Gelfand pair, weakly symmetric pair, and spherical pair

Question

I am a bit confused with the relations among Gelfand pairs, weakly symmetric pairs, and spherical pairs defined in the book "Harmonic analysis on commutative spaces" written by professor Joseph A. Wolf.

For convenience, let me recall the definitions in this book, and just consider connected groups $G$.

Definition 1 Let $G$ be a connected Lie group, and $K$ a compact subgroup. If the algebra $L^1(K\backslash G/K)$ is commutative under convolution, then $(G,K)$ is called a Gelfand pair.

Definition 2 Let $G$ be a connected Lie group, and $K$ a compact subgroup. If there exists an automorphism $\sigma$ of $G$ such that $\sigma(g)\in Kg^{-1}K$ for all $g\in G$, then $(G,K)$ is called a weakly symmetric pair.

Definition 3 Let $G$ be a complex reductive linear algebraic group, and $H$ a reductive subgroup. Denote by $\mathfrak{g}$ and $\mathfrak{h}$ the Lie algebras of $G$ and $H$ respectively. If there exists a borel subalgebra $\mathfrak{b}$ in $\mathfrak{g}$ such that $\mathfrak{b}+\mathfrak{h}=\mathfrak{g}$, then $(G,H)$ is called a spherical pair.

Now on page 281 of Wolf's book, there are two results: Theorem 12.6.10 and Theorem 12.6.11.

Let $G_\mathbb{C}$ be a connected complex reductive algebraic group, and $H_\mathbb{C}$ a reductive algebraic subgroup. Suppose that $G$ is a real form of $G_\mathbb{C}$ such that $H:=G\cap H_\mathbb{C}$ is a compact real form of $H_\mathbb{C}$. Then $(G_\mathbb{C},H_\mathbb{C})$ is a spherical pair if and only if $(G,H)$ is a weakly symmetric pair (by Theorem 12.6.10) if and only if $(G,H)$ is a Gelfand pair (by Theorem 12.6.11).

Thus, suppose that we have a real reductive group $G$ with its compact subgroup $K$, then $(G,K)$ is a Gelfand pair if and only if $(G,K)$ is a weakly symmetric pair. But I do not think that the two definitions are equivalent. As far as I know, weakly symmetric pairs are Gelfand pairs, but there exist Gelfand pairs which are not weakly symmetric pairs.

Hence, I think that I am probably misunderstanding the definitions, the theorems, or the relations among three pairs. I shall be grateful if experts may give any comments.

Answer 1

I like this question! I originally thought it was much less subtle, and so posted some ill informed guesses in the comments.

Although the notions of "weakly symmetric" and "Gelfand pair" differ in general (Lauret - Commutative spaces which are not weakly symmetric), they coincide for reductive groups. This is Theorem 7.3 of Akhiezer and Vinberg - Weakly symmetric spaces and spherical varieties.