L-indistinguishability for covering groups of algebraic tori

A global packet may simultaneously contain an automorphic representation and a non-automorphic representation. The global $\mathcal{S}$-group is expected, and known in some cases, to specify the automorphic representations in each global packet. For a covering group of an algebraic torus, it is not obvious from the definition whether the analogue of a global $\mathcal{S}$-group has finite order. In this paper, we verify this finiteness for a Brylinski-Deligne covering group of a torus.