By Shai M. J. Haran
In this quantity the writer extra develops his philosophy of quantum interpolation among the genuine numbers and the p-adic numbers. The p-adic numbers include the p-adic integers Zp that are the inverse restrict of the finite earrings Z/pn. this provides upward push to a tree, and likelihood measures w on Zp correspond to Markov chains in this tree. From the tree constitution one obtains particular foundation for the Hilbert area L2(Zp,w). the genuine analogue of the p-adic integers is the period [-1,1], and a likelihood degree w on it offers upward thrust to a distinct foundation for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For certain (gamma and beta) measures there's a "quantum" or "q-analogue" Markov chain, and a different foundation, that inside of sure limits yield the genuine and the p-adic theories. this concept will be generalized variously. In illustration idea, it's the quantum common linear workforce GLn(q)that interpolates among the p-adic team GLn(Zp), and among its actual (and advanced) analogue -the orthogonal On (and unitary Un )groups. there's a comparable quantum interpolation among the true and p-adic Fourier rework and among the genuine and p-adic (local unramified a part of) Tate thesis, and Weil particular sums.
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Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations (Lecture Notes in Mathematics) by Shai M. J. Haran