Harish–Chandra c-function - Imported Knowlpedia Content

Definition

Let $G$ be a connected noncompact semisimple Lie group with finite center, $G=KAN$ an Iwasawa decomposition, and $\overline N$ the opposite nilpotent subgroup. Write $H(g)\in\mathfrak a$ for the Iwasawa projection and let $\rho$ be half the sum of the positive restricted roots, counted with multiplicity. The Harish–Chandra $c$ -function is

c(\lambda)=\int_{\overline N} e^{-(i\lambda+\rho)(H(\overline n))}\,d\overline n

where the integral converges, then extended meromorphically to $\mathfrak a_{\mathbb C}^*$ . The Haar measure is normalized so that $c(-i\rho)=1$ .

Asymptotic meaning

For regular $\lambda$ , the elementary spherical function $\varphi_\lambda$ has, deep in the positive Weyl chamber, an expansion whose leading exponential terms are indexed by the restricted Weyl group. The coefficient of the term with spectral parameter $w\lambda$ is $c(w\lambda)$ . This asymptotic characterization and the integral definition agree with the stated normalizations Helgason, Chapter IV, §§6–7.

Product formula and Plancherel density

The Gindikin–Karpelevich formula factors $c(\lambda)$ into rank-one factors, one for each indivisible positive restricted root; those factors are explicit ratios of gamma functions involving the root multiplicities. On the real spectral axis, the spherical Plancherel measure is, up to the compatible normalizing constant,

|c(\lambda)|^{-2}\,d\lambda.

Thus zeros and poles of the meromorphic continuation encode both intertwining-operator phenomena and the analytic weight in spherical Fourier inversion Helgason, Chapter IV, §7.

Example and scope

In real rank one, the product formula has a single rank-one factor, so $c(\lambda)$ is a quotient of gamma functions. For higher rank, the factors assemble according to the restricted root system.

The displayed integral fixes one common sign convention for $\varphi_\lambda$ . Replacing $\lambda$ by $-\lambda$ , changing the positive chamber, or rescaling Haar measure changes the printed formula and normalization but not the resulting Plancherel theory. This scalar spherical $c$ -function should be distinguished from generalized matrix-valued $c$ -functions for nonspherical $K$ -types.

References

Sigurdur Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, American Mathematical Society, 2000. AMS record. Relevant: Chapter IV, §§6–7 on spherical-function asymptotics, the $c$ -function, and the Plancherel formula.
Harish-Chandra, “Spherical Functions on a Semisimple Lie Group I,” American Journal of Mathematics 80 (1958), 241–310, and “II,” 553–613. Part I JSTOR record. Relevant: the asymptotic expansion and Plancherel density for elementary spherical functions.