Correction to ‘ Quotients of Unipotent Translations

The statements of Main Theorem 1.1 and Theorem 2.1 of the author's paper [Trans. Amer. are affected similarly.) These restrictions can be removed if the conclusions of the results are weakened to allow for the possible existence of transitive, proper subgroups of G. In this form, the results can be extended to the setting where G is a product of real and p-adic Lie groups. There are two errors in the proof of Theorem 2.1 of [W]. To eliminate these mistakes, the statements of Theorem 2.1 and Main Theorem 1.1 should assume that Γ is discrete and G is connected. Although it includes the main cases of interest, the restriction to connected groups and discrete subgroups is not entirely satisfying. For more general Lie groups, however, there are two problems in the proof: (1) it was tacitly assumed that Aff(Λ\G) is second countable, but this may not be the case if G/G • is not finitely generated; and (2) some proper subgroups of G may be transitive on Γ\G, so X may not project onto all of G in Step 1 on p. 582. The first mistake can be eliminated by using the topology of convergence in measure. The second problem can be eliminated by hypothesis (as above), but it can be better resolved by weakening the conclusion to account for the transitive proper subgroups (see Cor. A.2 below). The improved argument that eliminates Problem (1) applies to any locally compact group, not just Lie groups (see A.1). Because M. Ratner's Classification of Invariant Measures (1.2) is now known to be true not just for Lie groups, but also for direct products of real and p-adic Lie groups [MT], [R], this allows us to extend Main Theorem 1.1 to this more general setting (see A.3). Before stating our results, let us present a simplified definition of central double-coset quotients. The original definition imposed a more complicated restriction on K, because the author did not realize that noncompact groups are unnecessary (when Γ\G has finite volume). Definition (cf. [W, p. 578]). Let U be a subgroup of a locally compact group G, and let Γ\G be a finite-volume homogeneous space of G. Suppose 1. Λ is a closed subgroup of G containing Γ; and 2. K is a compact subgroup of Aff(Λ\G) that centralizes U. Then the natural U-action on Λ\G/K is a quotient of the U-action on Γ\G. It is …