Harish-Chandra's formula for spherical functions

Remarks

The function $g \mapsto e^{(i \lambda - \rho)(H(g))}$ is left $K$-invariant and transforms by a character under right translation by $A N$. Averaging its right translates over $K$ gives a bi-$K$-invariant function, and the functional equation $\int_K \varphi(x k y) \, dk = \varphi(x) \varphi(y)$ is checked with the Iwasawa decomposition. For the converse one shows that spherical functions are joint eigenfunctions of the invariant differential operators on $G / K$, and Harish-Chandra's homomorphism identifies the possible eigenvalues with $W$-orbits in $\mathfrak a^*_\C$.

Examples: $\varphi_{-i\rho} = 1$, since then $i \lambda - \rho = 0$. The function $\Xi = \varphi_0$ is Harish-Chandra's $\Xi$-function, used to define Harish-Chandra's Schwartz space. The bounded spherical functions are the $\varphi_\lambda$ with $\operatorname{Im} \lambda$ in the convex hull of $W \rho$, and the positive definite ones include all $\varphi_\lambda$ with $\lambda \in \mathfrak a^*$ real.

Going deeper

For a student

The formula is the starting point of the spherical transform $\hat f(\lambda) = \int_G f(g) \varphi_{-\lambda}(g) \, dg$ on $G / K$. The asymptotic behaviour of $\varphi_\lambda$ at infinity is governed by the Harish-Chandra c-function. For $G = \mathrm{SL}(2, \R)$ the functions $\varphi_\lambda$ are Legendre functions of $\cosh$ of the hyperbolic distance.