Harish-chandra Modules for Quantum Symmetric Pairs

Let 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 ,B in the maximally split case. Finite-dimensional U -modules are shown to be Harish-Chandra as well as the 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 Bmodules and their classical counterparts is established up to the action of almost B-invariant elements. Let g be a semisimple Lie algebra and let g be the Lie subalgebra fixed by an involution θ of g. There is an extensive theory concerning the Harish-Chandra modules associated to the pair g, g. These are g-modules which behave nicely with respect to the restriction of the action of g to g. One of the main motivations in understanding such modules is their close connection and impact on the study of real Lie group representations. Harish-Chandra modules also are a class of infinitedimensional g-modules that have a reasonable amount of structure and thus are amenable to study. In this way they are similar to the other infinite-dimensional g-modules that have been examined thoroughly, the so-called category O modules. These modules behave nicely with respect to the Cartan subalgebra of g. Let U denote the Drinfeld-Jimbo quantization of the enveloping algebra of g. In the quantum case, there is already a well developed theory of category O modules (see for example [Jo]). However, much less is known about infinite-dimensional quantum modules that correspond to the classical Harish-Chandra modules. One of the main reasons for this difference is that there is an obvious quantum analog of the Cartan subalgebra of g, while the analogs of the invariant Lie subalgebra are less apparent. In [L2], we introduced one-sided coideal algebrasB = Bθ as quantum analogs of the enveloping algebra of the fixed Lie subalgebra under the maximally split form of an involution θ. These analogs generalize the known examples already in the literature in the maximally split case. Using these analogs in this paper, we lay the groundwork for the study of quantum Harish-Chandra modules. In the first part of the paper, we prove elementary results about quantum Harish-Chandra modules associated to U,B. As in the classical case, every finitedimensional simple U -module comes equipped with a positive definite conjugate linear form. One checks that B behaves nicely with respect to this form which allows us to decompose finite-dimensional U -modules into a direct sum of finite-dimenReceived by the editors October 22, 1999 and, in revised form, November 19, 1999. 2000 Mathematics Subject Classification. Primary 17B37. The author was supported by NSF grant no. DMS-9753211. c ©2000 American Mathematical Society