  Determination of quasi-invariant generalized functions is important for a
variety of problems in representation theory, notably character theory and
restriction problems. In this note, we review some new and easy-to-use
techniques to show vanishing of quasi-invariant generalized functions,
developed in the recent work of the authors (Uniqueness of Ginzburg-Rallis
models: the Archimedean case, Trans. Amer. Math. Soc. 363, (2011), 2763-2802).
The first two techniques involve geometric notions attached to submanifolds,
which we call metrical properness and unipotent $\chi$-incompatibility. The
third one is analytic in nature, and it arises from the first occurrence
phenomenon in Howe correspondence. We also highlight how these techniques
quickly lead to two well-known uniqueness results, on trilinear forms and
Whittaker models.
