Harish-Chandra integral
The Harish-Chandra integral is a concept from integral calculus that originated in the study of harmonic analysis on Lie groups. Closely related is the Harish-Chandra formula which is used to evaluate the integral.
The integrals are named after the Indian mathematician Harish-Chandra, who proved in 1957 the so-called Harish-Chandra formula. Today, the integral and its associated formula find applications in many fields, such as representation theory, random matrix theory and quantum field theory.
A special case is the formula for integrals over the unitary group which was independently discovered in 1980 by Claude Itzykson and Jean-Bernard Zuber and applied to quantum field theory. The integral over the unitary group is also referred to as the Harish-Chandra–Itzykson–Zuber integral.
Definition
Let be a connected, semisimple compact Lie group and let be the Haar probability measure. Let be its Lie algebra and be the Cartan subalgebra.The Harish-Chandra integral is the function
for, where denotes the adjoint representation, and is the killing form.
Harish-Chandra formula
Let be connected and semisimple, and let denote the positive root system of. Thenwhere
- is the Weyl group acting on,
- is a polynomial function called the discriminant,
- is the signature,
- is an inner product that extends the killing form to polynomial functions defined in the following way: If and are polynomial functions on the real Lie algebra, write in real coordinates, where. The associated differential operator is