Macdonald identities

Remarks

Macdonald found these identities in 1972 for each affine root system, in the language of affine root systems and theta functions, without Lie algebras. In the form above, they are the case $\lambda = 0$ of the Kac–Weyl character formula, since the trivial module has character $1$. The real roots have multiplicity $1$. The imaginary roots are the nonzero multiples of the null root $\delta$; in the untwisted case their multiplicity is the rank of the underlying simple Lie algebra. The Weyl group of an affine Lie algebra is a semidirect product of a finite Weyl group and a lattice of translations. So the right side becomes a sum over a lattice, which is a theta-type series.

Example: for the affine algebra of $\mathfrak{sl}_2$, write $q = e^{-\delta}$ and $z = e^{-\alpha_1}$. The identity becomes the Jacobi triple product $$\prod_{n \ge 1} (1 - q^n)(1 - z q^{n-1})(1 - z^{-1} q^n) = \sum_{m \in \Z} (-1)^m z^m q^{m(m-1)/2}.$$ Specializing further gives Jacobi's identity $\prod_{n \ge 1} (1 - q^n)^3 = \sum_{m \ge 0} (-1)^m (2m + 1)\, q^{m(m+1)/2}$, an expansion of the cube of the Dedekind eta function up to a power of $q$.

Going deeper

For a student

The left side is a product over the positive roots of an affine Lie algebra, and the right side is an alternating sum over its Weyl group. Because the affine Weyl group contains a lattice of translations, the right side is a theta function of that lattice. The left side is close to a power of the Dedekind eta function. Specializing all simple roots suitably gives Macdonald's formulas for $\eta(\tau)^{\dim \mathfrak{g}}$ when $\mathfrak{g}$ is a finite-dimensional simple Lie algebra. The Lie algebra proof via the Kac–Weyl character formula explains why these identities exist and how they generalize to characters of other modules.