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.



Gelfand pair, weakly symmetric pair, and spherical pair
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](https://doi.org/10.1112/S0024609397003925)), they coincide for reductive groups. This is Theorem 7.3 of [Akhiezer and Vinberg - Weakly symmetric spaces and spherical varieties](https://doi.org/10.1007/BF01236659).


