Appears in collection : 2025 - T1 - WS2 - Tempered representations and K-theory

In 1990, Kashiwara and Lusztig independently discovered the theory of crystal bases for complex semisimple Lie algebras. This theory says that if we deform the universal enveloping algebra by the Drinfled-Jimbo quantisation procedure, and let the quantisation parameter go to infinity, then the structure theory of the finite-dimensional representations becomes extremely simple. This allows an easy understanding of basic problems like tensor decompositions and branching rules. In this talk, I will explain a dual phenomenon, namely the crystallisation of the algebra of functions on a compact Lie group.

By the quantum duality principle, this has implications for unitary representations of complex semisimple Lie groups. If time permits, we will discuss the case of $\operatorname{SL}(2,\mathbb{C})$.