Representations of Quantum Coordinate Algebras at Generic $q$ and Wiring Diagrams

This paper is devoted to the representation theory of quantum coordinate algebra $\mathbb{C}_q[G]$, for a semisimple Lie group $G$ and a generic parameter $q$. By inspecting the actions of normal elements on tensor modules, we generalize a result of Levendorski and Soibelman in [22] for highest weight modules. For a double Bruhat cell $G^{w_1,w_2}$, we describe the primitive spectra $\mathrm{prim}\,\mathbb{C}_q[G]_{w_1,w_2}$ in a new fashion, and construct a bundle of $(w_1,w_2)$ type simple modules onto $\mathrm{prim}\,\mathbb{C}_q[G]_{w_1,w_2}$, provided $\mathrm{Supp}(w_1)\cap\mathrm{Supp}(w_2)=\varnothing$ or enough pivot elements. The fibers of the bundle are shown to be products of the spectrums of simple modules of 2-dimensional quantum torus $L_q(2)$. As an application of our theory, we deduce an equivalent condition for the tensor module to be simple, and construct some simple modules for each primitive ideal when $G=SL_3(\mathbb{C})$. This completes the Dixmier's program for $\mathbb{C}_q[SL_3]$. The wiring diagrams, introduced by Fomin and Zelevinsky in their study of total positivity (cf. [3,9]), is the main tool to compute the action of generalized quantum minors on tensor modules in the type A case. We obtain a quantum version of Lindstr\"{o}m's lemma, which plays an important role in transforming representation problems into combinatorial ones of wiring diagrams.