Arithmetical Investigations: Representation Theory, by Shai M. J. Haran

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 comprise 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 unique foundation for the Hilbert area L2(Zp,w). the true analogue of the p-adic integers is the period [-1,1], and a likelihood degree w on it supplies upward thrust to a unique foundation for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For designated (gamma and beta) measures there's a "quantum" or "q-analogue" Markov chain, and a different foundation, that inside of definite limits yield the true and the p-adic theories. this concept could be generalized variously. In illustration idea, it's the quantum common linear staff GLn(q)that interpolates among the p-adic workforce GLn(Zp), and among its genuine (and complicated) analogue -the orthogonal On (and unitary Un )groups. there's a comparable quantum interpolation among the genuine and p-adic Fourier remodel and among the true and p-adic (local unramified a part of) Tate thesis, and Weil particular sums.

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