Equidistribution of Expanding Degenerate Manifolds in the Space of Lattices

The paper’s main theorem (Theorem 1.2) gives a necessary-and-sufficient equidistribution criterion for gt-translates of real-analytic submanifolds in the expanding horosphere U+ of SLn, in terms of exactly the three obstructions stated by the model: (1) Aφ ∈ Wn−1(d,n−d), (2) a restriction-of-scalars arithmetic obstruction over a real number field K with rm=n and r≥d, and (3) an exterior-square obstruction when n is even and d=2 encoded by Aext ∈ W(n−2)/2(2n−3,N). The paper sets up the same parametrization Lφ={(x, x̃Aφ)} after permuting coordinates and works with the same diagonal flow gt and expanding U+, exactly as in the model’s setup. It proves non-escape of mass via quantitative non-divergence (Kleinbock–Margulis) and then, upon failure of equidistribution, twists by the centralizer to obtain invariance under a nontrivial unipotent subgroup, applies Ratner’s measure classification, and reduces to a finite list of algebraic scenarios which are shown to be precisely (1)–(3); conversely, each obstruction is shown to prevent equidistribution. These steps correspond to Theorem 1.1 and Theorem 1.2 and the technical lemmas in Sections 6–7, including the explicit construction and use of Aext and the exterior-square representation. See the statement of Theorem 1.2 and the definitions of Wr and Aext, the non-escape-of-mass criterion (Theorem 1.1), the twisting by the centralizer and unipotent invariance, the linearization and representation-theoretic reduction, and the final equivalence proof tying Proposition 5.1 to (1)–(3) via Lemmas 6.2, 6.4, and 6.5 . The model’s outline matches this architecture almost point-for-point. The only substantive flaw in the model’s write-up is a technical mismatch in B1: it asserts an exterior-power (∧d) bound with a term e−t‖q‖, whereas on ∧d the negative weight arising from replacing e1 by an ei≥d+1 is e−dt (not e−t). This can be repaired by carrying out B1 in the standard representation, as done in Lemma 6.2 of the paper, which yields the correct balance e(n−1)t‖Aq+p‖ versus e−t‖q‖ and the familiar en t scaling. With this minor correction, the model’s argument aligns with the paper’s and reaches the same “if and only if” criterion.