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Peter Michor
Peter Michor
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For reductive groups and smooth actions the answer seems to be positive, if I remember correctly; this follows from Luna's slice theorem:

  • D. Luna, Slices ́etales, Sur les groupes alg ́ebriques, Soc. Math. France, Paris, 1973, pp. 81– 105. Bull. Soc. Math. France, Paris, M ́emoire 33.

There seems to be even the statement: Each orbit contains exactly one closed orbit in its closure.
Added later: At least for a representation. See Claudio Gorodski's more detailed answer.

By the way, an orbit a smooth Lie group action on a smooth manifold is always an initial manifold, see here, or also theorem 6.4 in this book.

For reductive groups and smooth actions the answer seems to be positive, if I remember correctly; this follows from Luna's slice theorem:

  • D. Luna, Slices ́etales, Sur les groupes alg ́ebriques, Soc. Math. France, Paris, 1973, pp. 81– 105. Bull. Soc. Math. France, Paris, M ́emoire 33.

There seems to be even the statement: Each orbit contains exactly one closed orbit in its closure.

By the way, an orbit a smooth Lie group action on a smooth manifold is always an initial manifold, see here, or also theorem 6.4 in this book.

For reductive groups and smooth actions the answer seems to be positive, if I remember correctly; this follows from Luna's slice theorem:

  • D. Luna, Slices ́etales, Sur les groupes alg ́ebriques, Soc. Math. France, Paris, 1973, pp. 81– 105. Bull. Soc. Math. France, Paris, M ́emoire 33.

There seems to be even the statement: Each orbit contains exactly one closed orbit in its closure.
Added later: At least for a representation. See Claudio Gorodski's more detailed answer.

By the way, an orbit a smooth Lie group action on a smooth manifold is always an initial manifold, see here, or also theorem 6.4 in this book.

added 270 characters in body
Source Link
Peter Michor
Peter Michor
  • 25.3k
  • 2
  • 64
  • 112

For reductive groups and smooth actions the answer seems to be positive, if I remember correctly; this follows from Luna's slice theorem:

  • D. Luna, Slices ́etales, Sur les groupes alg ́ebriques, Soc. Math. France, Paris, 1973, pp. 81– 105. Bull. Soc. Math. France, Paris, M ́emoire 33.

There seems to be even the statement: Each orbit contains exactly one closed orbit in its closure.

By the way, an orbit a smooth Lie group action on a smooth manifold is always an initial manifold, see here, or also theorem 6.4 in this book.

For reductive groups and smooth actions the answer seems to be positive, if I remember correctly; this follows from Luna's slice theorem:

  • D. Luna, Slices ́etales, Sur les groupes alg ́ebriques, Soc. Math. France, Paris, 1973, pp. 81– 105. Bull. Soc. Math. France, Paris, M ́emoire 33.

There seems to be even the statement: Each orbit contains exactly one closed orbit in its closure.

For reductive groups and smooth actions the answer seems to be positive, if I remember correctly; this follows from Luna's slice theorem:

  • D. Luna, Slices ́etales, Sur les groupes alg ́ebriques, Soc. Math. France, Paris, 1973, pp. 81– 105. Bull. Soc. Math. France, Paris, M ́emoire 33.

There seems to be even the statement: Each orbit contains exactly one closed orbit in its closure.

By the way, an orbit a smooth Lie group action on a smooth manifold is always an initial manifold, see here, or also theorem 6.4 in this book.

Source Link
Peter Michor
Peter Michor
  • 25.3k
  • 2
  • 64
  • 112

For reductive groups and smooth actions the answer seems to be positive, if I remember correctly; this follows from Luna's slice theorem:

  • D. Luna, Slices ́etales, Sur les groupes alg ́ebriques, Soc. Math. France, Paris, 1973, pp. 81– 105. Bull. Soc. Math. France, Paris, M ́emoire 33.

There seems to be even the statement: Each orbit contains exactly one closed orbit in its closure.