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**Automorphic pseudodifferential analysis and higher level Weyl calculi.**
*(English)*
Zbl 1018.11018

Progress in Mathematics (Boston, Mass.). 209. Basel: Birkhäuser. vii, 246 p. (2003).

The group \(\text{SL} (2,\mathbb R)\) acts on the Poincaré upper half plane \(\mathcal H\) by linear fractional transformations. Given a discrete subgroup \(\Gamma\) of \(\text{SL} (2,\mathbb R)\), automorphic functions are \(\Gamma\)-invariant functions on \(\mathcal H\). A nonholomorphic modular form is an automorphic function on \(\mathcal H\) that is a generalized eigenfunction of the Laplace-Beltrami operator \(\Delta\). Automorphic functions can also be regarded as objects defined on \(\mathbb R^2\), rather than on \(\mathcal H\), which are invariant under the linear action of \(\Gamma\). Such an interpretation requires the use of distributions instead of functions, so that automorphic functions on \(\mathcal H\) correspond to automorphic distributions on \(\mathbb R^2\). This monograph studies a class of pseudodifferential operators in one variable whose Weyl symbols are automorphic distributions on \(\mathbb R^2\), that is, distributions invariant under the linear action of \(\Gamma\). By using the spectral theory of the Euler operator on \(L^2 (\Gamma \backslash \mathbb R^2)\), it can be shown that automorphic distributions are linear superpositions of the Eisenstein distributions and the cusp distributions. The main goal of this book is to construct a multiplication table for the associated operators.

The space of automorphic functions on the upper half plane can naturally be turned into a noncommutative algebra by adopting the quantum approach. This is achieved by associating operators to automorphic functions and considering the composition of those operators. The resulting composition formulas reveal various important objects in the theory of nonholomorphic modular forms such as zeta functions, Hecke operators, \(L\)-functions, and convolution of \(L\)-functions. One advantage of using distributions is that the algebraic structure on the space of automorphic distributions can be defined solely in terms of Weyl calculus of operators. The main result in this book expresses the composition of any two Eisenstein distributions in terms of the image of a canonical Bézout distribution, which is related to the Poincaré-Selberg series, under a quite interesting, but simple, operator.

The space of automorphic functions on the upper half plane can naturally be turned into a noncommutative algebra by adopting the quantum approach. This is achieved by associating operators to automorphic functions and considering the composition of those operators. The resulting composition formulas reveal various important objects in the theory of nonholomorphic modular forms such as zeta functions, Hecke operators, \(L\)-functions, and convolution of \(L\)-functions. One advantage of using distributions is that the algebraic structure on the space of automorphic distributions can be defined solely in terms of Weyl calculus of operators. The main result in this book expresses the composition of any two Eisenstein distributions in terms of the image of a canonical Bézout distribution, which is related to the Poincaré-Selberg series, under a quite interesting, but simple, operator.

Reviewer: Min Ho Lee (Cedar Falls)