On the Helgason-Johnson Bound

Let G be a simple non-compact linear Lie group. ? any irreducible unitary representation of with infinitesimal character ? whose continuous part is ?. The beautiful Helgason-Jonson bound in 1969 says that the norm ? upper bounded by ?(G), which stands for half sum positive roots G. current paper aims to give framework sharpen Helgason—Johnson when infinite-dimensional. We have explicit results exceptional groups. Ingredients proof include Parthasarathy’s Dirac operator inequality, Vogan pencil, and unitarily small convex hull introduced Salamanca-Riba Vogan.