Harish-Chandra modules for quantum symmetric pairs

Let U U denote the quantized enveloping algebra associated to a semisimple Lie algebra. This paper studies Harish-Chandra modules for the recently constructed quantum symmetric pairs U U , B B in the maximally split case. Finite-dimensional U U -modules are shown to be Harish-Chandra as well as the B B -unitary socle of an arbitrary module. A classification of finite-dimensional spherical modules analogous to the classical case is obtained. A one-to-one correspondence between a large class of natural finite-dimensional simple B B -modules and their classical counterparts is established up to the action of almost B B -invariant elements.