Dimension growth and Gelfand-Kirillov dimension of representations of quantum groups (2512.14385v1)

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.

• The paper establishes an asymptotic equivalence between the growth of simple module dimensions at roots of unity and the GK-dimension of generic highest weight modules.
• It computes explicit minimal nonzero values for various Lie types, notably revealing a significant deviation in type B quantum cases.
• It develops a quantum generalization of block decomposition, identifying new structures such as quantum cuspidal modules and non-dual-closed root subsystems.

Dimension Growth and Gelfand-Kirillov Dimension in Quantum Group Representations

Overview

This paper conducts a detailed investigation into two algebraic invariants within the representation theory of quantum groups: (1) the dimension growth of simple modules for the De Concini–Kac quantum group at roots of unity, and (2) the Gelfand-Kirillov (GK) dimension of simple highest weight modules for the quantum group at generic parameter $q$. These invariants, although naturally defined in distinct representation-theoretic regimes ($q$ a root of unity and $q$ transcendental, respectively), are shown to be closely related. The authors analyze how their interplay unveils new quantum phenomena and sharp distinctions relative to the classical enveloping algebra context, particularly highlighting the critical role of representations with non-integral weights.

Quantum Groups and Representation-Theoretic Invariants

The two central objects are the Drinfeld–Jimbo quantum group $U_q(\mathfrak{g})$ and its specialization $U_\zeta$ at a root of unity. The structure of $U_q$ mirrors $U(\mathfrak{g})$ in characteristic zero, while $U_\zeta$ provides a quantum analogue to representations of Lie algebras over fields of positive characteristic. The study leverages parallelism between the generic and root-of-unity cases to explain quantum/classical discrepancies.

The paper rigorously analyzes simple highest weight modules in both settings. For $U_\zeta$, the focus is on the dimension of simple modules $L_\zeta(\Lambda)$ as a function of the order $\ell$ of the root of unity. The growth rate of $\dim L_\zeta(\Lambda)$ as $\ell\rightarrow \infty$ is quantified and compared to the GK-dimension of the corresponding generic highest weight module $L_q(\Lambda)$.

Main Results

1. Asymptotic Correspondence between Dimension Growth and GK-Dimension

The authors establish that, for a simple module $L_\zeta(\Lambda)$ with fixed weight $\Lambda$,

$$\lim_{\ell\to\infty} \frac{\log \dim L_\zeta(\Lambda)}{\log \ell} = \operatorname{GKdim}_{U_q} L_q(\Lambda),$$

with the limit taken over all admissible orders $\ell$ of the root of unity [(2512.14385), Theorem 1]. This formula provides a precise bridge between the two regimes and enables a dimension-theoretic transfer between the quantum group at roots of unity and the generic $q$ case, analogous to results in the classical setting.

2. Explicit Computation and Minimal Nonzero Values

Through an explicit computation of the dimension growth, the paper determines the minimal nonzero value of both invariants for all simple Lie types. The main findings are:

• For all simple $L_q(\Lambda)$ outside the set of finite-dimensional modules, the minimal nonzero GK-dimension in type $A$, $C$, and most other types coincides with the classical value $h^\vee-1$ ($h^\vee$ the dual Coxeter number).
• For type $B_n$, a sharp deviation arises: the minimal GK-dimension in the quantum case equals $n$ (rank of $\Phi$), rather than the classical value $2n-2$.
• The set of possible minimal values aligns with the structure of maximal root subsystems and is realized precisely by regular blocks associated with maximal root subsystems (see Table 1 and Theorem 5.6 of (2512.14385)).

3. Block Decomposition and Integral Weyl Groups

The paper develops a quantum generalization of Soergel's structural description of blocks in the category $\mathcal{O}$, addressing the module-theoretic implications of the quantum parameter. In particular, it is shown that, unlike in the classical theory, the assignment from indecomposable blocks to integral root subsystems can produce non-dual-closed subsystems in the quantum case. This breaking of classical dual-closedness appears as soon as non-linear weights are considered.

Furthermore, the block decomposition is shown to be governed by integral Weyl groups and their corresponding root subsystems, and their combinatorics are analyzed in detail.

4. Quantum Cuspidal Modules

As an application of the dimension formulas, the authors settle the structure–theoretic existence of quantum cuspidal weight modules. Simple infinite-dimensional weight modules with constant weight multiplicities (quantum analogues of classical cuspidal modules) exist at generic $q$ only when the underlying semisimple Lie algebra consists of simple components of type $A$, $B$, or $C$. This provides a striking representation-theoretic distinction between the quantum and classical cases, especially in type $B$, where the quantum theory presents new constraints [(2512.14385), Section 6].

Implications and Outlook

This work significantly refines the understanding of dimension growth phenomena and the behaviour of GK-dimension in quantum group representation theory. The determination of the precise asymptotic link between dimensions at root-of-unity parameter and generic $q$ reinforces and extends the conceptual framework underlying quantum and modular representation theory. The identification of genuinely quantum effects in the block–root subsystem correspondence and the classification of quantum cuspidal modules exposes subtleties absent from the classical setting and suggests avenues for ongoing inquiry, including:

• Further exploration of the role of non-integral and non-linear weights in category $\mathcal{O}$ and their connections to quantum group automorphisms, dualities, and categorical structures.
• Investigation of combinatorial invariants attached to quantum groups, with a focus on their connections to Hecke algebra and Kazhdan–Lusztig theory, incorporating Lusztig's $a$-function.
• Analysis of the modular–quantum analogy at the level of associated varieties, primitive ideals, and support varieties for quantum algebras.

Conclusion

The paper "Dimension growth and Gelfand-Kirillov dimension of representations of quantum groups" (2512.14385) rigorously establishes a correspondence between dimension growth in quantum group representations at roots of unity and the GK-dimension of representations at generic $q$. The results clarify new quantum phenomena absent in classical theory, provide explicit dimension formulas for all Lie types, and settle the structure of quantum cuspidal modules. These findings contribute foundational tools and insights for quantum group theory, category $\mathcal{O}$, and beyond.