − v P ′ ( u ) P ( v ) − v 2 − 4 u − v P ( u ) P ′ ( v ) , so we prov ed the first formula (3.3). In a similar wa y we can prov e the second formula (3.4). The lemma is prov ed. □ The ab o v e lemma sho ws that g ij λ ( y ) are quadratic p olynomials in y 1 , . . . , y ℓ and λ , which may not dep end linearly on λ , so in general y 1 , . . . , y ℓ are not p encil generators of A W . Since deg y j = deg λ = 1 , j = 1 , . . . , ℓ, w e ma y attempt to construct prop er generators of the form z j = y j + c j λ, j = 1 , . . . , ℓ, (3.7) suc h that z 1 , . . . , z ℓ form a set of pencil generators of A W . The follo wing theorem sho ws that w e can indeed find p encil generators in this w a y . AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 21 Theorem 3.2. F or any fixe d numb er 0 ≤ m ≤ ℓ , ther e exist p encil gener ators z 1 , . . . , z ℓ of the form (3.7) , with c onstants c 1 , . . . , c ℓ define d by the gener ating function P 0 ( u ) = ℓ X j =1 c j u ℓ − j = u ℓ − ( u + 2) m ( u − 2) ℓ − m . (3.8) Pr o of. (cf. [10]) It suffices to find a p olynomial P 0 ( u ) = P ℓ j =1 c j u ℓ − j suc h that, after the shift P ( u ) → P ( u ) − λP 0 ( u ) , P ( v ) → P ( v ) − λP 0 ( v ) , the right hand side of (3.3) and (3.4) dep end at most linearly on λ . This condition is equiv alen t to the follo wing equation for P 0 : − ℓP 1 ( u ) P 1 ( v ) + u 2 − 4 u − v P ′ 1 ( u ) P 1 ( v ) − v 2 − 4 u − v P 1 ( u ) P ′ 1 ( v ) = 0 , (3.9) where P 1 ( u ) = u ℓ − P 0 ( u ). F rom the pro of of Theorem 3.4 of [10] w e kno w that w e can take P 1 ( u ) = ( u + 2) m ( u − 2) ℓ − m (3.10) for an y fixed 0 ≤ m ≤ ℓ . In order to pro v e that g = ( g ij ( z )) =  g λ (d z i , d z j )  | λ =0 , η = ( η ij ( z )) =  ∂ ∂ λ g λ (d z i , d z j )  (3.11) form a flat p encil of metrics, we need to show that the determinant of ( η ij ( z )) do es not v anishes at generic p oint of the orbit space M ( C ℓ , ω 1 ) of the affine W eyl group. Define the following generating function of the new co ordinates z 1 , . . . , z ℓ that are introduced in (3.7): Q ( u ) = ℓ X j =1 z j u ℓ − j , (3.12) then in the coordinates z 1 , . . . , z ℓ the metric η can be represented b y ℓ X j,k =1 η  d z j , d z k  u ℓ − j v ℓ − k = − ℓ ( Q ( u ) P 1 ( v ) + Q ( v ) P 1 ( u )) + u 2 − 4 u − v ( Q ′ ( u ) P 1 ( v ) + P ′ 1 ( u ) Q ( v )) − v 2 − 4 u − v ( Q ′ ( v ) P 1 ( u ) + P ′ 1 ( v ) Q ( u )) . (3.13) T o prov e the non-degeneracy of the metric η , let us adopt the metho d of calculation for the metric η giv en in the pro of of Theorem 3.4 of [10] to the present case. F or any fixed 0 ≤ m ≤ ℓ , consider the linear c hange of co ordinates ( z 1 , . . . , z ℓ ) 7→ ( τ 1 , . . . , τ ℓ ) defined b y ℓ X j =1 z j u ℓ − j = ℓ − m X j =1 τ j ( u + 2) m ( u − 2) ℓ − m − j − ℓ X j = ℓ − m +1 τ j ( u + 2) ℓ − j ( u − 2) j − 1 . (3.14) By inserting the expressions for Q ( u ) , Q ( v ) in to b oth sides of (3.13), we obtain an identit y in the v ariables u, v . Dividing this identit y by P 1 ( u ) and P 1 ( v ) yields a new identit y relating tw o rational 22 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG functions in u (treating v as a parameter), which p ossess p oles at u = ± 2. Comparing the regular and singular parts of this expression at u = 2 and u = − 2 leads to explicit formulae for the matrix elemen ts of  η ( dτ i , dτ j )  . This matrix exhibits a block diagonal form  W 1 0 0 W 2  , (3.15) with an ti-triangular matrices W 1 =        R 1 R 2 · · · R ℓ − m R 2 R 3 · · · 0 . . . . . . . . . R ℓ − m − 1 R ℓ − m · · · 0 R ℓ − m 0 · · · 0        , W 2 =        S 1 S 2 · · · S m S 2 S 3 · · · 0 . . . . . . . . . S m − 1 S m · · · 0 S m 0 · · · 0        , (3.16) they ha v e entries R s = 4 sτ s + (1 − δ s,ℓ − m )( s + 1) τ s +1 , S r = 4 r τ ℓ − m + r − 4(1 − δ r,m ) r τ ℓ − m + r +1 (3.17) for 1 ≤ s ≤ ℓ − m, 1 ≤ r ≤ m . Thus by a simple computation w e get det( η ij ( τ )) = ( ( − 1) ℓ 2 − (2 m +1) ℓ +2 m 2 2 4 ℓ m m ( ℓ − m ) ℓ − m ( τ ℓ − m ) ℓ − m ( τ ℓ ) m , m  = 0 , ℓ ; ( − 1) ℓ ( ℓ − 1) 2 4 ℓ ℓ ℓ ( τ ℓ ) ℓ , m = 0 , ℓ. (3.18) So the matrix ( η ij ( τ )) is non-degenerate on M ℓ,m := M ( C ℓ , ω 1 ) \ ( { τ ℓ = 0 } ∪ { τ ℓ − m = 0 } ) (3.19) when m  = 0 , ℓ , and on M ℓ, 0 = M ℓ,ℓ := M ( C ℓ , ω 1 ) \ { τ ℓ = 0 } when m = 0 , ℓ . Th us  g ij ( z )  and  η ij ( z )  form a flat p encil of metrics on M m , and z 1 , . . . , z ℓ are p encil generators of A W . The theorem is prov ed. □ Corollary 3.3. In the c o or dinates τ 1 , . . . , τ ℓ , the c omp onents of the metric g =  g ij ( τ )  and the c ontr avariant c omp onents Γ ij k ( τ ) of its L evi-Civita c onne ction ar e quasi-homo gene ous p olynomials with de gr e es deg g ij ( τ ) = i + j, deg Γ ij k ( τ ) = i + j − k , (3.20) and we have deg τ j = j . 