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

By Shai M. J. Haran

In this quantity the writer additional 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 jewelry Z/pn. this offers upward push to a tree, and chance measures w on Zp correspond to Markov chains in this tree. From the tree constitution one obtains particular foundation for the Hilbert house L2(Zp,w). the genuine analogue of the p-adic integers is the period [-1,1], and a chance degree w on it provides upward push to a unique 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 distinct foundation, that inside definite limits yield the genuine and the p-adic theories. this concept could be generalized variously. In illustration idea, it's the quantum basic linear workforce GLn(q)that interpolates among the p-adic team GLn(Zp), and among its actual (and complicated) 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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