Let G be a connected semi-simple Lie group with torsion-free fundamental group. We show that the twisted equivariant KK-theory \(KK_{\bullet }^{G}(G/K, \tau _G^G)\) of G has a ring structure induced from the renowned ring structure of the twisted equivariant K-theory \(K^{\bullet }_{K}(K, \tau _K^K)\) of a maximal compact subgroup K. We give a geometric description of representatives in \(KK_{\bullet }^{G}(G/K, \tau _G^G)\) in terms of equivalence classes of certain equivariant correspondences and obtain an optimal set of generators of this ring. We also establish various properties of this ring under some additional hypotheses on G and give an application to the quantization of q-Hamiltonian G-spaces in an appendix. We also suggest conjectures regarding the relation to positive energy representations of LG that are induced from certain unitary representations of G in the noncompact case.

设G为具有无扭转基本群的连通半单李群。我们表明，扭曲等变KK -理论\（KK _ {\子弹} ^ {G}（G / K，\ tau蛋白_G ^ G）\）的ģ具有从扭曲等变的著名的环结构诱导的环结构ķ -极大紧子群K 的理论\(K^{\bullet }_{K}(K, \tau _K^K)\)。我们根据某些等变对应的等价类给出\(KK_{\bullet }^{G}(G/K, \tau _G^G)\)中代表的几何描述，并获得该生成器的最优集合戒指。我们还在一些额外的假设下建立了这个环的各种属性ģ并给应用程序的量化q -Hamiltonian ģ -spaces的附录。我们还提出了关于与LG 的正能量表示关系的猜想，这些关系是从非紧致情况下G 的某些幺正表示导出的。