3.3. Flat co ordinates of the metric η . In this subsection, we are to show that the flat co ordinates of the metric η = ( η ij ( z )) defined in the last subsection are algebraic functions of τ 1 , . . . , τ ℓ . Since the explicit form of the matrix  η ( dτ i , dτ j )  giv en b y (3.15)–(3.17) coincides with the  W 2 0 0 W 3  blo c k of the matrix  η ij ( τ )  giv en in (3.26) of [10], we can adopt directly the results of Lemma 3.8, Lemma 3.9 and Theorem 3.11 of [10], by setting the parameter k that app ears there to b e zero, to c haracterize properties of flat co ordinates of η . W e first p erform c hanges of co ordinates to simplify the matrix ( η ij ( τ )). AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 23 Lemma 3.4 (cf. Lemma 3.8 of [10]) . Ther e exists a system of c o or dinates w 1 , . . . , w ℓ of the form w j = τ j + ℓ − m X s = j +1 c j s τ s , 1 ≤ j ≤ ℓ − m − 1 , w j = τ j + ℓ X s = j +1 h j s τ s , ℓ − m + 1 ≤ j ≤ ℓ − 1 , w ℓ − m = τ ℓ − m , w ℓ = τ ℓ . with c ertain c onstants c j s , h j s , such that the the matrix  η (d w i , d w j )  stil l p ossesses blo ck diagonal form (3.15)–(3.16) with the entries r eplac e d by R s = 4 sw s , S r = 4 r w ℓ − m + r , 1 ≤ s ≤ ℓ − m, 1 ≤ r ≤ m. The follo wing lemma simplifies the expression of the metric η further. Lemma 3.5 (cf. Lemma 3.9 of [10]) . In the new c o or dinates v 1 , . . . , v ℓ define d by v 1 = w 1 ( w ℓ − m ) − 1 2( ℓ − m ) , v ℓ = ( w ℓ ) 1 2 m , v s = w s ( w ℓ − m ) − s ℓ − m , 2 ≤ s ≤ ℓ − m − 1 , v ℓ − m = ( w ℓ − m ) 1 2( ℓ − m ) , v ℓ − m +1 = w ℓ − m +1 ( w ℓ ) − 1 2 m , v r = w r ( w ℓ ) − r + m − ℓ m , ℓ − m + 2 ≤ r ≤ ℓ − 1 . the metric η has the expr ession  B 1 0 0 B 2  , (3.21) wher e B 1 , B 2 ar e anti-triangular matric es of the form B 1 =            0 0 0 0 · · · 0 2 0 H 3 H 4 · · · H ℓ − m − 1 H ℓ − m 0 H 4 H 5 · · · H ℓ − m . . . . . . . . . · · · 0 H ℓ − m − 1 H ℓ − m 0 H ℓ − m 2            , B 2 =            0 0 0 0 · · · 0 2 0 H ℓ − m +3 H ℓ − m +4 · · · H ℓ − 1 H ℓ 0 H ℓ − m +4 H ℓ − m +5 · · · H ℓ . . . . . . . . . · · · 0 H ℓ − 1 H ℓ 0 H ℓ 2            , 24 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG with H s = 4 s ( v ℓ − m ) − 2 v s , H ℓ − m = 4( ℓ − m )( v ℓ − m ) − 2 , H ℓ − m + j = 4 j ( v ℓ ) − 2 v ℓ − m + j , H ℓ = 4 m ( v ℓ ) − 2 , 3 ≤ s ≤ ℓ − m − 1 , 3 ≤ j ≤ m − 1 . Remark 3.1. When m = 0 (r esp. m = ℓ ), the matrix B 2 (r esp. B 1 )do es not app e ar in (3.21) . When m = 1 (r esp. m = ℓ − 1 ), we have B 2 = 1 (r esp. B 1 = 1 ). When m = 2 (r esp. m = ℓ − 2 ), the matrix B 2 (r esp. B 1 ) has the form  0 2 2 0  . Theorem 3.6 (cf. Theorem 3.11 of [10]) . One c an cho ose flat c o or dinates of the metric η of the form t 1 = v 1 + v ℓ − m h 1 ( v 2 , . . . , v ℓ − m − 1 ) , t α = v ℓ − m ( v α + h α ( v α +1 , . . . , v ℓ − m − 1 )) , 2 ≤ α ≤ ℓ − m − 1 , t ℓ − m = v ℓ − m , t ℓ − m +1 = v ℓ − m +1 + v ℓ h ℓ − m +1 ( v ℓ − m +2 , . . . , v ℓ − 1 ) , t β = v ℓ ( v β + h β ( v β +1 , . . . , v ℓ − 1 )) , ℓ − m + 2 ≤ β ≤ ℓ − 1 , t ℓ = v ℓ . Her e h ℓ − m − 1 = h ℓ − 1 = 0 , h α ar e quasi-homo gene ous p olynomials of de gr e e ℓ − m − α ℓ − m for 1 ≤ α ≤ ℓ − m − 2 , and h β ar e quasi-homo gene ous p olynomials of de gr e e ℓ − β m for ℓ − m + 1 ≤ β ≤ ℓ − 2 . F rom the ab o ve-men tioned construction of the flat co ordinates t 1 , . . . , t ℓ , we know that they are quasi-homogeneous functions of z 1 , . . . , z ℓ with degrees d α = deg t α = 2( ℓ − m − α ) + 1 2( ℓ − m ) , 1 ≤ α ≤ ℓ − m ; (3.22) d β = deg t β = 2( ℓ − β ) + 1 2 m , ℓ − m + 1 ≤ β ≤ ℓ. (3.23) These num be rs satisfy a duality relation whic h is similar to that of [8] and [10]. T o describ e this relation, let R b e the Dynkin diagram of type C ℓ . F or any giv en in teger 0 ≤ m ≤ ℓ , we separate R into t wo componnets, the first one is formed by the first ℓ − m vertices, and the second one is formed b y the remaining m v ertices. On each component, w e ha v e an in v olution β 7→ β ∗ giv en b y the reflection with respect to its center. Then we hav e d β + d β ∗ = 1 , β = 1 , . . . , ℓ, (3.24) and η αβ ( t ) is a nonzero constant if and only if β = α ∗ . W e ha v e the following corollaries. Corollary 3. 