Analyzing the Weyl Construction for Dynamical Cartan Subalgebras

When the reduced twisted C^∗-algebra C_r^∗(G, c) of a non-principal groupoid G admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of C_r^∗(G, c). In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid S of G. In this paper, we study the relationship between the original groupoids S, G and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum B of the Cartan subalgebra C_r^∗(S, c). We then show that the quotient groupoid G/S acts on B, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map G → G/S admits a continuous section, then the Weyl twist is also given by an explicit continuous 2-cocycle on G/S × B.