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LSpice
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See the paper Self-maps of homogeneous spaces by Srinivas and Paranjape, Inventiones mathematicae  , 19981989, vol. 2. That paper addresses precisely this question.

See the paper by Srinivas and Paranjape, Inventiones mathematicae  , 1998, vol 2. That paper addresses precisely this question.

See the paper Self-maps of homogeneous spaces by Srinivas and Paranjape, Inventiones mathematicae, 1989, vol. 2. That paper addresses precisely this question.

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Venkataramana
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See the paper by Srinivas and Paranjape, Inventiones mathematicae , 1998, vol 2. That paper addresses precisely this question.