Sobolev Norms of Automorphic Functionals

It is well known that Frobenius reciprocity is one of the central tools in the representation theory. In this paper, we discuss Frobenius reciprocity in the theory of automorphic functions. This Frobenius reciprocity was discovered by Gel’fand, Fomin, and PiatetskiShapiro in the 1960s as the basis of their interpretation of the classical theory of automorphic functions in terms of the representation theory (eventually, of adelic groups, see [7, 8, 9]). Later, Ol’shanski gave a more transparent proof of it (see [14]). However, in the subsequent rapid development of the theory of automorphic functions, Frobenius reciprocity was barely noticeable. We believe that this is due to the incompleteness of the above-mentioned results. In this paper, we prove a general theorem (see Theorem 1.1), which we view as a quantitative version of Frobenius reciprocity. We then illustrate it by looking into the example of SL(2,R). We think that these methods will play a more prominent role in the theory of automorphic functions.