7. In the flat c o or dinates t 1 , . . . , t ℓ +1 , the matrix  η αβ ( t )  has the form  A 1 0 0 A 2  , (3.25) AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 25 wher e A 1 , A 2 ar e ( ℓ − m ) × ( ℓ − m ) and m × m matric es r esp e ctively, and they have the form A 1 =       2 4( ℓ − m ) . . . 4( ℓ − m ) 2       , A 2 =       2 4 m . . . 4 m 2       (3.26) when m, ℓ − m  = 0 , 1 , 2 ; when m = 0 (r esp. m = ℓ ), the matrix A 2 (r esp. A 1 ) do es not app e ar; when m = 1 (r esp. m = ℓ − 1 ), we have A 2 = 1 (r esp. A 1 = 1 ); when m = 2 or m = ℓ − 2 , we have A 2 =  0 2 2 0  or A 1 =  0 2 2 0  . Corollary 3.8. In the flat c o or dinates t 1 , . . . , t ℓ , the entries of the matric es  g αβ ( t )  ,  Γ αβ γ ( t )  ar e quasi-homo gene ous p olynomials of t 1 , . . . , t ℓ , 1 t ℓ − m , 1 t ℓ of de gr e es d α + d β and d α + d β − d γ . 3.4. The generalized F robenius manifold structures. F or eac h fixed 0 ≤ m ≤ ℓ , from Theo- rems 1.1, 3.2 it follows that there is a generalized F rob enius manifold structure of charge d = 1 on M D ( C ℓ , ω 1 ) defined by the flat pencil of metrics g , η giv en in (3.2). The unit vector field and the Euler v ector field are giv en b y e = η ♯ ( ω e ) = − η ♯ (d log y 1 ) , E = ℓ X α =1 d α t α ∂ ∂ t α , (3.27) with d α defined by (3.22) and (3.23). The structure constants of the generalized F rob enius manifold are p olynomials in t 1 , . . . , t ℓ , 1 t ℓ − m , 1 t ℓ . W e can determine the p otential of the generalized F rob enius manifolds b y the relation ∂ 2 F ∂ t α ∂ t β ( t ) = 1 2 − d α − d β η αξ η β ζ g ξζ ( t ) , α, β = 1 , . . . , ℓ. Remark 3.2. F r om the ab ove c onstruction, we se e that the gener alize d F r ob enius manifold structur es on M ℓ,m and M ℓ,ℓ − m ar e e quivalent. 3.5. Examples. In this subsection, we give some examples to illustrate the ab o ve constructure of generalized F rob enius manifold structures associated to ( C ℓ , ω 1 ). Example 3.1. L et ( R, ω ) = ( C 2 , ω 1 ) . We have the W a ( R ) -invariant λ -F ourier p olynomials y 1 = e 2 π ix 1 + λe 2 π i ( x 1 − x 2 ) + λe − 2 π i ( x 1 − x 2 ) + λ 2 e − 2 π ix 1 , y 2 = e 2 π ix 2 + e 2 π i (2 x 1 − x 2 ) + λ 2 e − 2 π ix 2 + λ 2 e − 2 π i (2 x 1 − x 2 ) . The c ontr avariant metric on V ⊗ C is given by  (d x i , d x j )  =  ( α ∨ i , α ∨ j )  − 1 =  1 1 1 2  , which induc es the metric  g ij λ ( y )  = − ( y 1 ) 2 + 2 λy 2 + 8 λ 2 − y 1 y 2 + 4 λy 1 − y 1 y 2 + 4 λy 1 − 2( y 2 ) 2 + 4( y 1 ) 2 − 8 λy 2 ! . 26 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG F r om The or em 3.2 we know that we have thr e e differ ent choic es p encil gener ators by taking m = 0 , 1 , 2 , and g = ( g ij ) = ( g ij 0 ) is indep endent of the choic e of p encil gener ators. Note that the gener alize d F r ob enius structur e for m = 0 , 2 ar e isomorphic, so we only ne e d to c onsider the c ase of m = 0 and m = 1 . Case 1. By taking m = 0 , we get the p encil gener ators z 1 = y 1 + 4 λ, z 2 = y 2 − 4 λ, and the metric η = ( η ij ( z )) =  8 z 1 + 2 z 2 4 z 2 4 z 2 − 32 z 1 − 24 z 2  . The variables τ α , w α , v α that ar e intr o duc e d in the last subse ction satisfy the r elations τ 1 = z 1 , τ 2 = 2 z 1 + z 2 ; w 1 = τ 1 − 1 6 τ 2 = v 1 v 2 , w 2 = τ 2 = ( v 2 ) 4 . The flat c o or dinates ar e given by t 1 = v 1 , t 2 = v 2 . In these flat c o or dinate the flat p encil of metrics has the form  η αβ ( t )  =  0 2 2 0  ,  g αβ ( t )  =   − ( t 2 ) 9 +9 t 1 ( t 2 ) 6 − 27( t 1 ) 2 ( t 2 ) 3 +27( t 1 ) 3 108( t 2 ) 3 ( t 2 ) 6 − 24 t 1 ( t 2 ) 3 − 18( t 1 ) 2 72( t 2 ) 2 ( t 2 ) 6 − 24 t 1 ( t 2 ) 3 − 18( t 1 ) 2 72( t 2 ) 2 3 t 1 − ( t 2 ) 3 12 t 2   . We have the p otential F = 1 48 ( t 1 ) 3 t 2 − 1 48 ( t 1 ) 2 ( t 2 ) 2 + 1 1440 t 1 ( t 2 ) 5 − 1 36288 ( t 2 ) 8 of the gener alize d F r ob enius manifold. The Euler ve ctor field and the unity ar e given by E = 3 4 t 1 ∂ ∂ t 1 + 1 4 t 2 ∂ ∂ t 2 , e = − 4 6 t 1 t 2 + ( t 2 ) 4  (3 t 1 + 2( t 2 ) 3 ) ∂ ∂ t 1 + 3 t 2 ∂ ∂ t 2  . Case 2. By taking m = 1 , we have the p encil gener ators z 1 = y 1 , z 2 = y 2 + 4 λ, and the metric η =  η ij ( y )  =  2 z 2 8 z 1 8 z 1 8 z 2  . The variables τ α , w α , v α that ar e intr o duc e d in the last subse ction satisfy the r elations