\indent In a recent paper, Rains and Vazirani used Hecke algebra techniques to develop $(q,t)$-generalizations of a number of well-known vanishing identities for Schur functions. However, their approach does not work directly at $q=0$ (the Hall-Littlewood level). We discuss a technique that is more combinatorial in nature, and allows us to obtain generalizations of some of their results at $q=0$ as well as a finite-dimensional analog of a recent summation formula of Warnaar. We will also briefly explain how these results are related to $p$-adic representation theory. Finally, we will explain how this method can be extended to give an explicit construction of Hall-Littlewood polynomials of type $BC$.