Haran, Shai M. J. Arithmetical investigations. Representation theory, orthogonal polynomials, and quantum interpolations. (English) Zbl 1151.11004 Lecture Notes in Mathematics 1941. Berlin: Springer (ISBN 978-3-540-78378-7/pbk). xii, 217 p. (2008). This book, which grew out of lectures given by the author at Kyushu University, was intended as a primer to the earlier book [S. Haran, The mysteries of real prime. Oxford: Clarendon Press (2001; Zbl 1014.11001)]. However, many new themes are included.In contrast to the usual line of research in number theory and non-archimedean analysis, the author looks at real counterparts of natural \(p\)-adic constructions, and then introduces \(q\)-structures interpolating between them. The starting point is the relation between \(p\)-adic analysis and analysis on trees. The sets \(\mathbb P^1(\mathbb Q_p)\), \(\mathbb Z_p\), and \(\mathbb P^1(\mathbb Q_p)/\mathbb Z_p^{*}\) are interpreted as boundaries of trees. There is a bijection between Markov chains on a tree and probability measures on its boundary. Considering natural measures on the above sets, the author gets the corresponding Markov chains, and then looks for their real counterparts. Here the state space is just \(\mathbb N\times \mathbb N\). However, the author finds a number of analogies with the \(p\)-adic case, and defines a \(q\)-interpolation. Both for the trees and “non-trees” corresponding to the reals and \(q\)-structures, ladder systems are introduced. The harmonic analysis on the above objects leads to analogs of classical orthogonal polynomials.Developing his approach for higher dimensions, the author considers unitary representations of \(\text{GL}_d(\mathbb Z_p)\) connected with the \(p\)-adic Grassmann manifolds. In its turn, the \(p\)-adic real analogy (with \(q\)-interpolations) brings into consideration real Grassmann manifolds and \(q\)-Grassmann manifolds, \(q\)-Selberg measures, and the quantum group \(U_q(\text{su} (1,1))\). A number of problems is formulated.A preprint version of this book has been reviewed: M. J. S. Haran, Arithmetical investigations. Representation theory, orthogonal polynomials and quantum interpolations. With notes by Yoshinori Yamasaki. COE Lecture Note 2. Fukuoka: Kyushu University, The 21st Century COE Program “DMHF” (2006; Zbl 1120.11001). Reviewer: Anatoly N. Kochubei (Kyïv) Cited in 1 Document MSC: 11-02 Research exposition (monographs, survey articles) pertaining to number theory 11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) 11S85 Other nonanalytic theory 33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics 43A05 Measures on groups and semigroups, etc. 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) Keywords:boundary of a tree; Markov chain; Grassmann manifold; \(p\)-adic Grassmann manifold; \(q\)-Selberg measure; quantum group Citations:Zbl 1014.11001; Zbl 1120.11001 PDFBibTeX XMLCite \textit{S. M. J. Haran}, Arithmetical investigations. Representation theory, orthogonal polynomials, and quantum interpolations. Berlin: Springer (2008; Zbl 1151.11004) Full Text: DOI