Infinitesimal character

In mathematics, the infinitesimal character of an irreducible representation ${\displaystyle \rho }$ of a semisimple Lie group ${\displaystyle G}$ on a vector space ${\displaystyle V}$ is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagonalizing the representation. It therefore is a way of extracting something essential from the representation ${\displaystyle \rho }$ by two successive linearizations.