Demazure-Lusztig operators appear throughout the study of p-adic
Whittaker functions. They can be used to construct related objects, in
both the nonmetaplectic and the metaplectic, and the
finite-dimensional and affine setting. Brubaker, Bump and Licata used
them to describe Iwahori-Whittaker functions in the finite-dimensional
setting; Manish Patnaik to prove an analogue of the Casselman-Shalika
formula for affine Kac-Moody groups. In this talk, I will review how
metaplectic analogues of the classical operators can be used to prove
an analogue of Tokuyama's theorem. This links the constructions of
Whittaker functions as a sum over a highest weight crystal
(Brubaker-Bump-Friedberg and McNamara), and as a sum over a Weyl group
(Chinta-Offen and McNamara). Then I will discuss joint work with
Manish Patnaik that relates metaplectic Iwahori-Whittaker functions to
Demazure-Lusztig operators in the finite dimensional as well as in the
affine setting.