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 (or its complexification ).
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 . Then , where 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
The bilinear form is defined by applying the differential operator to then evaluating the result at the zero, i.e.
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