  In his seminal Lecture Notes in Mathematics published in 1981, Andrey
Zelevinsky introduced a new family of Hopf algebras which he called {\em
PSH-algebras}. These algebras were designed to capture the representation
theory of the symmetric groups and of classical groups over finite fields. The
gist of this construction is to translate representation-theoretic operations
such as induction and restriction and their parabolic variants to algebra and
coalgebra operations such as multiplication and comultiplication. The Mackey
formula, for example, is then reincarnated as the Hopf axiom on the algebra
side. In this paper we take substantial steps to adapt these ideas for general
linear groups over compact discrete valuation rings. We construct an analogous
bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra
for the case of general linear groups over finite fields. We prove several base
change results relating algebras over extensions of discrete valuation rings.
