people define haar measure to be left invariant,Weil define module of a automorphism to be the quoient of aX and X,where aX denote X changed under operation “a",if it is left invariant,should module always be trival?
A naive question on Haar measure and the module of automorphism
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fourier-analysis
1 Answers
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Haar measure is invariant on one side, so translate on the other side for this. It is a matter of convention which side to use for which. In the commutative case (or discrete case, or compact case) Haar measure is invariant on both sides, so then it is trivial.
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1would you plz give me some introduction briefly on the use of haar integral in number theory,especially how fourier analysis work in number theory,elementary level ,thanks! – 2011-06-26
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0@ Yoshinobu Osawa. Ramakrishnan has a book on just this. http://www.amazon.com/Fourier-Analysis-Number-Graduate-Mathematics/dp/0387984364 – 2011-06-26
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1in what cases module is trival?Weil in his book say that either module <1 or is the power of an elememt in K – 2011-06-26
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0sure the number field or local fields are commutative – 2011-06-26
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0@YoshinobuOsawa: Surely this is exactly my question!! If *any* abelian group is unimodular, then how can one define he module function on number fields? Since you asked this question it has been 4,5 months; may I respectfully ask if you already found the answer to this? And are you already familiar with the Haar measures? Thanks and regards here. – 2011-11-18
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0Indeed, A.Weil only considers the module of locally compact commutative groups. It seems quite contradictory to me... Any help is appreciated, thanks. – 2011-11-18