--- title: Elliptic Weyl Group Invariants url: https://www.emergentmind.com/topics/elliptic-weyl-group-invariants type: topic --- # Elliptic Weyl Group Invariants 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 [2004.03587], [2007.16033], [2305.10604], [1910.02463], [1103.0951], [2411.12500]. ## 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 \(F\). Writing \(\operatorname{rad} I\simeq \mathbb{R}a\oplus \mathbb{R}\delta\), one takes \(G=\mathbb{Z}a\) and \(Q\cap \operatorname{rad} I=\mathbb{Z}a\oplus \mathbb{Z}\delta\) [2004.03587]. An elliptic root system is a subset \(R\subset F\) of non-isotropic vectors, \(I(\alpha,\alpha)\neq 0\), satisfying four conditions: \(Q(R)\) is a full lattice in \(F\); \(I(\alpha,\beta)\in \mathbb{Z}\) for all \(\alpha,\beta\in R\); for each \(\alpha\in R\), the reflection \[ w_\alpha(u)=u-I(u,\alpha)\,\alpha \] preserves \(R\); and \(R\) is not a disjoint union of two orthogonal subsystems. Passing to the hyperbolic extension \(F^h\), one defines \[ w_\alpha(u)=u-I^h(u,\alpha)\,\alpha \] in \(O(F^h)\), and the elliptic Weyl group \(W\subset O(F^h)\) is generated by all such reflections. The group sits in an exact sequence \[ 0\to \mathbb{Z}\to W\to W_{\mathrm{fin}}\to 1, \] where \(W_{\mathrm{fin}}\) is the finite Weyl group of the underlying finite root system [2004.03587]. The analytic invariant theory is formulated on \[ Y=\{x\in \operatorname{Hom}_{\mathbb{R}}(F^h,\mathbb{C})\mid x(a)=-2\pi i,\ \operatorname{Re}x(\delta)>0\}, \] \[ H=\{t\in \operatorname{Hom}_{\mathbb{R}}(\operatorname{rad} I,\mathbb{C})\mid t(a)=-2\pi i,\ \operatorname{Re}t(\delta)>0\}, \] with projection \(\pi:Y\to H\). The Weyl group acts holomorphically on \(Y\), and one studies the graded algebra \[ S^W=\bigoplus_{m\ge 0} S^W_m,\qquad S^W_m=\{f\in O(Y)\mid E\cdot f=m\,f,\ f(w\cdot x)=f(x)\ \forall w\in W\}, \] where \(E\) is the Euler field characterized by \(E(a^*)=0\) and \(E(\Lambda^*)=\mathrm{const.}\) Bernstein–Sato–Svarcman, Looijenga, and Wirthmüller imply that \(S^W\) is a free polynomial algebra over \(O(H)\) on \(\ell+1\) homogeneous generators \(x_0,\dots,x_\ell\) of positive rational degrees \(d_0\le \cdots \le d_\ell\). The codimension of the elliptic system is the number of exponents equal to the maximal one, namely \(|\{i\mid d_i=d_\ell\}|\) [2004.03587]. This framework fixes the meaning of “elliptic Weyl group invariants” in the strict sense of elliptic root systems: they are \(W\)-invariant holomorphic functions with specified Euler weight on the domain \(Y\), organized as the graded algebra \(S^W\). ## 2. Rank-one elliptic quasi-invariants from analytic elliptic cohomology A distinct but closely related elliptic realization arises for the rank-one Weyl group \(W\simeq \mathbb{Z}/2\mathbb{Z}\) in the setting of complex-analytic equivariant elliptic cohomology. For \(G=SU(2)\) with maximal torus \(T\simeq U(1)\), the Weyl group \(W=N_G(T)/T\simeq \mathbb{Z}/2\mathbb{Z}\) acts on \(T\) by inversion. One constructs a tower of \(T\)-spaces, in fact \(G\)-spaces, \[ F_0\to F_1\to \cdots \to F_m\to \cdots \] by the fibre–cofibre or Ganea construction applied to the basic fibration \(G/T\to BT\to BG\). Concretely, \[ F_m(G,T)\simeq (G/T)*G*\cdots *G, \] the \((m+1)\)-fold join of \(G/T\) with \(m\) copies of \(G\), and equivalently \[ F_m\cong \Sigma\,E_{2m}(T), \] where \(E_n(T)=T^{*(n+1)}\) is Milnor’s \(n\)-universal \(T\)-bundle. The space of \(m\)-quasi-invariants is the Borel homotopy quotient \[ X_m(G,T):=EG\times_G F_m(G,T), \] with fibration \[ F_m\to X_m\to BG. \] The spaces \(F_m(G,T)\) are called the \(m\)-quasi-flag manifolds and the spaces \(X_m(G,T)\) the spaces of \(m\)-quasi-invariants [2305.10604]. Fixing a Tate curve \(E_q\approx \mathbb{C}^*/q^{\mathbb{Z}}\), with parameter \(q\), and writing \(A=O_{\mathrm{an}}(\mathbb{C}^*)\), the \(T\)-equivariant elliptic cohomology of a finite \(T\)-CW-complex \(X\) is a coherent sheaf \(E_T^*(X)\) on \(M_T=E_q\), or equivalently an \(A_q\)-module. One has \[ E_T^*(E_nT)\cong O_{E_q}/\mathcal{J}^{n+1}, \] where \(\mathcal{J}\subset O_{E_q}\) is the ideal of functions vanishing at \(1\in E_q\). Since \(F_m\cong \Sigma E_{2m}(T)\), Theorem 7.3 gives \[ E_T^*(F_m)\cong O_{E_q}\times_{O_{E_q}/\mathcal{J}^{2m+1}} O_{E_q}, \] and, in algebraic form, \[ \widetilde E_T^*(F_m)\cong A\times_{A/\langle \Theta\rangle^{2m+1}} A, \] where \(\langle \Theta\rangle\subset A\) is generated by the classical Jacobi theta-function \[ \Theta(z)=(1-z)\prod_{n>0}(1-q^n z)(1-q^n z^{-1}). \] The involution \(s:z\mapsto z^{-1}\) extends to \(A_q\) by \[ s\cdot a(z)=a(z^{-1}),\qquad s\cdot \xi=\xi^{-1}, \] and Theorem 7.5 identifies \[ \widetilde E_T^*(F_m)=A e_+\oplus A\,\Theta^{2m+1} e_- \subset A[W], \] with \(e_\pm=(1\pm s)/2\). Taking \(W\)-invariants yields \[ \widetilde E_G^*(F_m)\cong A^W\oplus A^W\cdot (\Theta(z)-\Theta(z^{-1}))\,\vartheta(z)^{2m}\subset A, \] where \[ \vartheta(z)=(z^{1/2}-z^{-1/2})\prod_{n>0}(1-q^n z)(1-q^n z^{-1}). \] Equivalently, \[ Q_m^{\mathrm{ell}}(W):=A^W\oplus (\Theta(z)-\Theta(z^{-1}))\cdot \vartheta(z)^{2m}\cdot A^W\subset A \] is the ring of rank-one elliptic quasi-invariants [2305.10604]. The quasi-periodicity relations are \[ \Theta(qz)=-z^{-1}\Theta(z),\qquad \vartheta(qz)=+z^{-1}\vartheta(z). \] In complex-analytic elliptic cohomology, \(\Theta(z)\) and \(\vartheta(z)\) are regarded as characteristic classes of line bundles, and the Looijenga line bundle \(\mathcal{L}\to E_q\) has transition \(\xi\cdot v=qz^2v\), with trivial Weyl action and \[ H^0(E_q,\mathcal{L}^n)\cong \{f(z)\in A\mid f(qz)=q^{-n}z^{-2n}f(z)\}. \] The dependence on the quasi-invariance parameter \(m\) is entirely through the power \(2m+1\) of \(\mathcal{J}\) or \(\Theta\), and through \(\vartheta(z)^{2m}\) in the \(W\)-invariant summand. As \(m\) increases, one imposes higher vanishing order along the divisor \(\{1\}\); in the limit \(m\to \infty\) one recovers the full equivariant elliptic cohomology of a point, while \(m=0\) gives the “polynomial” invariants \(A^W\) [2305.10604]. ## 3. Weyl-invariant Jacobi forms and freeness phenomena For an even positive-definite lattice \(L\) of rank \(\ell\), with bilinear form \((\ ,\ )\), and a subgroup \(G\subset O(L)\), a holomorphic function \[ \varphi:\mathbb{H}\times (L\otimes \mathbb{C})\to \mathbb{C} \] is a \(G\)-invariant weak Jacobi form of weight \(k\in \mathbb{Z}\) and index \(m\in \mathbb{Z}_{>0}\) if it satisfies modularity under \(SL_2(\mathbb{Z})\), the elliptic transformation law \[ \varphi(\tau,z+x\tau+y)= \exp\!\bigl(-\pi i\,m\,((x,x)\tau+2(x,z))\bigr)\,\varphi(\tau,z), \] \(G\)-invariance \(\varphi(\tau,g\cdot z)=\varphi(\tau,z)\), and a Fourier expansion \[ \varphi(\tau,z)=\sum_{n\ge 0}\sum_{\ell\in L^*} c(n,\ell)e^{2\pi i(n\tau+(\ell,z))}. \] These forms assemble into the bigraded algebra \[ J^w(L)^G=\bigoplus_{k\in \mathbb{Z}}\bigoplus_{m\ge 0} J^w_{k,m}(L)^G \] over \(M_*(SL_2(\mathbb{Z}))\) [2007.16033]. A differential criterion for algebraic structure is given by the Jacobi-Jacobian. If \[ \varphi_j\in J^w_{k_j,m_j}(L)^G,\qquad j=1,\dots,\ell+1, \] then \[ J(\varphi_1,\dots,\varphi_{\ell+1})= \det\!