MACDONALD’S FORMULA IMPLIES THE MIRKOVIC-VILONEN DIMENSION FORMULA

Mirkovic-Vilonen: The dimension of Nπ^{λ}K/K ∩ Kπ^{μ}K/K is ⟨ρ, μ + λ⟩, and the number of irreducible components of top dimension is m_{μ}(λ), the multiplicity of the weight λ in the character E_{μ}.

We will deduce this using Macdonald’s formula. It is enough (by, e.g., the Weil conjectures) to show that

lim_{q→∞} \frac{#(Nπ^{λ}K/K ∩ Kπ^{μ}K/K)(𝔽_{q})}{q^{⟨ρ,μ+λ⟩}} = m_{μ}(λ).

The numerator in the left hand side is

∫_{G} 1_{A_{O}NK}(π^{−λ}y) 1_{Kπ^{−μ}K}(y^{−1}) dy = (1_{A_{O}NK} ∗ 1_{Kπ^{−w_{0}μ}K})(π^{−λ})
= (1_{Kπ^{−w_{0}μ}K}^{∨} · 1_{A_{O}NK})(π^{−λ})
= 1_{Kπ^{−w_{0}μ}K}^{∨}(π^{−λ}) δ_{B}^{1/2}(π^{−λ})
= 1_{Kπ^{−w_{0}μ}K}^{∨}(π^{−w_{0}λ}) q^{⟨ρ,λ⟩}.

By Macdonald’s formula, this is the coefficient of π^{−w_{0}λ} in

\frac{q^{⟨ρ , μ+λ⟩}}{W_{−w_{0}μ}(q^{−1})} ∑_{w∈W} w( ∏_{α>0} \frac{1 − q^{−1}π^{−α^{∨}}}{1 − π^{−α^{∨}}} ) · π^{−w w_{0}μ}.

Divide this by q^{⟨ρ , μ+λ⟩} and take the limit as q → ∞. The Weyl character formula implies that we get m_{−w_{0}μ}(−w_{0}λ) = m_{μ}(λ). This completes the proof.