  We describe a method, based on the theory of Macdonald-Koornwinder
polynomials, for proving bounded Littlewood identities. Our approach provides
an alternative to Macdonald's partial fraction technique and results in the
first examples of bounded Littlewood identities for Macdonald polynomials.
These identities, which take the form of decomposition formulas for Macdonald
polynomials of type $(R,S)$ in terms of ordinary Macdonald polynomials, are
$q,t$-analogues of known branching formulas for characters of the symplectic,
orthogonal and special orthogonal groups. In the classical limit, our method
implies that MacMahon's famous ex-conjecture for the generating function of
symmetric plane partitions in a box follows from the identification of
$(\mathrm{GL}(n,\mathbb{R}),\mathrm{O}(n))$ as a Gelfand pair. As further
applications, we obtain combinatorial formulas for characters of affine Lie
algebras; Rogers-Ramanujan identities for affine Lie algebras, complementing
recent results of Griffin et al.; and quadratic transformation formulas for
Kaneko--Macdonald-type basic hypergeometric series.
