Let $ G$ be the group of rational points of a connected reductive group $ \mathbb G$ defined over a non-archimedean local field $ F$ . For simplicity sake, I assume that $ \mathbb G$ is split over $ F$ . Let $ T$ the group of $ F$ -points of a maximal split torus of $ \mathbb G$ . Let $ K$ be a good maximal compact subgroup of $ G$ fixing a special vertex of the apartment of $ T$ . Let $ \mu$ be the Haar measure on $ G$ giving volume $ 1$ to $ K$ . My question is:

Is there a closed formula for the measure $ \mu (KtK)$ , $ t\in T$ ?

In the case of $ {\mathbb G}={\rm GL}(N)$ , there is such a formula in Macdonald’s book “Symmetric functions and Hall polynomials”. But I need a reference in the general case.