© 2008
Arithmetical Investigations
Representation Theory, Orthogonal Polynomials, and Quantum Interpolations
- Editors
- (view affiliations)
- 8.1k Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 1941)
Advertisement
© 2008
Part of the Lecture Notes in Mathematics book series (LNM, volume 1941)
In this volume the author further develops his philosophy of quantum interpolation between the real numbers and the p-adic numbers. The p-adic numbers contain the p-adic integers Z_{p} which are the inverse limit of the finite rings Z/p^{n}. This gives rise to a tree, and probability measures w on Z_{p} correspond to Markov chains on this tree. From the tree structure one obtains special basis for the Hilbert space L_{2}(Z_{p},w). The real analogue of the p-adic integers is the interval [-1,1], and a probability measure w on it gives rise to a special basis for L_{2}([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For special (gamma and beta) measures there is a "quantum" or "q-analogue" Markov chain, and a special basis, that within certain limits yield the real and the p-adic theories. This idea can be generalized variously. In representation theory, it is the quantum general linear group GL_{n}(q)that interpolates between the p-adic group GL_{n}(Z_{p}), and between its real (and complex) analogue -the orthogonal O_{n} (and unitary U_{n} )groups. There is a similar quantum interpolation between the real and p-adic Fourier transform and between the real and p-adic (local unramified part of) Tate thesis, and Weil explicit sums.