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Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions concerning properties of orbits.

  1. If there are only finitely many $G$ orbits, then in p-adic case, there exists open orbits. Now we may ask when there is exactly one open orbit, and what's the real case? What about infinitely many orbits?

  2. what's the relation between the analytic topology and zariski topology on orbits? For example, if an orbit is closed in analytic topology, then it is also closed in zariski topology(over algebraic closure).

  3. How to parametrize those orbits?

  4. How to characterize open and closed orbits?

These obviously are difficult questions in general, and feel free to answer in any special case, like actions on flag varieties.

edit: As the above questions are too broad to answer. So let's try to focus on two special cases here most interested to me: $G$ a reductive group, $H$ a subgroup of $G$ which is the fixed points of some involution of $G$.

Case 1: $H\times H$ acts on $G$, one acts on left, the other on right.

Case 2: If $P$ is a parabolic subgroup, $H$ acts on $G/P$.

And we mainly concern question 1 and 4.

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Your question 3 is essentially asking "in what sense is there a quotient?" Whole books have been written on the subject... – Mariano Suárez-Alvarez Jun 3 '10 at 2:56
Way way too broad, methinks. – Marty Jun 3 '10 at 3:01
Please change your question into one that has reasonable scope. We aren't here to write books for you. The other parts of this question can become separate questions. – S. Carnahan Jun 3 '10 at 3:32

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