THE PLANCHEREL FORMULA, THE PLANCHEREL THEOREM, AND THE FOURIER TRANSFORM OF ORBITAL INTEGRALS REBECCA A. HERB (UNIVERSITY OF MARYLAND) AND PAUL J. SALLY, JR. (UNIVERSITY OF CHICAGO) Abstract. We discuss various forms of the Plancherel Formula and the Plancherel Theorem on reductive groups over local fields. Dedicated to Gregg Zuckerman on his 60th birthday 1. Introduction The classical Plancherel Theorem proved in 1910 by Michel Plancherel can be stated as follows: Theorem 1.1. Let f L2(R) and define φn : R C for n N by φn(y) = 1 n −n f(x)eiyxdx. The sequence φn is Cauchy in L2(R) and we write φ = limn→∞ φn (in L2). Define ψn : R C for n N by ψn(x) = 1 n −n φ(y)e−iyxdy. The sequence ψn is Cauchy in L2(R) and we write ψ = limn→∞ ψn (in L2). Then, ψ = f almost everywhere, and R |f(x)|2 dx = R |φ(y)|2 dy. This theorem is true in various forms for any locally compact abelian group. It is often proved by starting with f L1(R)∩L2(R), but it is really a theorem about square integrable functions. There is also a “smooth” version of Fourier analysis on R, motivated by the work of Laurent Schwartz, that leads to the Plancherel Theorem. Definition 1.2 (The Schwartz Space). The Schwartz space, S(R), is the collection of complex-valued functions f on R satisfying: (1) f C∞(R). (2) f and all its derivatives vanish at infinity faster than any polynomial. That is, lim|x|→∞ |x|kf (m) (x) = 0 for all k, m N. Fact 1.3. The Schwartz space has the following properties: Date: June 21, 2011. 1 Contemporary Mathematics Volume 557, 2011 c 2011 American Mathematical Society 3
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