  We propose group theory interpretation of the integral representation of the
quantum open Toda chain wave function due to Givental. In particular we
construct the representation of $U((\mathfrak{gl}(N))$ in terms of first order
differential operators in Givental variables. The construction of this
representation turns out to be closely connected with the integral
representation based on the factorized Gauss decomposition. We also reveal the
recursive structure of the Givental representation and provide the connection
with the Baxter $Q$-operator formalism. Finally the generalization of the
integral representation to the infinite and periodic quantum Toda wave
functions is discussed.
