Harish-Chandra theorem for two-parameter quantum groups

Abstract This paper is devoted to investigating the center of two-parameter quantum groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mrow> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> {\mathfrak{g}} is even, the Harish-Chandra homomorphism is an isomorphism, and in particular, the center of the quantum group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi>U</m:mi> <m:mo>˘</m:mo> </m:mover> <m:mrow> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\breve{U}_{r,s}(\mathfrak{g})} of the weight lattice type is a polynomial algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝕂</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msub> <m:mi>z</m:mi> <m:msub> <m:mi>ϖ</m:mi> <m:mn>1</m:mn> </m:msub> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>z</m:mi> <m:msub> <m:mi>ϖ</m:mi> <m:mi>n</m:mi> </m:msub> </m:msub> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {\mathbb{K}[z_{\varpi_{1}},\dots,z_{\varpi_{n}}]} , where canonical central elements <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>z</m:mi> <m:mi>λ</m:mi> </m:msub> </m:math> {z_{\lambda}} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>λ</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi mathvariant="normal">Λ</m:mi> <m:mo>+</m:mo> </m:msup> </m:mrow> </m:math> {\lambda\in\Lambda^{+}} ) are turned out to be uniformly expressed. For the rank <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> {\mathfrak{g}} to be odd, we figure out a new invertible extra central generator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>z</m:mi> <m:mo>*</m:mo> </m:msub> </m:math> {z_{*}} , which does not survive in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {U_{q}(\mathfrak{g})} , then the center of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mover accent="true"> <m:mi>U</m:mi> <m:m