Before revision: Given two n \times n Hermitian matrices Y and \Lambda, the Harish-Chandra--Itzykson--Zuber (HCIZ) density on the unitary group \mathrm{U (n) is e^{ \mathrm{Tr (U\Lambda U^*Y)}d\mu(U)  where \mu is the Haar measure on \mathrm{U (n). (U\Lambda U^*Y)} is known as the HCIZ density.
After revision: Given two n \times n Hermitian matrices Y and \Lambda, the Harish-Chandra--Itzykson--Zuber (HCIZ) distribution on the unitary group \mathrm{U (n) is e^{ \mathrm{Tr (U\Lambda U^*Y)}d\mu(U) where \mu is the Haar measure on \mathrm{U (n). (U\Lambda U^*Y)} is known as the HCIZ density.