# Relation between Lie group characters and spherical functions on symmetric spaces

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)

• 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.

• I believe Takeuchi (1994, Thm 8.3) expresses $\varphi\circ\mathbf e$ as a linear combination of characters which involves more than just $\chi\circ\mathbf e$ unless we are in the group case (Remark (2) following). – Francois Ziegler Jul 18 '18 at 0:07
• Does the work on "Non abelian Fourier analysis" not cover this? – Suvrit Jul 18 '18 at 10:58
• @Suvrit Non abelian harmonic (Fourier) analysis is indeed the general framework (I should have mentioned the Peter-Weyl theorem along the way), but my question is more specific: in the case of $G/K$ a symmetric space, much more can be said than for any quotient $G/H$, e.g., the convolution algebra of $K$-bi-invariant functions is commutative and has this orthogonal basis of "zonal spherical functions" which looks very much like the case of class functions (aka central functions, conjugation-invariant) functions on $G$ itself. (contd.) – Gro-Tsen Jul 18 '18 at 12:11
• (contd.) For class functions on $G$ we have this very nice expression of characters given by Weyl's character formula and the formulæ of Kostant and Freudenthal, which gives a very good grip on the conjugation invariant part of $L^2(G)$. I am trying to understand the analogous situation for $K$-bi-invariant functions on $G$ (the "zonal" part of $L^2(G/K)$). So while non abelian harmonic analysis describes all of $L^2(G)$, it is much less precise (and less computational!) than what I am hoping for. – Gro-Tsen Jul 18 '18 at 12:15
• Thanks for the additional context -- the only other comment I have is whether there's anything of use for you in the book Analysis on symmetric cones – Suvrit Jul 18 '18 at 12:59