  We construct a bijection between admissible representations for an affine Lie
algebra $\mathfrak{g}$ at boundary admissible levels and $\mathbb{C}^\times$
fixed points in homogeneous elliptic affine Springer fibres for the Langlands
dual affine Lie algebra $\mathfrak{g}^\vee$. Using this bijection, we relate
the modularity of the characters of admissible representations to Cherednik's
Verlinde algebra construction coming from double affine Hecke algebras.
Finally, we show that the expected behaviors of simple modules under quantized
Drinfeld-Sokolov reductions are compatible with the reductions from affine
Springer fibres to affine Spaltenstein varieties. This yields (modulo some
conjectures) a similar bijection for irreducible representations of
$\mathcal{W}$-algebras, as well as an interpretation for their modularity
properties.
