Jacobian identities and polynomiality of invariant theta-function rings for the remaining affine Lie algebra types $D_l^{(1)}$ and $F_4^{(1)}$, $E_6^{(1)}$, $E_6^{(2)}$, $E_7^{(1)}$, $E_8^{(1)}$

For an affine Lie algebra $\mathfrak{g}$ with affine Weyl group $W$, Kac and Peterson announced in 1984 that the ring $\widetilde{Th}^{+}$ of $W$-invariant theta functions is a polynomial algebra over the ring of holomorphic functions on the upper half-plane, generated by the fundamental characters, and they tabulated the Jacobian determinants of the fundamental characters for all affine types except $F_4^{(1)}$, $E_6^{(1)}$, $E_6^{(2)}$, $E_7^{(1)}$ and $E_8^{(1)}$; their proofs were never published. Bernstein and Schwarzman (2006) published detailed proofs of a weaker statement for all affine root systems except $D_l^{(1)}$ and $A_{2l}^{(2)}$, and Iohara and Saito (2019/2020) completed the twisted type $A_{2l}^{(2)}$, identifying its Jacobians only up to an undetermined nonzero scalar. The open problem is to complete the program for the remaining types, namely $D_l^{(1)}$ and the five exceptional affine types: prove the full-strength polynomiality of $\widetilde{Th}^{+}$ and derive the explicit Jacobian formulas (including the missing Kac–Peterson Table J entries for the exceptional types and a published proof of the $D_l^{(1)}$ entry).

Background

In finite-dimensional invariant theory, Chevalley's theorem, extended to complex reflection groups by Shephard and Todd, states that for a finite group $G$ generated by reflections acting on a complex vector space $V$, the ring of polynomial invariants $S(V^{*})^{G}$ is a polynomial algebra generated by $\dim V$ algebraically independent homogeneous elements. The problem studied here is the affine analogue of this theorem, in which the finite reflection group is replaced by an infinite affine Weyl group and polynomials by theta functions.

An affine Lie algebra (also called an affine Kac–Moody algebra) of type $X_N^{(k)}$ and rank $l+1$ is an infinite-dimensional Lie algebra obtained from a finite-dimensional simple Lie algebra by a loop construction together with a central extension and a derivation; the untwisted types include $A_l^{(1)}$, $D_l^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$, $F_4^{(1)}$, and the twisted types, obtained by folding the Dynkin diagram by an automorphism of order $k$, include $A_{2l}^{(2)}$ and $E_6^{(2)}$. Its Cartan subalgebra $\mathfrak{h}$ carries the affine Weyl group $W$, the infinite group generated by the reflections $s_0,\dots,s_l$ in the simple real roots. The irreducible integrable highest weight modules $L(\Lambda)$ are indexed by dominant integral weights, and their normalized characters $\chi_{\Lambda}$ (generating functions of weight-space dimensions with the imaginary-root multiplicities divided out) are holomorphic functions on a domain $Y$ of $\mathfrak{h}$, a result due to Gorelik and Kac. The fundamental characters $\chi_{\Lambda_0},\dots,\chi_{\Lambda_l}$ are those attached to the fundamental weights.

These characters are theta functions: holomorphic functions on $Y$ with prescribed quasi-periodicity under a lattice subgroup of $W$ (the Heisenberg subgroup). Writing $\mathcal{O}_{\mathbb{H}}$ for the holomorphic functions of the upper half-plane coordinate $\tau$, the $W$-invariant theta functions form a non-negatively graded $\mathcal{O}_{\mathbb{H}}$-algebra $\widetilde{Th}^{+}$, spanned, modulo the central direction $\mathbb{C}\delta$ ($\delta$ denoting the imaginary root of $\mathfrak{g}$), by the normalized characters. The basic anti-invariant object is the denominator theta function $A_{\rho}$, the theta-function analogue of the classical Weyl denominator, where $\rho$ is the Weyl vector, defined by $\langle h_i,\rho\rangle=1$ for all simple coroots $h_i$ and by $\langle d,\rho\rangle=0$, $d$ being the derivation of the loop construction; $W$-anti-invariant theta functions of a given degree are multiples of $A_{\rho}$ by elements of $\mathcal{O}_{\mathbb{H}}$.

