Odd levels, the remaining non-reduced affine types, and the conceptual origin of the hidden $SL_2(\mathbb{Z})$-symmetry in $BC_l^{(2)}$ characters

Iohara and Saito (2025) proved that, for the twisted affine Lie algebra of type $BC_l^{(2)}$ at positive even level, adjoining certain $\psi$-twisted characters to the ordinary characters promotes the classical theta-subgroup symmetry of Kac and Peterson to an action of the full modular group $SL_2(\mathbb{Z})$ on the character span, and they observed that these functions are identified with characters and super-characters of the affine Lie superalgebra $B^{(1)}(0,l)$. This task addresses three linked questions they leave open: whether an analogous $SL_2(\mathbb{Z})$-structure or interpretation exists at odd levels; what conceptual mechanism explains the promotion of a congruence-subgroup symmetry to the full modular group upon adjoining twisted characters; and whether the phenomenon extends to the other three non-reduced affine root systems, associated with the affine Lie superalgebras $A^{(4)}(0,2l)$, $A^{(2)}(0,2l-1)$ and $C^{(2)}(l+1)$.

Background

An affine Kac-Moody Lie algebra is an infinite-dimensional Lie algebra obtained from a finite root system by adding an 'affine' direction represented by an imaginary root $\delta$; the twisted types, such as $BC_l^{(2)}$ (Kac's notation $A_{2l}^{(2)}$, Macdonald's $BC_l$), arise from nontrivial diagram automorphisms (symmetries of the Dynkin diagram) of a finite-dimensional simple Lie algebra. Its integrable highest-weight modules are indexed by dominant integral weights $\Lambda$ of a fixed level (central charge) $k$, and their characters $\chi_\lambda$ can be regarded as analytic functions of variables $(\tau,\underline{z},t)$ with $\tau$ in the complex upper half-plane. A root system is called non-reduced if it contains both a root $\alpha$ and its multiple $2\alpha$. Macdonald classified the affine root systems allowing non-reduced ones and proved the identities for products of the Dedekind eta-function $\eta(\tau)=q^{1/24}\prod_{n\geq1}(1-q^n)$ (with $q=e^{2\pi i\tau}$) attached to each type; Kac (1978) introduced a class of infinite-dimensional Lie superalgebras containing the four affine Lie superalgebras $B^{(1)}(0,l)$, $A^{(4)}(0,2l)$, $A^{(2)}(0,2l-1)$ and $C^{(2)}(l+1)$, noteworthy because their universal enveloping algebras are integral domains, whose real roots, after forgetting the $\mathbb{Z}/2\mathbb{Z}$-grading, are precisely the four non-reduced affine root systems $BCC_l$, $C^{\vee}BC_l$, $BB_l^{\vee}$ and $C^{\vee}C_l$.

Kac and Peterson (1984) established the modular behavior of affine characters: for an untwisted affine Lie algebra the full modular group $SL_2(\mathbb{Z})$ acts on the span of the characters at each fixed level, while for a twisted affine Lie algebra only a congruence subgroup does; for $BC_l^{(2)}$ this subgroup is the theta subgroup $\Gamma_\theta$, generated by $S\colon\tau\mapsto-1/\tau$ and $T^2\colon\tau\mapsto\tau+2$.

