Positive representations of non-simply-laced split real quantum groups

TL;DR

This paper constructs positive principal series representations for certain non-simply-laced split real quantum groups, revealing relations with Langlands dual groups and embedding into q-tori polynomials.

Contribution

It introduces new positive representations for non-simply-laced split real quantum groups and explores their relation to Langlands dual groups and modular double structures.

Findings

Representations parametrized by R^r for types B, C, F4, G2.

Generators of dual groups relate via transcendental relations.

Embeddings into q-tori polynomials established.

Abstract

We construct the positive principal series representations for $U_q(g_R)$ where $g$ is of type $B_n$, $C_n$, $F_4$ or $G_2$, parametrized by $R^r$ where $r$ is the rank of $g$. We show that under the representations, the generators of the Langlands dual group $U_{\tilde{q}}({}^Lg_R)$ are related to the generators of $U_q(g_R)$ by the transcendental relations. We define the modified quantum group of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the $q$-tori polynomials.