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