Given a symmetric pair (g,k), a pair of complex reductive Lie algebras, one has its quantum deformation (U_q(g), U^i(k)), called a quantum symmetric pair (QSP for short). A QSP consists of a quantum group U_q(g) and its certain coideal subalgebra U^i(k), called an i-quantum group.

Although the i-quantum group U^i(k) is a quantum deformation of the Lie algebra k, it is different from the quantum group U_q(k), in general. As a result, the representation theory of i-quantum groups has a different flavor than that of quantum groups. In these lectures, we focus on non-Levi branching rules, i.e., the k-module structure of various g-modules. Such problems cannot be solved by means of quantum groups because U_q(k) is not a subalgebra of U_q(g).

In the first part of the lectures, we review representation theory of quantum groups quickly.

Then, we introduce QSPs and their integrable modules.

We show that the integrable modules for a QSP leads us to a Peter-Weyl type decomposition of the dual of the i-quantum group.

In the remaining parts, we introduce some interesting type-dependent results.

We focus on type AI, AII, BII, and DII; the corresponding symmetric pairs are (gl_n, so_n), (gl_2n, sp_2n), (so_2n+1, so_2n), and (so_2n, so_2n-1).

For these pairs, we can understand the branching rules in terms of some combinatorial objects.

Such combinatorial objects include semistandard Young tableaux, King's symplectic tableaux, Kashiwara-Nakashima's orthogonal tableaux, and Gelfand-Tsetlin patterns.