In this talk I will explain a construction by K. Rietsch of the mirrors of homogeneous spaces. The latter include Grassmannians, quadrics and flag varieties, and their mirrors can be expressed using Lie theory. More precisely the mirrors of homogeneous spaces live on so-called `Richardson varieties', which possess a cluster structure, and the mirror superpotential is defined on these Richardson varieties.

I will start by detailing Rietsch's general construction, then I will present some recent results by Marsh-Rietsch on Grassmannians, as well as joint work with Rietsch (resp. Rietsch and Williams) on Lagrangian Grassmannians (resp. quadrics). In particular I will show how the restriction of the superpotential on each cluster chart is a Laurent polynomial, which changes as we change cluster charts