Iohara and Saito (2025) discovered a hidden enlargement of this symmetry for $BC_l^{(2)}$. Besides the ordinary characters they consider the $\psi$-twisted characters: here $\psi\colon W\to\{\pm1\}$ is the 'nice' sign character of the affine Weyl group $W$ (the group generated by the reflections in the real roots) that equals $-1$ on the simple reflection of the short simple root $\alpha_l$ and $+1$ on the other simple reflections, 'nice' meaning that it is induced by a $\mathbb{Z}/2\mathbb{Z}$-valued linear function $\bar\psi$ on the root lattice; the twisted character is the Weyl-Kac-type alternating formula with an extra sign (here $\varepsilon(w)$ is the sign of the Weyl group element $w$, i.e. $(-1)$ raised to the length of $w$), $\operatorname{ch}^{\psi}L(\Lambda)=\frac{\sum_{w\in W}\varepsilon(w)\psi(w)e^{w(\Lambda+\rho)}}{\sum_{w\in W}\varepsilon(w)\psi(w)e^{w(\rho)}},$ equivalently the character with the factor $(-1)^{\bar\psi(\Lambda-\lambda)}$ inserted on each weight space. Working with two coordinate systems on the character domain, attached to the two special indices $0$ and $l$ of $BC_l^{(2)}$ (an index $i$ is special when $\delta-a_i\alpha_i$ is a root) and denoted by maps $\varphi^{(\mathrm{I})}$ and $\varphi^{(\mathrm{II})}$, they prove that at positive even level $k$ the group $SL_2(\mathbb{Z})$ acts on the span of $\{\chi_\lambda\circ(\varphi^{(\mathrm{I})})^{-1},\ \chi_\lambda\circ(\varphi^{(\mathrm{II})})^{-1},\ \chi_\lambda^{\psi}\circ(\varphi^{(\mathrm{II})})^{-1}\}_{\lambda\in P_{k,+}}$ (Theorem 5.1) and, surprisingly, also on the span of the $\psi$-twisted characters alone in the $(\mathrm{I})$ coordinates (Theorem 5.2). They further observe that at even level these functions coincide with the characters and super-characters (characters with parity signs) of integrable highest-weight modules over the affine Lie superalgebra $B^{(1)}(0,l)$, with $\bar\psi$ becoming the parity map on its root lattice. The proofs are direct computations with theta series and twisted Weyl-group invariants in the spirit of Kac-Peterson.

The authors themselves formulate what remains unexplained: the construction and the superalgebra identification exist only at even level ('For an odd $k$, the authors are not aware of any such interpretation'); the appearance of the full modular group is 'quite mysterious' and its raison d'etre 'to be revealed'; and for the three other non-reduced affine types 'there might be something similar'.

Problem Statement

The problem is to complete the modular theory of the character spans of the non-reduced affine root systems, taking the even-level $BC_l^{(2)}$ theorem of Iohara and Saito as the established prototype. It consists of three linked parts, all stated as unresolved by the source.

(1) Odd levels of $BC_l^{(2)}$. For each odd positive integer $k$, determine whether the span of the characters and $\psi$-twisted characters of the integrable highest-weight modules of level $k$ over the affine Lie algebra of type $BC_l^{(2)}$ (expressed through the two special coordinate systems, as in Theorems 5.1 and 5.2) admits an action of $SL_2(\mathbb{Z})$ extending the Kac-Peterson $\Gamma_\theta$-action, and whether these functions admit an interpretation as (super-)characters of integrable modules over an affine Lie superalgebra, as they do at even level.

(2) Conceptual origin. Give a conceptual derivation of the even-level phenomenon: a framework - for instance from the representation theory of $B^{(1)}(0,l)$, from the modularity theory of vertex-operator superalgebras (the superalgebra analogue of vertex operator algebras, the chiral algebras of conformal field theory, whose module categories are governed by modular-group actions), or from the arithmetic of the non-reduced root system - from which the $SL_2(\mathbb{Z})$-actions of Theorems 5.1 and 5.2 follow as corollaries.

(3) The remaining non-reduced types. Carry out the analogue of the $BC_l^{(2)}$ analysis for the other three non-reduced affine root systems and their associated affine Lie superalgebras: for $C^{\vee}BC_l$ with $A^{(4)}(0,2l)$, for $BB_l^{\vee}$ with $A^{(2)}(0,2l-1)$, and for $C^{\vee}C_l$ with $C^{(2)}(l+1)$, construct the appropriate twisted characters, determine which subgroup of $SL_2(\mathbb{Z})$ acts on the character span at each positive level, and decide whether an identification with (super-)characters of the corresponding affine Lie superalgebra holds.

The three parts concern one phenomenon - the promotion of a congruence-subgroup symmetry to the full modular group upon adjoining twisted characters, for algebras whose real root systems are non-reduced - so a single framework answering part (2) is expected to govern parts (1) and (3); nevertheless each part has independent standing.

References

1. K. Iohara and Y. Saito, Hidden $SL_2(\mathbb{Z})$-symmetry in $BC_l^{(2)}$, The Quarterly Journal of Mathematics 76 (2025) 859-881, DOI 10.1093/qmath/haaf033, https://doi.org/10.1093/qmath/haaf033; preprint arXiv:2502.03316, https://arxiv.org/abs/2502.03316
2. K. Iohara and Y. Saito, Macdonald Identities: Revisited, Journal of Lie Theory 35 (2025) 191-205, DOI 10.5802/jolt.1379, https://doi.org/10.5802/jolt.1379; preprint arXiv:2409.07317, https://arxiv.org/abs/2409.07317
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