τ 1 = 1 2 z 1 + 1 4 z 2 , τ 2 = − 1 2 z 1 + 1 4 z 2 ; w 1 = τ 1 = ( v 1 ) 2 , w 2 = τ 2 = ( v 2 ) 2 . AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 27 The flat c o or dinates ar e given by t 1 = v 1 , t 2 = v 2 . In these flat c o or dinate the flat p encil of metrics has the form  η αβ  =  1 0 0 1  ,  g ij ( t )  = − 1 4 (( t 1 ) 2 + ( t 2 ) 2 ) − 1 2 t 1 t 2 − 1 2 t 1 t 2 − 1 4 (( t 1 ) 2 + ( t 2 ) 2 ) ! . The p otential of the gener alize d F r ob enius manifold has the form F = − 1 48 ( t 1 ) 4 − 1 8 ( t 1 ) 2 ( t 2 ) 2 − 1 48 ( t 2 ) 4 , and the Euler ve ctor field and the unity ar e given by E = 1 2 t 1 ∂ ∂ t 1 + 1 2 t 2 ∂ ∂ t 2 , e = − 2 ( t 1 ) 2 − ( t 2 ) 2  t 1 ∂ ∂ t 1 − t 2 ∂ ∂ t 2  . Example 3.2. L et ( R, ω ) = ( C 3 , ω 1 ) . We have the W a ( R ) -invariant λ -F ourier p olynomials y 1 = e 2 π ix 1 + λe 2 π i ( x 1 − x 2 ) + λe − 2 π i ( x 1 − x 2 ) + λe 2 π i ( x 2 − x 3 ) + λe − 2 π i ( x 2 − x 3 ) + λ 2 e − 2 π ix 1 , y 2 = e 2 π ix 2 + e 2 π i (2 x 1 − x 2 ) + e 2 π i ( x 1 + x 2 − x 3 ) + e 2 π i ( x 1 − x 2 + x 3 ) + λe 2 π i ( x 1 − x 3 ) + λe − 2 π i ( x 1 − 2 x 2 + x 3 ) + λe − 2 π i ( x 1 − x 3 ) + λe 2 π i ( x 1 − 2 x 2 + x 3 ) + λ 2 e − 2 π ix 2 + λ 2 e − 2 π i (2 x 1 − x 2 ) + λ 2 e − 2 π i ( x 1 − x 2 + x 3 ) + λ 2 e − 2 π i ( x 1 + x 2 − x 3 ) , y 3 = e 2 π ix 3 + e 2 π i (2 x 1 − x 3 ) + e 2 π i (2 x 2 − x 3 ) + e 2 π i (2 x 1 − 2 x 2 + x 3 ) + λ 2 e − 2 π ix 3 + λ 2 e − 2 π i (2 x 1 − 2 x 2 + x 3 ) + λ 2 e − 2 π i (2 x 1 − x 3 ) + λ 2 e − 2 π i (2 x 2 − x 3 ) . The c ontr avariant metric on V ⊗ C is given by  (d x i , d x j )  =  ( α ∨ i , α ∨ j )  − 1 =   1 1 1 1 2 2 1 2 3   , which induc es the metric ( g ij λ ( y )) =    − ( y 1 ) 2 + 2 λy 2 + 12 λ 2 − y 1 y 2 + 8 λy 1 + 3 λy 3 − y 1 y 3 + 4 λy 2 − y 1 y 2 + 8 λy 1 + 3 λy 3 − 2( y 2 ) 2 + 8( y 1 ) 2 + 2 y 1 y 3 − 8 λy 2 − 2 y 2 y 3 + 4 y 1 y 2 − 12 λy 3 − y 1 y 3 + 4 λy 2 − 2 y 2 y 3 + 4 y 1 y 2 − 12 λy 3 − 3( y 3 ) 2 + 4( y 2 ) 2 − 8 y 1 y 3    . F r om The or em 3.2 we know that we have four differ ent choic es of p encil gener ators by take m = 0 , 1 , 2 , 3 , and g = ( g ij ) = ( g ij 0 ) is indep endent of the choic e of p encil gener ators. Note that the gener alize d F r ob enius manifold structur e for m = 0 is isomorphic to the one for m = 3 , and the gener alize d F r ob enius manifold structur e for m = 1 is isomorphic to the one for m = 2 , so we only ne e d to c onsider the c ase of m = 0 and of m = 1 . Case 1 . By taking m = 0 we get the p encil gener ators z 1 = y 1 + 6 λ, z 2 = y 2 − 12 λ, z 3 = y 3 + 8 λ, 28 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG and the metric η =  η ij ( z )  =    12 z 1 + 2 z 2 − 4 z 1 + 6 z 2 + 3 z 3 8 z 1 + 4 z 2 + 6 z 3 − 4 z 1 + 6 z 2 + 3 z 3 − 112 z 1 − 56 z 2 − 12 z 3 48 z 1 − 8 z 2 − 36 z 3 8 z 1 + 4 z 2 + 6 z 3 48 z 1 − 8 z 2 − 36 z 3 64 z 1 + 96 z 2 + 96 z 3    . The variables τ α , w α , v α satisfy the r elations τ 1 = z 1 , τ 2 = 4 z 1 + z 2 , τ 3 = 4 z 1 + 2 z 2 + z 3 ; w 1 = τ 1 − 1 6 τ 2 + 1 30 τ 3 = v 1 v 3 , w 2 = τ 2 − 1 4 τ 3 = v 2 ( v 3 ) 4 , w 3 = τ 3 = ( v 3 ) 6 . The flat c o or dinates satisfy the r elations t 1 = v 1 − 1 12 ( v 2 ) 2 v 3 , t 2 = v 2 v 3 , t 3 = v 3 . In these flat c o or dinate the flat metric has the form ( η αβ ( t )) =   0 0 2 0 12 0 2 0 0   . The p otential of the gener alize d F r ob enius manifold has the expr ession F = 1 24 ( t 1 ) 2 t 2 t 3 − 1 48 ( t 1 ) 2 ( t 3 ) 2 − 1 216 t 1 ( t 2 ) 3 ( t 3 ) 2 − 1 288 t 1 ( t 2 ) 2 t 3 + 1 1440 t 1 t 2 ( t 3 ) 4 − 1 60480 t 1 ( t 3 ) 7 + 1 4320 ( t 2 ) 5 ( t 3 ) 3 − 1 6912 ( t 2 ) 4 + 1 17280 ( t 2 ) 3 ( t 3 ) 3 − 1 34560 ( t 2 ) 2 ( t 3 ) 6 + 1 345600 t 2 ( t 3 ) 9 − 1 7603200 ( t 3 ) 12 , and the Euler ve ctor field and the unity ar e given by E = 5 6 t 1 ∂ ∂ t 1 + 1 2 t 2 ∂ ∂ t 2 + 1 6 t 1 ∂ ∂ t 1 , e = − 120 10( t 2 ) 2 + 120 t 1 t 3 + 20 t 2 ( t 3 ) 3 + ( t 3 ) 6  2 t 1 + t 2 t 3 + 1 10 ( t 3 ) 5  ∂ ∂ t 1 +2  t 2 + ( t 3 ) 3  ∂ ∂ t 2 + 2 t 3 ∂ ∂ t 3  . Case 2. By taking m = 1 we get the p encil gener ators z 1 = y 1 + 2 λ, z 2 = y 2 + 4 λ, z 3 = y 3 − 8 λ, and and the metric η =  η ij ( z )  =   4 z 1 + 2 z 2 12 z 1 + 2 z 2 + 3 z 3 − 8 z 1 + 4 z 2 + 2 z 3 12 z 1 + 2 z 2 + 3 z 3 − 16 z 1 + 8 z 2 − 4 z 3 − 16 z 1 − 24 z 2 − 4 z 3 − 8 z 1 + 4 z 2 + 2 z 3 − 16 z 1 − 24 z 2 − 4 z 3 − 64 z 1 − 32 z 2 − 32 z 3   . AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 29 