\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 $bc$.