Siegel domains

Fix an isomorphism T0→ defined over k, where is the multiplicative group. If k is a number field, embed ℝ*+ in, identifying t ∈ with the element (tv) ∈ such that tv = t if v is archimedean, tv = 1 if v is finite. Then is identified with a subgroup of which is split in the extension T0 since is simply connected. We denote by AM0 the unique connected subgroup of T0 which projects onto. If k is a function field, fix once and for all a place V0 of k and a uniformising parameter ∞ at v0. The group generated by ∞ can be identified with a subgroup of and with a subgroup of. By the fact that is contained in ZM0 (see 1.1.3 (2)), we see that there exists a subgroup of ZM0 such that the projection pr : G → G() establishes an isomorphism between this group and a subgroup of finite index of. Fix such a subgroup once and for all, and denote it by AM0- Up to replacing it by

which satisfies the same conditions, we can suppose that AM0 is invariant under the action of W.

More generally, if P = MU is a standard parabolic subgroup of G, we set. Let logM be the composition of the map log : with the projection.