Notes on Chevalley Groups and Root Category II: Compact Lie Groups and Representations

TL;DR

This work develops a bridge from quiver-derived root categories to classical representation theory of compact Lie groups. By constructing a compact real form from root-category data and identifying a maximal compact subgroup within Lusztig's reductive framework, the authors recover the Peter–Weyl and Plancherel theorems in a setting informed by the coordinate ring ${\mathbf{O}}_A$ and the modified quantum group ${\dot{\mathbf{U}}}$. The approach yields explicit realizations of core structures (Hilbert spaces of matrix coefficients, unitary highest-weight representations) and clarifies how universal covering and center data interact with compact forms. Overall, the paper demonstrates that the classical representation theory of compact Lie groups can be derived from Lusztig’s quantum and root-category machinery, providing explicit constructions and proofs. The results unify Hall-algebra-derived Lie theory with analytic harmonic analysis on compact groups, highlighting a path from quiver representations to concrete Peter–Weyl and Plancherel decompositions.

Abstract

This paper is a continuation of [5]. Using the root categories, we define the compact real forms of the complex semisimple Lie algebras, and maximal compact subgroups of the Chevalley groups over $\mathbb{C}$. In [7], Lusztig used the modified quantum group $\dot{\mathbf{U}}$ and its canonical basis to obtain the reductive group and its coordinate ring $\mathbf{O}_A$, in particular the tensor product decomposition of $\mathbf{O}_A$. By combining these two kinds of structures, we explore in this paper how the classical theory of the compact Lie groups, such as Peter-Weyl theorem and Plancherel theorem, can be recovered completely.

Theorems & Definitions (96)

• Proposition 2.1
• Definition 2.2
• Theorem 2.3
• Lemma 2.4
• Theorem 2.5
• Corollary 2.6
• Definition 2.7
• Lemma 2.8
• proof
• Theorem 2.9