**Setup:** Let $G/K$ be an irreducible compact Riemannian symmetric space, where $G$ is a simply connected compact real Lie group, and $K$ is the maximal compact connected subgroup of some noncompact real form of $G$.

If $\chi$ is the character of some finite-dimensional irreducible representation $V$ of $G$, which is spherical (meaning that the set $V^K$ of $K$-fixed vectors in $V$ is nonzero), I understand that $V^K$ is one-dimensional, and we can define the unique (up to a constant) zonal spherical function $\varphi$ associated to $\chi$ by: $\varphi(g) := \int_{k\in K} \chi(gk)$ [Helgason 1984, theorem IV.4.2]. This is left and right $K$-invariant, in other words, it can be seen as a function on the double coset space $K\backslash G/K$.

Now $K\backslash G/K$ has a nice description: if $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ is the Cartan decomposition of the Lie algebra of $G$ into that of $K$ and complement, and $\mathfrak{a}$ a maximal abelian subspace of $\mathfrak{p}$, then $H\mapsto\mathbf{e}(H) := \exp(2\pi H)$ defines a bijection from a Weyl alcove in $\mathfrak{a}$ to $K\backslash G/K$ [Helgason 1978, theorem VII.8.6]. This suggests that we look at $\varphi\circ\mathbf{e}$ on $\mathfrak{a}$, which is invariant under the Weyl group of the restricted root system.

My questions are about $\varphi\circ\mathbf{e}$, its relation with $\chi\circ\mathbf{e}$ on $\mathfrak{a}$, and their Fourier transforms: essentially, I think I understand the case of a compact Lie group and I am trying to get a grasp on the analogous situation for a symmetric space:

**Questions:**

What can we say about the relation between the functions $\varphi\circ\mathbf{e}$ and $\chi\circ\mathbf{e}$ on $\mathfrak{a}$ (I used to think that they coincide, but evidently it's much more complicated)?

More precisely, if we introduce the Fourier transform, a.k.a., the weight decomposition, of $\varphi\circ\mathbf{e}$ (i.e., writing $\varphi(\mathbf{e}(H)) = \sum_\Lambda \widehat\varphi(\Lambda)\,\mathbf{e}(\langle\Lambda,H\rangle)$ for $H\in\mathfrak{a}$ and $\Lambda$ ranging over the lattice dual to $\{H\in\mathfrak{a} : \mathbf{e}(H)\in K\}$, I am talking about the values $\widehat\varphi(\Lambda)$). The question of understanding $\widehat\chi$ is well answered by highest weight theory and the various formulæ of Weyl, Kostant and Freudenthal ($\widehat\chi(\Lambda)$ is the multiplicity of the weight $\Lambda$ in $V$). What analogous theory or formulæ exist for $\widehat\varphi$, if any?

Specifically, does the support of $\widehat\varphi$ equal that of $\widehat\chi$ restricted to (the dual space to) $\mathfrak{a}$? Are the values of $\widehat\varphi$ even integers (up to a common constant)? Do they relate to the multiplicities $\widehat\chi(\Lambda)$ of $\chi$? And most importantly: how can $\widehat\varphi$ be computed algorithmically?

These questions were asked in a fairly naïve way, I am starting to realize that they are not so naïve, but even though I have found a few references, I still struggle to form a picture of what is known about $\widehat\varphi$, and whether it can reasonably be computed algorithmically.

References:

Helgason 1978:

*Differential Geometry, Lie Groups and Symmetric Spaces*Helgason 1984:

*Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators, and Spherical Functions)*

Additional references found after asking the question:

Takeuchi 1994:

*Modern Spherical Functions*(monograph translated from the 1975 Japanese original) [pointed out by Francois Ziegler in the comments]Vretare 1976: "Elementary Spherical Functions on Symmetric Spaces" (

*Math. Scand.***39**343–358) [seems to answer a few questions, such as that of the set of appearing weights]Clerc 1988: "Fonctions Sphériques des Espaces Symétriques Compacts" (

*Trans. Amer. Math. Soc***306**421–431)

I imagine that since none of these references give any sort of analogue of Weyl's character formula, this means that none is known (or perhaps even that none is *possible* for some reason?). I would still like to know more.

Analysis on symmetric cones$\endgroup$ – Suvrit Jul 18 '18 at 12:59