\begin{pmatrix} m_1\varphi_1 & \cdots & m_{\ell+1}\varphi_{\ell+1}\\[4pt] \partial_{z_1}\varphi_1 & \cdots & \partial_{z_1}\varphi_{\ell+1}\\ \vdots & & \vdots\\ \partial_{z_\ell}\varphi_1 & \cdots & \partial_{z_\ell}\varphi_{\ell+1} \end{pmatrix} \] is a weak Jacobi form of weight \(\sum_j k_j+\ell\) and index \(\sum_j m_j\), invariant under \(G\) up to \(\det\). It vanishes exactly, with multiplicity one, on all reflection mirrors \[ \{(\tau,z):(r,z)\in \mathbb{Z}\tau+\mathbb{Z}\} \] for primitive \(r\in L\) with reflection in \(G\), and \[ J\not\equiv 0 \iff \varphi_1,\dots,\varphi_{\ell+1} \] are algebraically independent over \(M_*(SL_2(\mathbb{Z}))\). For an irreducible root system \(R\), the theta-block \[ P_R(\tau,z)=\prod_{r>0}\frac{\vartheta(\tau,(r,z))}{\eta(\tau)} \] lies in \(J^w_{-|R|/2,\ h^\vee}(L_R)\) and vanishes precisely with multiplicity one along every reflection mirror of \(W\) [2007.16033]. These ingredients yield an automorphic proof of Wirthmüller’s theorem. If \(R\) is any irreducible root system of rank \(\ell\) other than \(E_8\), then \[ J^w(L_R)^W \] is a free bigraded algebra over \(M_*(SL_2(\mathbb{Z}))\) on exactly \(\ell+1\) generators \(\varphi_{k_j,m_j}\), where \(m_j\) are the Dynkin labels of the highest root and \(k_j=d_j+1\) with \(d_j\) the exponents of \(W(R)\). For \(A_2\), \[ J^w(L_{A_2})^{W(A_2)}\cong M_*(SL_2(\mathbb{Z}))[\Phi_{0,1},\Phi_{2,1},\Phi_{6,2}], \] and for \(D_4\) one obtains five generators of weights \(0,2,4,4,6\) and indices \(1,1,1,1,2\), freely generating over \(M_*(SL_2(\mathbb{Z}))\) [2007.16033]. The exceptional case \(E_8\) is explicitly non-free. K. Sakai constructed nine algebraically independent weak Jacobi forms \[ A_m\in J^w_{k_m,m}(L_{E_8})^W,\qquad m=1,2,3,4,5, \] \[ B_n\in J^w_{\ell_n,n}(L_{E_8})^W,\qquad n=2,3,4,6, \] with \((k_m,\ell_n)=(4,6,6,8,10;8,10,12,14)\). If \[ g=\frac{J(A_1,\dots,A_5,B_2,B_3,B_4,B_6)}{\Phi_{E_8}}\in M_{172}(SL_2(\mathbb{Z})), \] then every \(\varphi\in J^w_{k,m}(L_{E_8})^W\) satisfies \[ g^{\,m-1}\varphi(\tau,z)=P(A_1,\dots,A_5,B_2,B_3,B_4,B_6) \] for a unique polynomial \(P\in M_*(SL_2(\mathbb{Z}))[X_1,\dots,X_9]\). Thus \(J^w(L_{E_8})^W\) is finitely generated over \(M_*(SL_2(\mathbb{Z}))\), but subject to infinitely many relations, one for each index [2007.16033]. ## 4. Good basic invariants, flat coordinates, and Frobenius structures The algebra \(S^W\) for an elliptic Weyl group admits a refined system of generators defined through Taylor expansion along a suitable splitting. An admissible triplet \((g,\zeta,L)\) consists of a semisimple element \(g\in W\), a primitive \(d_\ell\)-th root of unity \(\zeta\) such that \(g\cdot x_i=\zeta^{d_i}x_i\) on a chosen set of basic invariants, and an \((\ell+1)\)-dimensional subspace \(L\subset F\) splitting \(\operatorname{rad} I\), stable under \(g\), and containing no root. In coordinates \((z_0,\dots,z_\ell)\) adapted to \(L\oplus \operatorname{rad} I\), any invariant \(f\in S^W\) has a convergent Taylor series along \(L\), and the map \[ \psi_{g,\zeta,L}:S^W \xrightarrow{\mathrm{Taylor}} O(H)[z_0,\dots,z_\ell] \] is an isomorphism of graded \(O(H)\)-algebras [2004.03587]. A set of basic invariants \(\{x_0,\dots,x_\ell\}\) is good with respect to \((g,\zeta,L)\) if \[ \psi(x_i)=z_i,\qquad i=0,\dots,\ell. \] Equivalently, in coordinates adapted to \(L\), \[ \frac{\partial x_i}{\partial z_j}\Big|_L=\delta_{ij}, \] and all higher derivatives vanish unless \(\sum_k b_k d_k=d_i\). Existence holds for any elliptic root system and any choice of Coxeter transformation \(c\) and root-free splitting \(L\) [2004.03587]. In codimension one, meaning \(d_{\ell-1}