Elliptic Weyl Group Invariants

• Elliptic Weyl group invariants are invariant functions from elliptic root systems that form graded algebras, playing a key role in analytic elliptic cohomology.
• They underpin Weyl-invariant weak Jacobi forms and determine Frobenius structures via good invariants and flat coordinates, linking algebra with geometry.
• These invariants drive character formulas and geometric constructions, leading to applications such as explicit elliptic K3 surfaces and mirror symmetry descriptions.

Elliptic Weyl group invariants occur in several closely related settings: as invariant functions for elliptic Weyl groups attached to elliptic root systems, as Weyl-invariant weak Jacobi forms, and as elliptic or quasi-invariant objects arising from finite Weyl groups in analytic elliptic cohomology. In the sources considered here, these invariants appear as generators of polynomial or bigraded algebras, as $W$-fixed parts of elliptic-cohomological pullbacks, and as flat coordinates for Frobenius structures; they also enter character formulas on the elliptic set of a finite Weyl group and explicit geometric constructions such as elliptic K3 surfaces (Satake, 2020, Wang, 2020, Berest et al., 2023, Reeder, 2019, Satake et al., 2011, Bonnafé, 2024).

1. Elliptic root systems and the invariant ring $S^W$

For the elliptic-root-system formulation, one begins with a real vector space $F$ of dimension $\ell+2$, equipped with a symmetric bilinear form

$I:F\times F\to \mathbb{R}$

whose radical

$$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$

has dimension $2$. A marking is a one-dimensional sublattice $G\subset \operatorname{rad} I$ such that $G\cap Q\simeq \mathbb{Z}$, where $Q$ is a full lattice in $S^W$0. Writing $S^W$1, one takes $S^W$2 and $S^W$3 (Satake, 2020).

An elliptic root system is a subset $S^W$4 of non-isotropic vectors, $S^W$5, satisfying four conditions: $S^W$6 is a full lattice in $S^W$7; $S^W$8 for all $S^W$9; for each $F$0, the reflection

$F$1

preserves $F$2; and $F$3 is not a disjoint union of two orthogonal subsystems. Passing to the hyperbolic extension $F$4, one defines

$F$5

in $F$6, and the elliptic Weyl group $F$7 is generated by all such reflections. The group sits in an exact sequence

$F$8

where $F$9 is the finite Weyl group of the underlying finite root system (Satake, 2020).

The analytic invariant theory is formulated on

$\ell+2$0

$\ell+2$1

with projection $\ell+2$2. The Weyl group acts holomorphically on $\ell+2$3, and one studies the graded algebra

$\ell+2$4

where $\ell+2$5 is the Euler field characterized by $\ell+2$6 and $\ell+2$7 Bernstein–Sato–Svarcman, Looijenga, and Wirthmüller imply that $\ell+2$8 is a free polynomial algebra over $\ell+2$9 on $I:F\times F\to \mathbb{R}$0 homogeneous generators $I:F\times F\to \mathbb{R}$1 of positive rational degrees $I:F\times F\to \mathbb{R}$2. The codimension of the elliptic system is the number of exponents equal to the maximal one, namely $I:F\times F\to \mathbb{R}$3 (Satake, 2020).

This framework fixes the meaning of “elliptic Weyl group invariants” in the strict sense of elliptic root systems: they are $I:F\times F\to \mathbb{R}$4-invariant holomorphic functions with specified Euler weight on the domain $I:F\times F\to \mathbb{R}$5, organized as the graded algebra $I:F\times F\to \mathbb{R}$6.

2. Rank-one elliptic quasi-invariants from analytic elliptic cohomology

A distinct but closely related elliptic realization arises for the rank-one Weyl group $I:F\times F\to \mathbb{R}$7 in the setting of complex-analytic equivariant elliptic cohomology. For $I:F\times F\to \mathbb{R}$8 with maximal torus $I:F\times F\to \mathbb{R}$9, the Weyl group $$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$0 acts on $$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$1 by inversion. One constructs a tower of $$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$2-spaces, in fact $$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$3-spaces,

$$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$4

by the fibre–cofibre or Ganea construction applied to the basic fibration $$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$5. Concretely,

$$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$6

the $$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$7-fold join of $$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$8 with $$\operatorname{rad} I=\{x\in F\mid I(x,y)=0\ \forall y\in F\}$$9 copies of $2$0, and equivalently

$2$1

where $2$2 is Milnor’s $2$3-universal $2$4-bundle. The space of $2$5-quasi-invariants is the Borel homotopy quotient

$2$6

with fibration

$2$7

The spaces $2$8 are called the $2$9-quasi-flag manifolds and the spaces $G\subset \operatorname{rad} I$0 the spaces of $G\subset \operatorname{rad} I$1-quasi-invariants (Berest et al., 2023).

