Harish-Chandra Xi-function - Imported Knowlpedia Content

Definition

Let $G$ be a connected real semisimple Lie group with finite center, let $K$ be a maximal compact subgroup, and choose a minimal parabolic subgroup $P=MAN$ . In the compact realization of the unitary spherical principal-series representation induced from the trivial representation of $P$ , let $v_0$ be the normalized $K$ -fixed vector. The Harish-Chandra Xi-function is the matrix coefficient

\Xi(g)=\langle \pi_0(g)v_0,v_0\rangle,\qquad g\in G.

Equivalently, for the Iwasawa projection $H:G\to\mathfrak a$ and the half-sum $\rho$ of positive restricted roots,

\Xi(g)=\int_K e^{-\rho(H(gk))}\,dk,

with Haar probability measure on $K$ .

Basic properties

The function $\Xi$ is continuous, real-valued, positive, positive definite, and bi- $K$ -invariant, with $\Xi(e)=1$ and $\Xi(g^{-1})=\Xi(g)$ . It is the elementary spherical function with spectral parameter zero. Its decay along a positive Weyl chamber is exponential up to a polynomial factor; that estimate supplies the natural weight in the Harish-Chandra Schwartz space Knapp, Chapter VII.

Example

For a compact semisimple group one may take $K=G$ . The relevant induced representation has only the constant normalized $K$ -fixed vector, and $\Xi(g)=1$ for every $g$ . For a noncompact semisimple group, $\Xi$ instead decays at infinity and records the large-scale behavior of spherical matrix coefficients.

Conventions and scope

The letter $\Xi$ is also used for variants on reductive groups and for estimates comparable to the normalized spherical coefficient. The definition above fixes the standard connected semisimple, finite-center setting. Signs in the integral formula vary with the convention for the Iwasawa projection and positive roots; the matrix-coefficient definition is invariant under that bookkeeping.

References

Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VII on principal series and elementary spherical functions.