For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: there exists an operation $\ast: X\times X\to X$ satisfying $$x\ast x =x, \quad x\ast(x\ast y) = y,\quad x\ast(y\ast z) = (x\ast y)\ast (x\ast z) $$ for any $x,y,z\in X$. This operation is given explicitly on cosets by $$ (gG^\theta ,hG^\theta) \mapsto g\theta(g^{-1}h)G^\theta,$$ and when $X$ is a Riemannian symmetric space this is precisely ''reflecting $hG^\theta$ through $gG^\theta$.''

In fact, one gets a quandle structure (no longer involutive) for any homogeneous space $G/G^\theta$ where $\theta$ is a finite order automorphism of $G$. While I've cared about symmetric spaces and these more general homogeneous spaces for quite some time now, I've only just become aware of a name for this structure.

Is this structure important for the study of symmetric spaces? For example, can I see it playing a role in the harmonic analysis of $X$? What about the study of invariant differential operators or other representation-theoretic considerations associated to $X$?

I've certainly used the embedding $G/G^\theta\hookrightarrow G$ that is implicitly involved in defining $\ast$, but had never come across the (left-)distributive property in the definition, for example.

Looking around, I see that this structure can more-or-less be used to define symmetric spaces among smooth manifolds. This is pretty, but I'm not sure if it's helpful. Also, the quandle structure (without the involutive assumption) also exists for more general $G/G^\theta$. Does this imply interesting properties for these spaces among all $G$-spaces?

For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: there exists an operation $\ast: X\times X\to X$ satisfying $$x\ast x =x, \quad x\ast(x\ast y) = y,\quad x\ast(y\ast z) = (x\ast y)\ast (x\ast z) $$ for any $x,y,z\in X$. This operation is given explicitly on cosets by $$ (gG^\theta ,hG^\theta) \mapsto g\theta(g^{-1}h)G^\theta,$$ and when $X$ is a Riemannian symmetric space this is precisely “reflecting $hG^\theta$ through $gG^\theta$.”

In fact, one gets a quandle structure (no longer involutive) for any homogeneous space $G/G^\theta$ where $\theta$ is a finite order automorphism of $G$. While I've cared about symmetric spaces and these more general homogeneous spaces for quite some time now, I've only just become aware of a name for this structure.

Is this structure important for the study of symmetric spaces? For example, can I see it playing a role in the harmonic analysis of $X$? What about the study of invariant differential operators or other representation-theoretic considerations associated to $X$?

I've certainly used the embedding $G/G^\theta\hookrightarrow G$ that is implicitly involved in defining $\ast$, but had never come across the (left-)distributive property in the definition, for example.

Looking around, I see that this structure can more-or-less be used to define symmetric spaces among smooth manifolds. This is pretty, but I'm not sure if it's helpful. Also, the quandle structure (without the involutive assumption) also exists for more general $G/G^\theta$. Does this imply interesting properties for these spaces among all $G$-spaces?

For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: there exists an operation $\ast: X\times X\to X$ satisfying $$x\ast x =x, \quad x\ast(x\ast y) = y,\quad x\ast(y\ast z) = (x\ast y)\ast (x\ast z) $$ for any $x,y,z\in X$. This operation is given explicitly on cosets by $$ (gG^\theta ,hG^\theta) \mapsto g\theta(g^{-1}h)G^\theta,$$ and when $X$ is a Riemannian symmetric space this is precisely ''reflecting $hG^\theta$ through $gG^\theta$.''

In fact, one gets a quandle structure (no longer involutive) for any homogeneous space $G/G^\theta$ where $\theta$ is a finite order automorphism of $G$. While I've cared about symmetric spaces and these more general homogeneous spaces for quite some time now, I've only just become aware of a name for this structure.

Is this structure important for the study of symmetric spaces? For example, can I see this structure playing a role in the harmonic analysis of $X$? Differential operators of other representation theory associated to $X$?

Looking around, I see that this structure can more-or-less be used to define symmetric spaces among smooth manifolds. This is pretty, but I'm not sure if it's helpful. Also, the quandle structure (without the involutive assumption) also exists for more general $G/G^\theta$. Does this structure imply interesting properties for these spaces among all $G$-spaces?

For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: there exists an operation $\ast: X\times X\to X$ satisfying $$x\ast x =x, \quad x\ast(x\ast y) = y,\quad x\ast(y\ast z) = (x\ast y)\ast (x\ast z) $$ for any $x,y,z\in X$. This operation is given explicitly on cosets by $$ (gG^\theta ,hG^\theta) \mapsto g\theta(g^{-1}h)G^\theta,$$ and when $X$ is a Riemannian symmetric space this is precisely ''reflecting $hG^\theta$ through $gG^\theta$.''

In fact, one gets a quandle structure (no longer involutive) for any homogeneous space $G/G^\theta$ where $\theta$ is a finite order automorphism of $G$. While I've cared about symmetric spaces and these more general homogeneous spaces for quite some time now, I've only just become aware of a name for this structure.

Is this structure important for the study of symmetric spaces? For example, can I see it playing a role in the harmonic analysis of $X$? What about the study of invariant differential operators or other representation-theoretic considerations associated to $X$?

I've certainly used the embedding $G/G^\theta\hookrightarrow G$ that is implicitly involved in defining $\ast$, but had never come across the (left-)distributive property in the definition, for example.

Looking around, I see that this structure can more-or-less be used to define symmetric spaces among smooth manifolds. This is pretty, but I'm not sure if it's helpful. Also, the quandle structure (without the involutive assumption) also exists for more general $G/G^\theta$. Does this imply interesting properties for these spaces among all $G$-spaces?