Dimension growth and Gelfand-Kirillov dimension of representations of quantum groups

Abstract

We consider two algebraic invariants in the representation theory of quantized enveloping algebras: the dimension growth of simple modules for the De Concini-Kac quantum group at roots of unity, and the Gelfand-Kirillov dimension of simple highest weight modules for the quantum group at generic $q$. In spite of being defined for different values of the parameter $q$, these invariants reflect closely related features in the respective contexts. We show that several new phenomena appear in the quantum case and the representations with non-integral weights contribute to both invariants in a way that cannot be ignored. Building on this, we determine the minimal non-zero value of these invariants for each Lie type. As an application we show that quantum cuspidal modules at generic $q$ can occur only when the underlying semisimple Lie algebra has simple components of type $A$, $B$, or $C$, providing a more explicit representation-theoretic distinction with the classical case.