# Volumes of double cosets $KtK$

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.

## 1 Answer

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 $$\mu(K\varpi^\lambda K) = \frac{\sum_{y\in W\lambda W}q^{l(y)}}{\sum_{w\in W}q^{l(w)}}.$$

• Thanks a lot, this is exactly what I need. I also found a formula in Macdonald's "Spherical functions on a group of $p$-adic type" in terms of Poincaré polynomials. But of course this is the same formula. – Paul Broussous May 15 at 11:48
• After checking Tits's Corvalis article, the formula that Gross states is not proved by Tits. [Tits](3.3.1) is just the Bruhat-Tits decomposition $G=IW^{\rm aff}I$ and the formula $\vert IwI/I\vert =q^{l(w)}$, where $I$ is the Iwahori subgroup, $W^{\rm aff}$ the affine Weyl group and $l$ the length function. Gross just writes that the formula is a simple consequence of the BT decomposition. I do not think that it is that simple. – Paul Broussous May 15 at 12:12
• Ah, sorry about that. I didn't check the reference closely. I agree it's not immediate, but it is not too bad. If I have time later, I'll write up the argument. – WSL May 15 at 15:52