By W. W. Sawyer
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Extra info for A Concrete Approach to Abstract Algebra
1 Remark. L* jd*.
Thus we recover the following result, which may also be seen by inspection of the usual formulas for the Schrodinger representation as a representation on £2(1R). 10 Remark. The Schrodinger-Weil representation ir~w is unitarizable. e. 2. Models for infinitesimal representations and unitarizability 47 For 7l-ta we come up with II WI+211 2 = l+2( -2- ko l+2 ) 2 + -2-1 IlwI11 and for ir ko with IIW-C/+2) II 2 = l+2( -2- ko l+2 + -2- ) - 1 Ilw-III 2 showing that in both cases we have to require ko gives the following result.
Summarizing we obtain the following result. 1 Theorem. The above construction gives a 1-1 correspondence between the irreducible unitary projective representations ofSL(2, R) with multiplier>.. and the irreducible unitary representations of G J (R) with non-trivial central character 'ljJm. The corresponding non-archimedean result is as follows . 2 Theorem. Let F be a p-adic field. There is a 1-1 correspondence between the irreducible smooth projective representations ofSL(2, F) with multiplier>..
A Concrete Approach to Abstract Algebra by W. W. Sawyer