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

The Harish-Chandra integral is the function

$H(x,y):=\underset{G}{\int }{e}^{⟨{\mathrm{Ad}}_{g}x,y⟩}\mathrm{d}g$

for $x,y\in \mathfrak{𝔱}$, where $\mathrm{Ad}$ denotes the adjoint representation, and $⟨\cdot ,\cdot ⟩$ is the killing form.

Harish-Chandra formula

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

${\mathrm{\Delta }}_{\mathfrak{𝔱}}(x){\mathrm{\Delta }}_{\mathfrak{𝔱}}(y)\underset{G}{\int }{e}^{⟨{\mathrm{Ad}}_{g}x,y⟩}\mathrm{d}g=\frac{[[{\mathrm{\Delta }}_{\mathfrak{𝔱}},{\mathrm{\Delta }}_{\mathfrak{𝔱}}]]}{|W|}\sum _{w\in W}\epsilon (w)e^{⟨w(x),y⟩}$,

where

• $W$ is the Weyl group acting on $\mathfrak{𝔱}$,
• ${\mathrm{\Delta }}_{\mathfrak{𝔱}}(x):=\prod _{\alpha \in R^{+}}⟨\alpha ,x⟩$ is a polynomial function called the discriminant,
• $\epsilon (w):=(-1{)}^{|w|}$ is the signature,
• $[[{\mathrm{\Delta }}_{\mathfrak{𝔱}},{\mathrm{\Delta }}_{\mathfrak{𝔱}}]]$ is an inner product that extends the killing form to polynomial functions defined in the following way: If $p$ and $q$ are polynomial functions on the real Lie algebra $\mathfrak{𝔤}$, write $p(x)=\sum \beta c_{\beta }{x}^{\beta }$ in real coordinates $x_1,\dots ,x_n$, where $n=\dim \mathfrak{𝔤}$. The associated differential operator is

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

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

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