The Volume of the Fundamental Domain for Some Arithmetical Subgroups of Chevalley Groups

As usual, there is associated to gQ a connected algebraic group G of linear transformations of VC = VQ ⊗ QC. If H is some lattice in VQ satisfying (i) M = ∑ λ∈L M ∩ V (λ), (ii) (X α/n!)M ⊆ M for all α, then we let GZ = {g ∈ G | gM = M}. Let ω be a left invariant form on GR of highest degree which takes the value ±1 on (∧i=1H1) ∧ (∧α>0Xα) and let [dg] be the Haar measure associated to ω. Our purpose now is to show the following. If ξ(·) is the Riemann zeta function, Πi=1(ti + 1) is the Poincaré polynomial of GC, and c is the order of the fundamental group of GC then ∫