Volumes of double cosets $KtK$

Question

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.

Answer 1

Let $t=\varpi^\lambda$ where $\varpi$ is a uniformizer and $\lambda:\mathbb{G}_m\to T$ is a dominant weight. The assumption that $\lambda$ is dominant is harmless as we may conjugate by an appropriate representative $n\in K$ of a Weyl group element. Then $t$ determines a standard parabolic $P_\lambda\subset G$, and we may consider the partial flag variety $G/P_\lambda$.

If $q$ is the cardinality of the residue field, then Tits shows in 3.3.1 of his Corvalis article that $$ \mu(K\varpi^\lambda K) = \frac{\#(G/P_\lambda)(q)}{q^{\dim(G/P_\lambda)}}q^{\langle \lambda, 2\rho\rangle}. $$ Here $2\rho=\sum_{\alpha\in \Phi^+}\alpha$ is the sum of the positive roots. Gross shows in his article on the Satake isomorphism how these volumes are related to Kazhdan-Lusztig polynomials, which I would expect is related to the formulas of Macdonald. In this direction, this may also be expressed in terms of the extended affine Weyl group: let $l$ be the length function of the extended affine Weyl group $X_\ast(T)\rtimes W$, where $W$ is the Weyl group of $(G,T)$. Then $$ \mu(K\varpi^\lambda K) = \frac{\sum_{y\in W\lambda W}q^{l(y)}}{\sum_{w\in W}q^{l(w)}}.$$