Fixing a Tate curve $G\subset \operatorname{rad} I$2, with parameter $G\subset \operatorname{rad} I$3, and writing $G\subset \operatorname{rad} I$4, the $G\subset \operatorname{rad} I$5-equivariant elliptic cohomology of a finite $G\subset \operatorname{rad} I$6-CW-complex $G\subset \operatorname{rad} I$7 is a coherent sheaf $G\subset \operatorname{rad} I$8 on $G\subset \operatorname{rad} I$9, or equivalently an $G\cap Q\simeq \mathbb{Z}$0-module. One has

$G\cap Q\simeq \mathbb{Z}$1

where $G\cap Q\simeq \mathbb{Z}$2 is the ideal of functions vanishing at $G\cap Q\simeq \mathbb{Z}$3. Since $G\cap Q\simeq \mathbb{Z}$4, Theorem 7.3 gives

$G\cap Q\simeq \mathbb{Z}$5

and, in algebraic form,

$G\cap Q\simeq \mathbb{Z}$6

where $G\cap Q\simeq \mathbb{Z}$7 is generated by the classical Jacobi theta-function

$G\cap Q\simeq \mathbb{Z}$8

The involution $G\cap Q\simeq \mathbb{Z}$9 extends to $Q$0 by

$Q$1

and Theorem 7.5 identifies

$Q$2

with $Q$3. Taking $Q$4-invariants yields

$Q$5

where

$Q$6

Equivalently,

$Q$7

is the ring of rank-one elliptic quasi-invariants (Berest et al., 2023).

The quasi-periodicity relations are

$Q$8

In complex-analytic elliptic cohomology, $Q$9 and $S^W$00 are regarded as characteristic classes of line bundles, and the Looijenga line bundle $S^W$01 has transition $S^W$02, with trivial Weyl action and

$S^W$03

The dependence on the quasi-invariance parameter $S^W$04 is entirely through the power $S^W$05 of $S^W$06 or $S^W$07, and through $S^W$08 in the $S^W$09-invariant summand. As $S^W$10 increases, one imposes higher vanishing order along the divisor $S^W$11; in the limit $S^W$12 one recovers the full equivariant elliptic cohomology of a point, while $S^W$13 gives the “polynomial” invariants $S^W$14 (Berest et al., 2023).

3. Weyl-invariant Jacobi forms and freeness phenomena

For an even positive-definite lattice $S^W$15 of rank $S^W$16, with bilinear form $S^W$17, and a subgroup $S^W$18, a holomorphic function

$S^W$19

is a $S^W$20-invariant weak Jacobi form of weight $S^W$21 and index $S^W$22 if it satisfies modularity under $S^W$23, the elliptic transformation law

$S^W$24

$S^W$25-invariance $S^W$26, and a Fourier expansion

$S^W$27

These forms assemble into the bigraded algebra

$S^W$28

over $S^W$29 (Wang, 2020).

A differential criterion for algebraic structure is given by the Jacobi-Jacobian. If

$S^W$30

then

$S^W$31

is a weak Jacobi form of weight $S^W$32 and index $S^W$33, invariant under $S^W$34 up to $S^W$35. It vanishes exactly, with multiplicity one, on all reflection mirrors

$S^W$36

for primitive $S^W$37 with reflection in $S^W$38, and

$S^W$39

are algebraically independent over $S^W$40. For an irreducible root system $S^W$41, the theta-block

$S^W$42

lies in $S^W$43 and vanishes precisely with multiplicity one along every reflection mirror of $S^W$44 (Wang, 2020).