The variables τ α , w α , v α satisfy the r elations τ 1 = 3 4 z 1 + 1 8 z 2 − 1 16 z 3 , τ 2 = z 1 + 1 2 z 2 + 1 4 z 3 , τ 3 = − 1 4 z 1 + 1 8 z 2 − 1 16 z 3 ; w 1 = τ 1 − 1 6 τ 2 = v 1 v 2 , w 2 = τ 2 = ( v 2 ) 4 , w 3 = τ 3 = ( v 3 ) 2 . The flat c o or dinates ar e given by t 1 = v 1 , t 2 = v 2 , t 3 = v 3 . In these flat c o or dinate the flat metric has the form ( η αβ ( t )) =   0 2 0 2 0 0 0 0 1   . The p otential of the gener alize d F r ob enius manifold has the expr ession F = 1 48 ( t 1 ) 3 t 2 − 1 48 ( t 1 ) 2 ( t 2 ) 2 + 1 1440 t 1 ( t 2 ) 5 − 1 36288 ( t 2 ) 8 − 1 8 t 1 t 2 ( t 3 ) 2 + 1 96 ( t 2 ) 4 ( t 3 ) 2 − 1 48 ( t 3 ) 4 , and the Euler ve ctor field and the unity ar e given by E = 3 4 t 1 ∂ ∂ t 1 + 1 4 t 2 ∂ ∂ t 2 + 1 2 t 3 ∂ ∂ t 3 , e = − 12 6 t 1 t 2 + ( t 2 ) 4 − 6( t 3 ) 2  t 1 + 2 3 ( t 2 ) 3  ∂ ∂ t 1 + t 2 ∂ ∂ t 2 − t 3 ∂ ∂ t 3  . 3.6. Landau-Ginzburg sup erp otential. As for the M D ( A ℓ , ω ℓ ) case, in this subsection we are to represen t the generalized F rob enius manifold structures on M D ( C ℓ , ω 1 ) in terms of sup erp otentials. Consider the following rational functions of p : Λ( p ) = p 2 − 1 p 2 m   ℓ X j =1 a j p 2( ℓ − j )   , a 1 , . . . , a ℓ ∈ C , m = 0 , . . . , ℓ. (3.28) Let f M ℓ,m b e the space f M ℓ,m =  ( a 1 , . . . , a ℓ ) ∈ C ℓ  . W e define the following tw o tensors on f M ℓ,m as follo ws: ˜ η ( ∂ ′ , ∂ ′′ ) = X q : Λ ′ ( q )=0 Res p = q ∂ ′ (Λ) ∂ ′′ (Λ) Λ ′ ( p ) d p ( p 2 − 1) 2 , (3.29) ˜ c ( ∂ ′ , ∂ ′′ , ∂ ′′′ ) = X q : Λ ′ ( q )=0 Res p = q ∂ ′ (Λ) ∂ ′′ (Λ) ∂ ′′′ (Λ) Λ ′ ( p ) d p ( p 2 − 1) 2 , (3.30) where the summation runs ov er the critical p oin ts of Λ( p ), including the critical p oint at infinity for the m = ℓ case. 30 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG Lemma 3.9. F or any given m = 0 , . . . , ℓ , the tensor ˜ η is a flat metric on f M ℓ,m , and it has flat c o or dinates ˜ t α = 1 2( ℓ − m − α ) + 1 Res p = ∞  Λ 2( ℓ − m − α )+1 2( ℓ − m ) ( p ) d p p 2 − 1  , α = 1 , . . . , ℓ − m ; (3.31) ˜ t β = 1 2( ℓ − β ) + 1 Res p =0  Λ 2( ℓ − β )+1 2 m ( p ) d p p 2 − 1  , β = ℓ − m + 1 , . . . , ℓ. (3.32) Pr o of. Let k 1 , k 2 b e the ro ots of Λ( p ) whic h hav e the following expansions: k 1 = Λ 1 2( ℓ − m ) = a 1 2( ℓ − m ) 1  p + a 2 − a 1 2( ℓ − m ) a 1 1 p + O  1 p 2  , p → ∞ , (3.33) k 2 = Λ 1 2 m = ( − a ℓ ) 1 2 m  1 p + a ℓ − a ℓ − 1 2 ma ℓ p + O  p 2   , p → 0 . (3.34) W e assume that for z ∈ C , arg z ∈ ( − π , π ], then b y using (3.31), (3.32), we obtain 1 2 log p − 1 p + 1 =    P ℓ − m α =1 ˜ t α k 2( ℓ − m − α )+1 1 + O  1 /k 2( ℓ − m ) 1  , k 1 → ∞ , π 2 i + P ℓ β = ℓ − m +1 ˜ t β k 2( ℓ − β )+1 2 + O  1 /k 2 m 2  , k 2 → ∞ , (3.35) from whic h it follo ws that 1 p 2 − 1 ∂ p ( k , ˜ t ) ∂ ˜ t α =    1 k 2( ℓ − m − α )+1 1 + O  1 /k 2( ℓ − m ) 1  , k 1 → ∞ , O  1 /k 2 m 2  , k 2 → ∞ , for α = 1 , . . . , ℓ − m, (3.36) 1 p 2 − 1 ∂ p ( k , ˜ t ) ∂ ˜ t β =    O  1 /k 2( ℓ − m ) 1  , k 1 → ∞ , 1 k 2( ℓ − β )+1 2 + O  1 /k 2 m 2  , k 2 → ∞ , for β = ℓ − m + 1 , . . . , ℓ. (3.37) By using the implicit function theorem one can obtain the following relation: ∂ ˜ t α (Λ( p ( k , ˜ t )) = − Λ ′ ( p ) ∂ p ( k , ˜ t ) ∂ ˜ t α , dΛ = 2( ℓ − m ) k 2( ℓ − m ) − 1 1 d k 1 = 2 mk 2 m − 1 2 d k 2 . Th us, when m  = 0 , ℓ w e ha v e ˜ η ( ∂ ˜ t α , ∂ ˜ t β ) = −  Res p = ∞ + Res p =0  1 ( p 2 − 1) 2 ∂ p ∂ ˜ t α ∂ p ∂ ˜ t β dΛ , = −  Res k 1 = ∞ 2( ℓ − m ) ( p 2 − 1) 2 ∂ p ∂ ˜ t α ∂ p ∂ ˜ t β k 2( ℓ − m ) − 1 1 d k 1 + Res k 2 = ∞ 2 m ( p 2 − 1) 2 ∂ p ∂ ˜ t α ∂ p ∂ ˜ t β k 2 m − 1 2 d k 2  . (3.38) By using (3.36), (3.37), w e arrive at ( ˜ η αβ ) = ( ˜ η ( ∂ ˜ t α , ∂ ˜ t β )) =  ˜ A 1 ˜ A 2  , where ˜ A 1 , ˜ A 2 are ( ℓ − m ) × ( ℓ − m ) and m × m matrices, which are anti-diagonal matrices with an ti-diagonal elemen ts 2( ℓ − m ) and 2 m resp ectiv ely . When m = 0, the function Λ( p ) has a critical p oin t at p = 0; when m = ℓ , it has a critical point at p = ∞ . Thus, in the deriv ation of ˜ η ( ∂ ˜ t α , ∂ ˜ t β ) as in (3.38), we only need to tak e residue at p = ∞ for the m = 0 case, and to take residue at p = 0 for the m = ℓ case. In these t w o cases the matrix ( ˜ η αβ ) is an ti-diagonal with an ti-diagonal elemen ts 2 ℓ . The lemma is pro v ed. □ AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 31 W e are to sho w that the op eration of m ultiplication on the tangen t spaces of f M ℓ,m defined b y (2.23) yields a generalized F rob enius manifold structure on f M ℓ,m . Let ± q 1 , . . . , ± q ℓ b e the distinct critical p oin ts of Λ( p ), where we assume that q i  = ± 1. Then for m  = ℓ w e ha v e Λ ′ ( p ) = 2( ℓ − m ) a 1 p 2 m +1 ℓ Y i =1 ( p 2 − q 2 i ) . (3.39) When m = 0, the function Λ( p ) has a critical p oint at p = 0, which we denote by q ℓ = 0. F or the m = ℓ case, we can easily see, by making a change of v ariable p → 1 /p , that the form ulae (3.29), (3.30) yield the same F rob enius manifold structure as for the m = 0 case, so in what follo ws we assume that m  = ℓ . Lemma 3.10. The fol lowing r elations hold true: Λ ′′ ( q i ) = Λ ′′ ( − q i ) = c i,m p Λ ′ ( p ) p 2 − q 2 i     p = q i , (3.40) wher e c i,m = 2 − δ i,ℓ δ m, 0 . Pr o of. F rom (3.39) it follows that Λ ′′ ( p ) = Λ ′ ( p ) ℓ X i =1 2 p p 2 − q 2 i − 2 m + 1 p ! . So w e get Λ ′′ ( ± q i ) =    2 p Λ ′ ( p ) p 2 − q 2 i    p = ± q i , i  = ℓ or i = ℓ, m  = 0 − Λ ′ ( p ) p    p =0 , i = ℓ, m = 0 . = c i,m p Λ ′ ( p ) p 2 − q 2 i     p = ± q i . The lemma is pro v ed. □ Let u i = Λ( q i ) = Λ( − q i ) b e the critical v alues of Λ( p ), then we hav e ∂ u i Λ( p ) | p = ± q j = δ ij . By using the Lagrange interpolation form ula, we get ∂ u i Λ( p ) = c i,m ( p 2 − 1) p Λ ′ ( p ) ( q 2 i − 1)Λ ′′ ( q i )( p 2 − q 2 i ) , (3.41) whic h implies that ˜ η ( ∂ u i , ∂ u j ) = ℓ X k =1 Res p = ± q k ∂ u i Λ( p ) · ∂ u j Λ( p ) Λ ′ ( p ) d p ( p 2 − 1) 2 = ℓ X k =1 Res p = ± q k c 2 i,m p 2 Λ ′ ( p ) ( q 2 i − 1)( q 2 j − 1)Λ ′′ ( q i )Λ ′′ ( q j )( p 2 − q 2 i )( p 2 − q 2 j ) d p = c 2 i,m δ ij ( q 2 i − 1) 2 (Λ ′′ ( q i )) 2 Res p = ± q i p 2 Λ ′ ( p ) ( p 2 − q 2 i ) 2 d p = c i,m δ ij ( q 2 i − 1) 2 Λ ′′ ( q i ) . (3.42) 32 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG Similarly , w e ha v e ˜ c ( ∂ u i , ∂ u j , ∂ u k ) = c i,m δ ij δ ik ( q 2 i − 1) 2 Λ ′′ ( q i ) . (3.43) F rom (3.42) and (3.43) it follows that ∂ u i · ∂ u j = δ ij ∂ u i , i, j = 1 , . . . , ℓ. Th us u 1 , . . . , u ℓ are canonical co ordinates of the multiplication, and the unit vector field and the Euler v ector field are giv en b y ˜ e = ℓ X i =1 ∂ u i , ˜ E = ℓ X i =1 u i ∂ u i . Lemma 3.11. The unit ve ctor field and the Euler ve ctor field c an also b e r epr esente d in the form ˜ e = − 2 grad ˜ η log( a 1 + · · · + a ℓ ) , (3.44) ˜ E = ℓ X α =1 a α ∂ a α = ℓ − m X α =1 2( ℓ − m − α ) + 1 2( ℓ − m ) ˜ t α ∂ ˜ t α + m X β =1 2( ℓ − β ) + 1 2 m ˜ t ℓ − m + β ∂ ˜ t ℓ − m + β , (3.45) Pr o of. By using the relation ∂ u i Λ( p ) = p 2 − 1 p 2 m ℓ X α =1 ( ∂ u i a α ) p 2( ℓ − α ) ! , w e obtain from (3.41) that ∂ u i a α = 2 c i,m × ( − 1) α − 1 ( ℓ − m ) a 1 σ α − 1 ( b q 2 i ) ( q 2 i − 1)Λ ′′ ( q i ) . It follo ws that ∂ u i ( a 1 + · · · + a ℓ ) = − 2 c i,m ( ℓ − m ) a 1 ( q i − 1) 2 Λ ′′ ( q i ) ℓ Y j =1 (1 − q 2 j ) = − c i,m ( q i − 1) 2 Λ ′′ ( q i ) ℓ X α =0 2( ℓ − m − α )( a α +1 − a α ) = − 2 c i,m ( q i − 1) 2 Λ ′′ ( q i ) ( a 1 + · · · + a ℓ ) , where w e set a ℓ +1 = a 0 = 0, thus (3.44) is prov e. The v alidit y of (3.45) comes from the homogeneit y of u i and ˜ t α . The lemma is prov ed. □ F rom Lemma 2.11 w e know that the 4-tensor ˜ c αβ γ ξ := ∂ ˜ t ξ ˜ c ( ∂ ˜ t α , ∂ ˜ t β , ∂ ˜ t γ ) is symmetric. W e can also v erify the homogeneity conditions (2.33) and (2.34), so M ℓ,m is a generalized F rob enius manifold with c harge d = 1. The intersection form of f M ℓ,m can be represen ted b y ˜ g ( ∂ ′ , ∂ ′′ ) = X q : Λ ′ ( q )=0 Res p = q ∂ ′ (Λ) · ∂ ′′ (Λ) Λ( p )Λ ′ ( p ) d p ( p 2 − 1) 2 , (3.46) AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 33 whic h is defined on f M ℓ,m \ Σ m , where Σ m := { p | Λ( p ) = 0 , Λ ′ ( p ) = 0 } . In terms of the canonical co ordinates, w e hav e ˜ g (d u i , d u j ) = u i ( q 2 i − 1) 2 Λ ′′ ( q i ) δ ij c i,m . In what follows, w e establish an isomorphism betw een the generalized F rob enius manifold structures defined on f M ℓ,m and M ℓ,m , whic h is in tro duced in (3.19). Theorem 3.12. L et the map h : f M ℓ,m → M ℓ,m , ( a 1 , . . . , a ℓ ) 7→ ( z 1 , . . . , z ℓ ) b e define d by z j = [ s ℓ − j ] ℓ X k =1 a k ( s − 2) ℓ − k ( s + 2) k − 1 ! , wher e [ s k ]( f ( s )) stands for the c o efficient of s k in the p olynomial f ( s ) . Then h gives an isomorphism b etwe en the gener alize d F r ob enius manifold structur es define d on f M ℓ,m and on M ℓ,m . Pr o of. Let s = 2 1+ p 2 1 − p 2 , then Λ( p ) giv en in (3.28) becomes e Λ( s ) = Λ( p ( s )) = P ℓ j =1 z j s ℓ − j ( s − 2) m ( s + 2) ℓ − m = Q ( s ) P 2 ( s ) , (3.47) where Q ( s ) is given in (3.12), and P 2 ( s ) is just P 1 ( s ) with m replaced by ℓ − m , see (3.10) and Remark 3.2. Denote b y s i = 2 1 + q 2 i 1 − q 2 i , i = 1 , . . . , ℓ the critical p oints of e Λ( s ), then for m = 1 , . . . , ℓ − 1 w e hav e d 2 e Λ d s 2 ( s i ) = d 2 e Λ d p 2 ( q i ) 4 ( s i − 2)( s i + 2) 3 , ∂ u i e Λ( s ) = ( s 2 − 4) d e Λ d s ( s ) ( s 2 i − 4)( s − s i ) d 2 e Λ d s 2 ( s i ) . W e can calculate the generating function of ˜ η (d z j , d z k ) as follows: ℓ X j,k =1 ˜ η (d z j , d z k ) r ℓ − j s ℓ − k = ℓ X i,j =1 ˜ η (d u i , d u j ) ∂ u i e Λ( r ) ∂ u j e Λ( s ) P 2 ( r ) P 2 ( s ) = ℓ X i =1 2( r 2 − 4)( s 2 − 4) d e Λ d s ( r ) d e Λ d s ( s ) ( s 2 i − 4) d 2 e Λ d s 2 ( s i )( r − s i )( s − s i ) P 2 ( r ) P 2 ( s ) = ℓ X i =1 2( r 2 − 4)( s 2 − 4) d e Λ d s ( r ) d e Λ d s ( s ) ( s 2 i − 4) d 2 e Λ d s 2 ( s i ) P 2 ( r ) P 2 ( s )  1 r − s i − 1 s − s i  1 s − r , 34 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG here we employ a similar argument used in the proof of Le mma 2.6, since ( s 2 − 4) dΛ d s ( s ) P 2 ( s ) is a p olynomial. F urthermore, we hav e ℓ X j,k =1 ˜ η (d z j , d z k ) r ℓ − j s ℓ − k = 2(( r 2 − 4) d e Λ d s ( r ) − ( s 2 − 4) d e Λ d s ( s )) r − s P 2 ( r ) P 2 ( s ) = 2  s 2 − 4 s − r  Q ′ ( s ) P 2 ( r ) − Q ( s ) P ′ 2 ( s ) P 2 ( s ) P 2 ( r )  − r 2 − 4 s − r  Q ′ ( r ) P 2 ( s ) − Q ( r ) P ′ 2 ( r ) P 2 ( r ) P 2 ( s )  = 2  r 2 − 4 r − s ( Q ′ ( r ) P 2 ( s ) + Q ( r ) P ′ 2 ( s )) − s 2 − 4 r − s ( Q ′ ( s ) P 2 ( r ) + Q ( s ) P ′ 2 ( r )) − ℓ ( Q ( r ) P 2 ( s ) + Q ( s ) P 2 ( r ))  . (3.48) Similarly , w e ha v e ℓ X j,k =1 ˜ g (d z j , d z k ) r ℓ − j s ℓ − k = − 2 ℓQ ( r ) Q ( s ) + 2 r 2 − 4 r − s Q ′ ( r ) Q ( s ) − 2 s 2 − 4 r − s Q ( s ) Q ′ ( s ) . (3.49) By comparing the form ulae (3.48), (3.49) with (3.13) and (3.3) resp ectively , we obtain h ∗ η = 1 2 ˜ η , h ∗ g 0 = 1 2 ˜ g . It is also easy to prov e that h ∗ ˜ e = e, h ∗ ˜ E = E . F or the m = 0 case, note that s ℓ = 2 is not a critical p oin t of e Λ( s ), so we hav e d 2 e Λ d p 2 ( p )      p =0 = 8 d e Λ d s ( s )      s =2 , ∂ u ℓ e Λ( s ) = ( s + 2) d e Λ d s ( s ) 4 d 2 e Λ d s 2 ( s )    s =2 . Th us the generating functions of ˜ η (d z j , d z k ) and ˜ g (d z j , d z k ) can b e calculated similarly as ab ov e, and w e can also reac h the conclusion of the theorem. The theorem is pro v ed. □ Remark 3.3. By using the change of variable p → 1 /p in the formulae (3.29) and (3.30) , we se e that the gener alize d F r ob enius manifolds f M ℓ,m and f M ℓ,ℓ − m ar e e quivalent. Remark 3.4. L et us r esc ale the flat c o or dinates ˜ t 1 , . . . , ˜ t ℓ to intr o duc e the new flat c o or dinates t α = 2 6( ℓ − m )+2 α − 1 4( ℓ − m ) ( ℓ − m ) ˜ t α , α = 1 , . . . , ℓ − m − 1 , t β = 2 6( ℓ − β )+4 m +3 4 m m ˜ t β , β = ℓ − m + 1 , . . . , ℓ − 1 , t ℓ = 2 3 4 m ˜ t ℓ , t ℓ − m = 2 4( ℓ − m ) − 1 4( ℓ − m ) ˜ t ℓ − m . Then in these new flat c o or dinates, the p otential ˜ F ( t ) for the F r ob enius manifold f M ℓ,m c oincides with the p otential F ( t ) for the F r ob enius manifold M ℓ,m which is c onstructe d in Se ction 3.4; the c omp onents of the flat metric ( ˜ η αβ ) ar e r elate d with that of the flat metric ( η αβ ) given in Cor ol lary 3.7 by  ˜ η αβ  =  √ 2 η αβ  . AFFINE WEYL GROUPS AND GENERALIZED FROBENIUS MANIFOLDS 35 4. The Cases of ( B ℓ , ω 1 ) and ( D ℓ , ω 1 ) 4.1. The Inv ariant λ -F ourier P olynomial Ring. Let R b e the root system of t ype B ℓ or D ℓ realized in the ℓ -dimensional Euclidean space V with orthonormal basis e 1 , . . . , e ℓ . T ake the simple ro ots as follows: B ℓ case: α 1 = e 1 − e 2 , . . . , α ℓ − 1 = e ℓ − 1 − e ℓ , α ℓ = e ℓ ; D ℓ case: α 1 = e 1 − e 2 , . . . , α ℓ − 1 = e ℓ − 1 − e ℓ , α ℓ = e ℓ − 1 + e ℓ . Then the fundamental weigh ts are giv en by ω i = α 1 + 2 α 2 + · · · + ( i − 1) α i − 1 + i ( α i + α i +1 + · · · + α ℓ ) , i = 1 , . . . , ℓ − 1 , ω ℓ = 1 2 ( α 1 + 2 α 2 + · · · + ℓα ℓ ) for the B ℓ case, and ω i = α 1 + 2 α 2 + · · · + ( i − 1) α i − 1 + i ( α i + · · · + α ℓ − 2 ) + 1 2 i ( α ℓ − 1 + α ℓ ) , i = 1 , . . . , ℓ − 2 , ω ℓ − 1 = 1 2  α 1 + 2 α 2 + · · · + ( ℓ − 2) α ℓ − 2 + 1 2 ℓα ℓ − 1 + 1 2 ( ℓ − 2) α ℓ  , ω ℓ = 1 2  α 