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 {\displaystyle G} be a connected, semisimple compact Lie group and let d g {\displaystyle \mathrm {d} g} be the Haar probability measure. Let g = Lie ( G ) {\displaystyle {\mathfrak {g}}=\operatorname {Lie} (G)} be its Lie algebra and t ⊂ g {\displaystyle {\mathfrak {t}}\subset {\mathfrak {g}}} be the Cartan subalgebra (or its complexification t ⊂ g C {\displaystyle {\mathfrak {t}}\subset {\mathfrak {g}}_{\mathbb {C} }} ). The Harish-Chandra integral is the function
H ( x , y ) := ∫ G e ⟨ Ad g x , y ⟩ d g {\displaystyle H(x,y):=\int _{G}e^{\langle \operatorname {Ad} _{g}x,y\rangle }\mathrm {d} g}
for x , y ∈ t {\displaystyle x,y\in {\mathfrak {t}}} , where Ad {\displaystyle \operatorname {Ad} } denotes the adjoint representation, and ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot {,}\cdot \rangle } is the killing form.
Harish-Chandra formula Let G {\displaystyle G} be connected and semisimple, and let R + {\displaystyle R^{+}} denote the positive root system of t {\displaystyle {\mathfrak {t}}} . Then
Δ t ( x ) Δ t ( y ) ∫ G e ⟨ Ad g x , y ⟩ d g = [ [ Δ t , Δ t ] ] | W | ∑ w ∈ W ε ( w ) e ⟨ w ( x ) , y ⟩ {\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
W {\displaystyle W} is the Weyl group acting on t {\displaystyle {\mathfrak {t}}} ,
Δ t ( x ) := ∏ α ∈ R + ⟨ α , x ⟩ {\displaystyle \Delta _{\mathfrak {t}}(x):=\prod _{\alpha \in R^{+}}\langle \alpha ,x\rangle } is a polynomial function called the discriminant,
ε ( w ) := ( − 1 ) | w | {\displaystyle \varepsilon (w):=(-1)^{|w|}} is the signature,
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