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Harish-Chandra's c-function

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Function named after Harish Chandra

In mathematics, Harish-Chandra's c-function is a function related to the intertwining operator between two principal series representations, that appears in the Plancherel measure for semisimple Lie groups. Harish-Chandra (1958a, 1958b) introduced a special case of it defined in terms of the asymptotic behavior of a zonal spherical function of a Lie group, and Harish-Chandra (1970) introduced a more general c-function called Harish-Chandra's (generalized) C-function. Gindikin and Karpelevich (1962, 1969) introduced the Gindikin–Karpelevich formula, a product formula for Harish-Chandra's c-function.

Gindikin–Karpelevich formula

The c-function has a generalization cw(λ) depending on an element w of the Weyl group. The unique element of greatest length s0, is the unique element that carries the Weyl chamber a + ∗ {\displaystyle {\mathfrak {a}}_{+}^{*}} onto − a + ∗ {\displaystyle -{\mathfrak {a}}_{+}^{*}} . By Harish-Chandra's integral formula, cs0 is Harish-Chandra's c-function:

c ( λ ) = c s 0 ( λ ) . {\displaystyle c(\lambda )=c_{s_{0}}(\lambda ).}

The c-functions are in general defined by the equation

A ( s , λ ) ξ 0 = c s ( λ ) ξ 0 , {\displaystyle \displaystyle A(s,\lambda )\xi _{0}=c_{s}(\lambda )\xi _{0},}

where ξ0 is the constant function 1 in L(K/M). The cocycle property of the intertwining operators implies a similar multiplicative property for the c-functions:

c s 1 s 2 ( λ ) = c s 1 ( s 2 λ ) c s 2 ( λ ) {\displaystyle c_{s_{1}s_{2}}(\lambda )=c_{s_{1}}(s_{2}\lambda )c_{s_{2}}(\lambda )}

provided

ℓ ( s 1 s 2 ) = ℓ ( s 1 ) + ℓ ( s 2 ) . {\displaystyle \ell (s_{1}s_{2})=\ell (s_{1})+\ell (s_{2}).}

This reduces the computation of cs to the case when s = sα, the reflection in a (simple) root α, the so-called "rank-one reduction" of Gindikin & Karpelevich (1962). In fact the integral involves only the closed connected subgroup G corresponding to the Lie subalgebra generated by g ± α {\displaystyle {\mathfrak {g}}_{\pm \alpha }} where α lies in Σ0. Then G is a real semisimple Lie group with real rank one, i.e. dim A = 1, and cs is just the Harish-Chandra c-function of G. In this case the c-function can be computed directly and is given by

c s α ( λ ) = c 0 2 − i ( λ , α 0 ) Γ ( i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + 1 + i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + m 2 α + i ( λ , α 0 ) ) , {\displaystyle c_{s_{\alpha }}(\lambda )=c_{0}{2^{-i(\lambda ,\alpha _{0})}\Gamma (i(\lambda ,\alpha _{0})) \over \Gamma ({1 \over 2}({1 \over 2}m_{\alpha }+1+i(\lambda ,\alpha _{0}))\Gamma ({1 \over 2}({1 \over 2}m_{\alpha }+m_{2\alpha }+i(\lambda ,\alpha _{0}))},}

where

c 0 = 2 m α / 2 + m 2 α Γ ( 1 2 ( m α + m 2 α + 1 ) ) {\displaystyle c_{0}=2^{m_{\alpha }/2+m_{2\alpha }}\Gamma \left({1 \over 2}(m_{\alpha }+m_{2\alpha }+1)\right)}

and α0=α/〈α,α〉.

The general Gindikin–Karpelevich formula for c(λ) is an immediate consequence of this formula and the multiplicative properties of cs(λ), as follows:

c ( λ ) = c 0 ∏ α ∈ Σ 0 + 2 − i ( λ , α 0 ) Γ ( i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + 1 + i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + m 2 α + i ( λ , α 0 ) ) , {\displaystyle c(\lambda )=c_{0}\prod _{\alpha \in \Sigma _{0}^{+}}{2^{-i(\lambda ,\alpha _{0})}\Gamma (i(\lambda ,\alpha _{0})) \over \Gamma ({1 \over 2}({1 \over 2}m_{\alpha }+1+i(\lambda ,\alpha _{0}))\Gamma ({1 \over 2}({1 \over 2}m_{\alpha }+m_{2\alpha }+i(\lambda ,\alpha _{0}))},}

where the constant c0 is chosen so that c(–iρ)=1 (Helgason 2000, p.447).

Plancherel measure

The c-function appears in the Plancherel theorem for spherical functions, and the Plancherel measure is 1/c times Lebesgue measure.

p-adic Lie groups

There is a similar c-function for p-adic Lie groups. Macdonald (1968, 1971) and Langlands (1971) found an analogous product formula for the c-function of a p-adic Lie group.

References

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