1 + 2 α 2 + · · · + ( ℓ − 2) α ℓ − 2 + 1 2 ( ℓ − 2) α ℓ − 1 + 1 2 ℓα ℓ  for the D ℓ case. T ake ω = ω 1 , then κ = 1, and the n umbers θ i = ( ω i , ω 1 ) are given by B ℓ case: θ i = 1 , θ ℓ = 1 2 , i = 1 , . . . , ℓ − 1; D ℓ case: θ i = 1 , θ ℓ − 2 = θ ℓ = 1 2 , i = 1 , . . . , ℓ − 2 . W e define ξ 1 , . . . , ξ ℓ b y the relation cω + x 1 α ∨ 1 + · · · + x ℓ α ∨ ℓ = ξ 1 e 1 + · · · + ξ ℓ e ℓ , then the basic generators of A W are giv en by y i = λσ i ( ζ 1 , . . . , ζ ℓ ) , y ℓ = λ 1 2 σ ℓ ( ˜ ζ 1 , . . . , ˜ ζ ℓ ) , i = 1 , . . . , ℓ − 1 for the B ℓ case, and by y i = λσ i ( ζ 1 , . . . , ζ ℓ ) , i = 1 , . . . , ℓ − 2 , y ℓ − 1 = 1 2 λ 1 2  σ ℓ ( ˜ ζ 1 + , . . . , ˜ ζ ℓ + ) + σ ℓ ( ˜ ζ 1 − , . . . , ˜ ζ ℓ − )  , y ℓ = 1 2 λ 1 2  σ ℓ ( ˜ ζ 1 + , . . . , ˜ ζ ℓ + ) − σ ℓ ( ˜ ζ 1 − , . . . , ˜ ζ ℓ − )  for the D ℓ case. Here w e use the notations ζ j = e 2 π iξ j + e − 2 π iξ j , j = 1 , . . . , ℓ, ˜ ζ j = e π iξ j + e − π iξ j , j = 1 , . . . , ℓ, ˜ ζ j ± = e π iξ j ± e − π iξ j , j = 1 , . . . , ℓ. 36 LINGRUI JIANG, SI-QI LIU, YINGCHAO TIAN, YOUJIN ZHANG 4.2. The Relation b et w een the case of ( B ℓ , ω 1 ) , ( D ℓ , ω 1 ) and that of ( C ℓ , ω 1 ) . Motiv ated by [17], w e perform the c hange of co ordinates y j 7→ ˆ y j = y j , j = 1 , . . . , ℓ − 1 , y ℓ 7→ ˆ y ℓ = ( y ℓ ) 2 − ℓ − 1 X k =1 2 ℓ − k y k − 2 ℓ λ, (4.1) in the ( B ℓ , ω 1 ) case, and the change of co ordinates y j 7→ ˆ y j = y j , j = 1 , . . . , ℓ − 2 , y ℓ − 1 7→ ˆ y ℓ − 1 = y ℓ − 1 y ℓ − 1 4 ℓ − 2 X k =1  2 ℓ − k − ( − 2) ℓ − k  y k − 1 4  2 ℓ − ( − 2) ℓ  λ, (4.2) y ℓ 7→ ˆ y ℓ = ( y ℓ ) 2 + ( y ℓ − 1 ) 2 − 1 2 ℓ − 2 X k =1  2 ℓ − k + ( − 2) ℓ − k  y k − 1 2  2 ℓ + ( − 2) ℓ  λ, in the ( D ℓ , ω 1 ) case, then in these new co ordinates the metric g λ coincides with the one given by (3.3) for the ( C ℓ , ω 1 ) case. Thus, the generalized F rob enius manifold structures that w e obtain in this wa y from ( B ℓ , ω 1 ) and ( D ℓ , ω 1 ) are isomorphic to the one that we obtain from ( C ℓ , ω 1 ). 5. Conclusions Starting from an irreducible reduced root system R in the Euclidean space V and a fixed weigh t ω , w e in tro duce the W a ( R )-inv ariant λ -F ourier polynomial ring A W in [14], and constructed a generalized F rob enius manifold structure on the orbit space of the asso ciated affine W eyl group W a ( R ) under the assumption of the existence of a set of so called p encil generators of A W . In this pap er, w e construct p encil generators for the root systems of t ype A ℓ , B ℓ , C ℓ and D ℓ with the choice ω ℓ for A ℓ and ω = ω 1 for the other ro ot systems. W e also sho w that in the A ℓ case the structure constants of the asso ciated F rob enius algebra are quasi-homogeneous p olynomials in the flat co ordinates of the flat metric η , and in other cases they are rational functions of the flat co ordinates. W e exp ect that this construction of generalized F rob enius manifolds also works for other choices of the weigh t ω , and also for the exceptional ro ot systems of t ype G 2 , F 4 , E 6 , E 7 and E 8 , and w e will consider these cases in subsequen t publications. It is also interested to study the possibility of constructing generalized F rob enius manifold structures on the orbit space of Jacobi groups [1 – 3]. Ac kno wledgemen t. This w ork is supp orted b y NSF C No. 12571266. References [1] M. Bertola, Jacobi groups, Jacobi forms and their applications, Ph.D. Thesis, SISSA, T rieste, 1999. [2] M. Bertola, F rob enius manifold structure on orbit space of Jacobi groups. I, Differential Geom. Appl. 13 (2000) 19–41. [3] M. Bertola, F rob enius manifold structure on orbit space of Jacobi groups. I I, Differential Geom. Appl. 13 (2000) 213–233. [4] N. Bourbaki, Lie groups and Lie algebras. Chapter 4–6. Elem. Math. (Berlin), Springer-V erlag, Berlin, 2002. [5] Z. 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Lingrui Jiang, Departmen t of Mathematical Sciences, Tsingh ua Universit y Beijing 100084, P .R. China jlr24@mails.tsingh ua.edu.cn Si-Qi Liu, Departmen t of Mathematical Sciences, Tsingh ua Universit y Beijing 100084, P .R. China liusq@tsingh ua.edu.cn Yingc hao Tian, Departmen t of Mathematical Sciences, Tsingh ua Universit y Beijing 100084, P .R. China tian yc23@mails.tsinghua.edu.cn Y oujin Zhang, Departmen t of Mathematical Sciences, Tsingh ua Universit y Beijing 100084, P .R. China y oujin@tsinghua.edu.cn