  The classical Peter-Weyl theorem describes the structure of the space of
functions on a semi-simple algebraic group. On the level of characters (in type
A) this boils down to the Cauchy identity for the products of Schur
polynomials. We formulate and prove the analogue of the Peter-Weyl theorem for
the current groups. In particular, in type A the corresponding characters
identity is governed by the Cauchy identity for the products of q-Whittaker
functions. We also formulate and prove a version of the Schur-Weyl theorem for
current groups. The link between the Peter-Weyl and Schur-Weyl theorems is
provided by the (current version of) Howe duality.
