Harish-Chandra theorem for two-parameter quantum groups

Abstract This paper is devoted to investigating the center of two-parameter quantum groups U r , s ⁢ ( ???? ) {U_{r,s}(\mathfrak{g})} via establishing the Harish-Chandra homomorphism. Based on the Rosso form and the representation theory of weight modules, we prove that when the rank ???? {\mathfrak{g}} is even, the Harish-Chandra homomorphism is an isomorphism, and in particular, the center of the quantum group U ˘ r , s ⁢ ( ???? ) {\breve{U}_{r,s}(\mathfrak{g})} of the weight lattice type is a polynomial algebra ???? ⁢ [ z ϖ 1 , … , z ϖ n ] {\mathbb{K}[z_{\varpi_{1}},\dots,z_{\varpi_{n}}]} , where canonical central elements z λ {z_{\lambda}} ( λ ∈ Λ + {\lambda\in\Lambda^{+}} ) are turned out to be uniformly expressed. For the rank ???? {\mathfrak{g}} to be odd, we figure out a new invertible extra central generator z * {z_{*}} , which does not survive in U q ⁢ ( ???? ) {U_{q}(\mathfrak{g})} , then the center of U ˘ r , s ⁢ ( ???? ) {\breve{U}_{r,s}(\mathfrak{g})} contains ???? ⁢ [ z ϖ 1 , … , z ϖ n ] ⊗ ???? ???? ⁢ [ z * 1 ℓ , z * - 1 ℓ ] {\mathbb{K}[z_{\varpi_{1}},\dots,z_{\varpi_{n}}]\otimes_{\mathbb{K}}\mathbb{K}% [z_{*}^{{\frac{1}{\ell}}},z_{*}^{{-\frac{1}{\ell}}}]} , where ℓ = 2 {\ell=2} , except ℓ = 4 {\ell=4} for D 2 ⁢ k + 1 {D_{2k+1}} .