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.^[1] 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.^[2]

Let ${\displaystyle G}$ be a connected, semisimple compact Lie group and let ${\displaystyle \mathrm {d} g}$ be the Haar probability measure. Let ${\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)}$ be its Lie algebra and ${\displaystyle {\mathfrak {t}}\subset {\mathfrak {g}}}$ be the Cartan subalgebra (or its complexification ${\displaystyle {\mathfrak {t}}\subset {\mathfrak {g}}_{\mathbb {C} }}$).

The Harish-Chandra integral is the function^[3][4]

${\displaystyle H(x,y):=\int _{G}e^{\langle \operatorname {Ad} _{g}x,y\rangle }\mathrm {d} g}$

for ${\displaystyle x,y\in {\mathfrak {t}}}$, where ${\displaystyle \operatorname {Ad} }$ denotes the adjoint representation, and ${\displaystyle \langle \cdot {,}\cdot \rangle }$ is the killing form.

Let ${\displaystyle G}$ be connected and semisimple, and let ${\displaystyle R^{+}}$ denote the positive root system of ${\displaystyle {\mathfrak {t}}}$. Then^[3]

${\displaystyle \Delta _{\mathfrak {t}}(x)\Delta _{\mathfrak {t}}(y)\int _{G}e^{\langle \operatorname {Ad} _{g}x,y\rangle }\mathrm {d} g={\frac {[\![\Delta _{\mathfrak {t}},\Delta _{\mathfrak {t}}]\!]}{|W|}}\sum \limits _{w\in W}\varepsilon (w)e^{\langle w(x),y\rangle }}$,

where

• ${\displaystyle W}$ is the Weyl group acting on ${\displaystyle {\mathfrak {t}}}$,
• ${\displaystyle \Delta _{\mathfrak {t}}(x):=\prod _{\alpha \in R^{+}}\langle \alpha ,x\rangle }$ is a polynomial function called the discriminant,
• ${\displaystyle \varepsilon (w):=(-1)^{|w|}}$ is the signature,
• ${\displaystyle [\![\Delta _{\mathfrak {t}},\Delta _{\mathfrak {t}}]\!]}$ is an inner product that extends the killing form to polynomial functions defined in the following way: If ${\displaystyle p}$ and ${\displaystyle q}$ are polynomial functions on the real Lie algebra ${\displaystyle {\mathfrak {g}}}$, write ${\displaystyle p(x)=\sum {\beta }c_{\beta }x^{\beta }}$ in real coordinates ${\displaystyle x_{1},\dots ,x_{n}}$, where ${\displaystyle n=\dim {\mathfrak {g}}}$. The associated differential operator is

${\displaystyle p(\partial )=\sum _{\beta }c_{\beta }{\frac {\partial ^{|\beta |}}{\partial x^{\beta }}}.}$

The bilinear form ${\displaystyle [[p,q]]}$ is defined by applying the differential operator ${\displaystyle p(\partial )}$ to ${\displaystyle q(x)}$ then evaluating the result at the zero, i.e.

${\displaystyle [[p,q]]:=p(\partial )q(x){\bigg |}_{x=0}.}$