Abstract: 

Lie-theoretic data is used to define many mathematical objects and categories. When the data is of type A, the same universal structure would often appear in these categories under various disguises.

This talk will try to advocate such a paradigm by explaining several intriguing bridges between the representation theory of p-adic groups and the realm of quantum algebras.

In the first part, I will show how the quantum affine Schur-Weyl duality can be used to bring together techniques and notions from separate settings.  As an example, these can be applied to obtain a branching law (Gan-Gross-Prasad conjectures), which bears significance in number theory.

In the second part, we will recall how deforming the Bernstein-Zelevinski ring of representations of GL_n lands us in the theory of canonical bases for quantum groups. Insight into this phenomenon allows for an introduction of new invariants to the p-adic setting.