The structure of $\widetilde{Th}^{+}$ was first attacked by Looijenga (1976) in his work on root systems and elliptic curves; his argument was found to be insufficient by Bernstein and Schwarzman, whose detailed 2006 papers prove a weaker form of the expected statement for all affine root systems except types $D_l^{(1)}$ and $A_{2l}^{(2)}$. Meanwhile Kac and Peterson (1984), as an application of their theory of modular transformations of affine characters, had announced the full statement — $\widetilde{Th}^{+}$ is a polynomial algebra over $\mathcal{O}_{\mathbb{H}}$ generated by the fundamental characters $\{\chi_{\Lambda_i}\}_{0\le i\le l}$ — and had presented a list (their Table J) of the Jacobians of the fundamental characters for all affine types except $F_4^{(1)}$, $E_6^{(1)}$, $E_6^{(2)}$, $E_7^{(1)}$ and $E_8^{(1)}$; their proofs were never published. Iohara and Saito (2019, published 2020) completed the previously excluded twisted type $A_{2l}^{(2)}$. Writing $J=\det(\partial_j\chi_{\Lambda_i})_{0\le i,j\le l}$ for the Jacobian determinant of the fundamental characters with respect to a basis of $\mathfrak{h}$, and $J^{0},\dots,J^{l}$ for the accompanying $i$-indexed family of determinants of the fundamental characters defined in [1, §3.2] (so that the $i=0$ member is $J$, i.e. $J^{0}=J$), their final result is

$J^{i}(\tau,z,t)\propto(-1)^{i}\eta(\tau)^{l-1}F^{(2l+3)}_{l+1-i}(\tau)A_{\rho}(\tau,z,t),$

where $\eta(\tau)$ is the Dedekind eta-function, $F^{(M)}_r(\tau)$ are explicit level-$M$ string functions — holomorphic modular forms arising as the $q$-series that multiply pure theta functions in character expansions; for $M=2l+3$ they coincide with $\eta$ times the characters of the $(2,2l+3)$ minimal-series representations of the Virasoro algebra — and $\propto$ means equality up to a non-zero scalar constant that the paper does not determine. Since the explicit factor $\eta^{l-1}F^{(2l+3)}_{l+1-i}$ never vanishes on the upper half-plane, the Jacobian is not identically zero; this forces the algebraic independence of the fundamental characters, and comparison of Poincaré series (graded rank generating functions) then yields the polynomiality of $\widetilde{Th}^{+}$. Thus, for each affine type, the entire statement reduces to identifying the Jacobian determinants explicitly, which is exactly the content that is missing for the remaining types.

Problem Statement

Complete the Kac–Peterson program on the Jacobians of fundamental characters and the structure of invariant theta-function rings for the affine Lie algebras of the remaining types. Specifically, for each of the affine types $D_l^{(1)}$ (with $l\ge4$), $F_4^{(1)}$, $E_6^{(1)}$, $E_6^{(2)}$, $E_7^{(1)}$ and $E_8^{(1)}$, establish with complete published proofs: (i) the announced Kac–Peterson statement that the ring $\widetilde{Th}^{+}$ of affine-Weyl-invariant theta functions is a polynomial algebra over $\mathcal{O}_{\mathbb{H}}$ generated by the fundamental characters $\chi_{\Lambda_0},\dots,\chi_{\Lambda_l}$ — for the exceptional types only the weaker Bernstein–Schwarzman form is currently available, and for $D_l^{(1)}$ not even that; and (ii) the explicit identification of each Jacobian determinant $J^{i}$ ($0\le i\le l$) as an explicitly determined non-zero constant times a product of powers of the Dedekind $\eta$-function and explicit modular-form (string-function) factors times the denominator theta function $A_{\rho}$ — supplying the Kac–Peterson Table J entries that do not exist for the five exceptional types and a published proof of the recorded but unproved $D_l^{(1)}$ entry. An answer must resolve all the listed types; partial treatments of individual types, while valuable, do not complete the problem as stated.

References

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