These ingredients yield an automorphic proof of Wirthmüller’s theorem. If $S^W$45 is any irreducible root system of rank $S^W$46 other than $S^W$47, then

$S^W$48

is a free bigraded algebra over $S^W$49 on exactly $S^W$50 generators $S^W$51, where $S^W$52 are the Dynkin labels of the highest root and $S^W$53 with $S^W$54 the exponents of $S^W$55. For $S^W$56,

$S^W$57

and for $S^W$58 one obtains five generators of weights $S^W$59 and indices $S^W$60, freely generating over $S^W$61 (Wang, 2020).

The exceptional case $S^W$62 is explicitly non-free. K. Sakai constructed nine algebraically independent weak Jacobi forms

$S^W$63

$S^W$64

with $S^W$65. If

$S^W$66

then every $S^W$67 satisfies

$S^W$68

for a unique polynomial $S^W$69. Thus $S^W$70 is finitely generated over $S^W$71, but subject to infinitely many relations, one for each index (Wang, 2020).

4. Good basic invariants, flat coordinates, and Frobenius structures

The algebra $S^W$72 for an elliptic Weyl group admits a refined system of generators defined through Taylor expansion along a suitable splitting. An admissible triplet $S^W$73 consists of a semisimple element $S^W$74, a primitive $S^W$75-th root of unity $S^W$76 such that $S^W$77 on a chosen set of basic invariants, and an $S^W$78-dimensional subspace $S^W$79 splitting $S^W$80, stable under $S^W$81, and containing no root. In coordinates $S^W$82 adapted to $S^W$83, any invariant $S^W$84 has a convergent Taylor series along $S^W$85, and the map

$S^W$86

is an isomorphism of graded $S^W$87-algebras (Satake, 2020).

A set of basic invariants $S^W$88 is good with respect to $S^W$89 if

$S^W$90

Equivalently, in coordinates adapted to $S^W$91,

$S^W$92

and all higher derivatives vanish unless $S^W$93. Existence holds for any elliptic root system and any choice of Coxeter transformation $S^W$94 and root-free splitting $S^W$95 (Satake, 2020).

In codimension one, meaning $S^W$96, good invariants acquire a canonical role. For admissible $S^W$97 of zero-type, the $S^W$98-span of good invariants is unique. One may choose the last coordinate $S^W$99 so that

$F$00

and then Proposition 8.3 states that

$F$01

This is the condition that $F$02 is the unit field of a Frobenius structure. Theorem 8.4 shows that in this basis the metric matrix is the constant anti-diagonal form

$F$03

so the good basic invariants coincide with Saito’s flat coordinates (Satake, 2020).

The same codimension-one hypothesis yields a unique Frobenius-manifold structure $F$04, up to rescaling, with unit $F$05 and normalized Euler field $F$06. In flat coordinates,

$F$07

and there is a potential $F$08 such that

$F$09

Because the $F$10 are good, their expansions

$F$11

encode the multiplication: the Taylor coefficients of the good invariants along $F$12 are precisely the structure constants of the Frobenius multiplication $F$13 in the flat basis (Satake, 2020).

5. Mirror symmetry, eta-products, and extended Weyl-group invariants

For the simple-elliptic cases $F$14 and $F$15, the invariant theory of the extended Weyl group appears directly as a Frobenius manifold that is identified with Gromov–Witten theory. Let $F$16 denote the algebra of holomorphic functions on the universal covering of the domain $F$17 of the elliptic root system of type $F$18, invariant under the extended Weyl group $F$19. By Saito’s extended affine theory, in both cases $F$20 is a free polynomial algebra on $F$21 generators. For type $F$22,

$F$23

with $F$24, $F$25, $F$26 for $F$27, and degrees

$F$28

For type $F$29,

$F$30

with $F$31, $F$32, $F$33 for $F$34, and weights

$F$35

The absence of algebraic relations is equivalent to the fact that the $F$36-action is reflection-free on the complement of the elliptic discriminant and that the ring of invariants of a reflection group is polynomial (Satake et al., 2011).

