[For the entire collection see Zbl 0561.00006.]
Let \(\tilde{\mathfrak g}\) be an affine Lie algebra, and \({\mathfrak g}\) be an affine subalgebra of \(\tilde{\mathfrak g}\). In this note, the authors consider the irreducible decomposition of an irreducible \(\tilde{\mathfrak g}\)-module L(\({\tilde \Lambda}\)) with highest weight \({\tilde \Lambda}\) with regards to the subalgebra \({\mathfrak g}\). Namely, they deal with the multiplicity (\({\tilde \Lambda}:\Lambda)\) of an irreducible highest weight \({\mathfrak g}\)-module \(L(\Lambda)\) appearing in \(L({\tilde \Lambda}\)), called the branching coefficient, for various pairs of Lie algebras \((\tilde{\mathfrak g},{\mathfrak g})\). In the decomposition of \(L({\tilde \Lambda}\)), the irreducible components appear as a ”string” \(L(\Lambda-n\delta)\) \((n\in {\mathbb{Z}})\) in the direction of the null root \(\delta\) of \({\mathfrak g}\), and there is an identity of characters: \[ ch_{L(\Lambda)}|_{{\mathfrak h}}=\sum_{\Lambda}E_{{\tilde \Lambda}\Lambda}(q) ch_{L(\Lambda)},\quad q=e^{-\delta},\quad E_{{\tilde \Lambda}\Lambda}(q)=\sum_{n\in {\mathbb{Z}}}({\tilde \Lambda}:\quad \Lambda -n\delta)q^ n. \] Here \({\mathfrak h}\) denotes the Cartan subalgebra of \({\mathfrak g}\) and \(\Lambda\) runs over a finite set of dominant integral weights of \({\mathfrak g}\) mod \(C\delta\) having the same level \(m=(\Lambda,\delta).\)
Following V. G. Kac and D. H. Peterson [Bull. Am. Math. Soc. 3, 1057-1061 (1980; Zbl 0457.17007), Adv. Math. 53, 125-264 (1984)], the characters are expressible as quotient of classical theta functions up to rational power of q, and \(e_{{\tilde \Lambda}\Lambda}(\tau)=E_{{\tilde \Lambda}\Lambda}(q)\times\) (some power of q) with \(q=e^{2\pi i\tau}\) becomes a modular function. Thus the problem is to determine the modular functions \(e_{{\tilde \Lambda}\Lambda}(\tau)\). The authors have determined \(e_{{\tilde \Lambda}\Lambda}(\tau)\) for pairs \(A^{(1)}_{2\ell -1}\supset C_{\ell}^{(1)}\), \(A_{2\ell}^{(1)}\supset A_{2\ell}^{(2)}\) and \(C_{2\ell}^{(1)}\supset C_{\ell}^{(1)}\) in terms of Hecke indefinite modular forms [Lett. Math. Phys. 6, 463-469 (1982; Zbl 0526.17008), Errata 7, 271 (1983); Adv. Stud. Pure Math. 4, 97-119 (1984; Zbl 0578.17015)].
Generalizing the techniques used there, in this note the authors establish a general duality relating the function \(e_{{\tilde \Lambda}\Lambda}(\tau)\) for two pairs of Lie algebras \((\tilde{\mathfrak g},{\mathfrak g})\) vs \((\tilde{\mathfrak g}^+,{\mathfrak g}^+)\). Here the roles of the rank \(\ell\) and the level m of the representation are interchanged between these two pairs and they have \(e_{{\tilde \Lambda}\Lambda}(\tau)=e^+_{{\tilde \Lambda}^+\Lambda^+}(\tau).\) They work out a list of such dual identities for pairs \(\tilde{\mathfrak g}\supset {\mathfrak g}\), where \({\mathfrak g}=\tilde {\mathfrak g}^{\sigma}\) is the invariant subalgebra of an involutive diagram automorphism \(\sigma\) of \(\tilde{\mathfrak g}\). This leads in particular to a duality of branching coefficients.