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A symmetric space is a connected Riemannian manifold in which at every point there exists a global self-isometry whose differential at the given point is minus identity.

2 votes
1 answer
174 views

Chains in $K\backslash G/B$ lying over a closed $K$-orbit

Let $G$ be a complex connected reductive Lie group, $\theta$ an involution, and $K = G^\theta$ the fixed-point subgroup. Then $K$ has finitely many orbits on $G/B$, one of which is open and (quite of …
Allen Knutson's user avatar
7 votes
0 answers
149 views

Eigenspaces and covering relations of twisted involutions

Let $\theta:G\to G$ be an involution of a complex connected reductive Lie group, preserving a maximal torus $T$ (which, for me, lies inside a $\theta$-invariant Borel $B$). Let $K = G^\theta$ be the f …
Allen Knutson's user avatar
6 votes
2 answers
314 views

Covering relations in $K\backslash G/B$

Let $G$ be a simply connected complex Lie group, $\theta$ an involution, and $K = G^\theta$ the fixed point subgroup. Pick a $\theta$-invariant Borel subgroup $B$. Then there is a natural map $K\backs …
Allen Knutson's user avatar