We describe work with F. Brumley on the asymptotics of Whittaker functions motivated by number theoretic questions. Whittaker functions arise from generic representations of Lie groups when imposing invariance properties with respect to a maximal unipotent subgroup. For $SL(2,R)$ they coincide with the K-Bessel functions. In general we are interested in ''computing'' them in the semiclassical limit by methods from the geometry of integrable systems and representation theory.