## Pseudodifferential Operators with Automorphic Symbols by André Unterberger

By André Unterberger

The major result of this e-book mix pseudo differential research with modular shape thought. The tools depend for the main half on particular spectral idea and the prolonged use of particular capabilities. the start line is a concept of modular distribution within the aircraft, on the way to be new to so much readers and relates lower than the Radon transformation to the classical one in every of modular type of the non-holomorphic sort. Modular different types of the holomorphic style are addressed too in a extra concise method, inside a basic scheme facing quantization concept and effortless, yet novel, representation-theoretic concepts.

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Basic modular distributions precisely kills the poles of Bε ( ρ−iλ 2 ) under consideration. On the other hand, the (ε) poles of hom2−ρ,−iλ to be taken care of are at ρ = 2 + 2ε − iλ, 6 + 2ε − iλ, . . : they are killed with the help of the other factor Γ( s+ε 2 − the entire function L∗ (s, N) in place of L(s, N). iλ −1 4 ) present if using What remains to be done is to show that N changes, for every integer N ≥ 1, to a multiple, under the operator TNdist . 23) so that TNdist N , h = 1 4 χ m, n=0 √ ∞ d −1 m n |t| −1−iλ 2iπ bmn d e −∞ F1−1 h N m ant , √ d t N dt.

8) g∈Γ ρ−1 where sρ is the distribution −(Im z) 2 δ0,i∞ supported in the hyperbolic line (0, i∞) from 0 to i∞: we restrict it, in what follows, to the space of functions ∂ in Π invariant under the map z → −z −1 , the ∂x -derivative of which is rapidly decreasing at infinity on (0, i∞). 9) to prove this latter formula, observe that, if M satisfies the conditions just listed, one has M(x + iy) = M −x + iy x2 + y 2 , so that ∂M 1 ∂M (iy) = − 2 ∂x y ∂x i y . 10) Let Σ be the union of the (locally finite) collection of g-transforms, with g ∈ Γ, of the line (0, i∞).

3). 2 Hecke distributions We introduce here Hecke operators acting on automorphic distributions: these will be related later to the more traditional notion of Hecke operator acting on automorphic functions. 1. 1) ad=N, d>0 b mod d and dist T−1 S , h = S , (x, ξ) → h(−x, ξ) . 2) Just as in the automorphic function environment, the linear span of the Hecke operators TNdist with N ≥ 1 makes up an algebra, which is generated, as such, by the operators Tpdist with p prime. Automorphic distributions which are dist left invariant, or change to their negatives, under T−1 , are said to be of even or 20 Chapter 1.