Abstract: This talk will focus on the link between geometry and representation theory of Lie groups in the context of operator algebras. Weyl character formula describes characters of irreducible representations of compact Lie groups. This formula can be obtained using geometric method, for example, from the Atiyah-Bott fixed-point theorem. Harish-Chandra character formula, the noncompact analogue of the Weyl character formula, are closely related in the context of geometry. We apply orbital integrals on K-theory of Harish-Chandra Schwartz algebra of a semisimple Lie group G, and then use geometric method to deduce Harish-Chandra character formulas for discrete series representations of G. This is joint work with Peter Hochs (arXiv:1701.08479).