Saito’s flat-structure or primitive-form theory endows $F$37 with a Frobenius manifold structure. The unit vector field is

$F$38

the pairing $F$39 is constant in the flat basis with only $F$40 nonzero, and the Euler field is

$F$41

For $F$42, the potential contains the theta-function coefficients

$F$43

while for $F$44 the potential contains fourteen functions $F$45 expressed in terms of

$F$46

including

$F$47

These choices satisfy associativity and homogeneity (Satake et al., 2011).

The mirror theorem identifies these Frobenius manifolds with the Gromov–Witten Frobenius manifolds of the orbifolds $F$48 and $F$49. In the $F$50 case one may take

$F$51

together with an explicit linear change of flat generators, so that the Gromov–Witten potential pulls back to the Saito potential $F$52. In the $F$53 case one takes

$F$54

for suitable constants $F$55, so that $F$56 becomes $F$57 (Satake et al., 2011).

At genus one, the eta-quotient

$F$58

is the generating function of the three-point Gromov–Witten invariants of $F$59, and under the mirror identification the coefficients $F$60 are the Fourier coefficients of the unique modular form of weight $F$61 and level $F$62 appearing as one of the basic invariant functions $F$63 in the Saito potential for $F$64. Equivalently,

$F$65

is both the genus-one Gromov–Witten potential of $F$66 and the $F$67-function on the elliptic-Weyl Frobenius manifold of type $F$68 (Satake et al., 2011).

6. Elliptic elements, character formulas, and geometric realizations

The adjective “elliptic” also has a representation-theoretic meaning for finite Weyl groups. For a simply connected compact Lie group $F$69 with maximal torus $F$70, weight lattice $F$71, root system $F$72, and Weyl group $F$73, one sets

$F$74

An element $F$75 is elliptic if $F$76, equivalently if $F$77 has no nonzero fixed weights in $F$78, its coset in $F$79 lies in a single $F$80-conjugacy class, and $F$81 is finite. It is elliptic-regular if, in addition, $F$82 acts freely on the set of all roots $F$83. In Reeder’s general formula for the character of $F$84 on the zero weight space, the weighted partition function $F$85 becomes trivial on the elliptic set: $F$86 because $F$87 when $F$88. On the elliptic-regular set this leads to a monomial product formula involving positive coroots and a constant equal to $F$89 or $F$90; for a Coxeter element one recovers Kostant’s formula for the trace, while for $F$91 the formula leads to a method for determining all representations for which the zero weight space is irreducible (Reeder, 2019).

A geometric realization of Weyl-group invariants appears in the $F$92 case. For $F$93 with coordinates $F$94 and the Weyl group $F$95, classical invariant theory gives

$F$96

where $F$97. The derived subgroup $F$98 has index $F$99, and

$\ell+2$00

where $\ell+2$01 is a skew-invariant of degree $\ell+2$02. Taking

$\ell+2$03

one obtains

$\ell+2$04

a hypersurface of degree $\ell+2$05 (Bonnafé, 2024).

On $\ell+2$06, the projection

$\ell+2$07

extends to a morphism whose pullback to the minimal resolution $\ell+2$08 is an elliptic fibration. After birational change of fiber coordinates, the affine equation becomes

$\ell+2$09

with

$\ell+2$10

$\ell+2$11

and discriminant

$\ell+2$12

up to a nonzero scalar factor. The singular-fiber configuration is

$\ell+2$13

The minimal resolution is a K3 surface with

$\ell+2$14

$\ell+2$15

and

$\ell+2$16

This construction shows that the invariant theory of $\ell+2$17, notably the six basic invariants and the Jacobian of the reflecting arrangement, leads via quotient and resolution to an explicit elliptic K3 surface (Bonnafé, 2024).

Taken together, these developments show that elliptic Weyl group invariants are not a single formal object but a family of rigorously connected structures: graded invariants $\ell+2$18 for elliptic Weyl groups, rank-one elliptic quasi-invariants in analytic elliptic cohomology, Weyl-invariant Jacobi forms and their freeness or non-freeness, flat invariants determining Frobenius manifolds, character-theoretic formulas on the elliptic set of a finite Weyl group, and explicit algebraic surfaces constructed from Weyl-group invariant rings (Satake, 2020, Berest et al., 2023, Wang, 2020, Satake et al., 2011, Reeder, 2019, Bonnafé, 2024).