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In his seminal Lecture Notes in Mathematics published in 1981, AndreyZelevinsky introduced a new family of Hopf algebras which he called {\emPSH-algebras}. These algebras were designed to capture the representationtheory of the symmetric groups and of classical groups over finite fields. Thegist of this construction is to translate representation-theoretic operationssuch as induction and restriction and their parabolic variants to algebra andcoalgebra operations such as multiplication and comultiplication. The Mackeyformula, for example, is then reincarnated as the Hopf axiom on the algebraside. In this paper we take substantial steps to adapt these ideas for generallinear groups over compact discrete valuation rings. We construct an analogousbi-algebra that contains a large PSH-algebra that extends Zelevinsky's algebrafor the case of general linear groups over finite fields. We prove several basechange results relating algebras over extensions of discrete valuation rings.

An inductive approach to representations of general linear groups over compact discrete valuation rings