M.Sc Student | Yehoshua Gal |
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Subject | Applications of Quantitative Estimates on Matrix Coefficients |

Department | Department of Mathematics |

Supervisor | Professor Amos Nevo |

Full Thesis text |

Matrix coefficients are to be found in any study of representation theory of simple Lie groups, as they provide a connection between the abstract world of representations and the concrete space of functions on the group. Let G denote a connected simple non-compact Lie group with finite center. This research thesis mainly dealt with bounding the norm of convolution operators on a function space associated with G and a lattice of G. The main tool we used was quantitative estimates for K-finite matrix coefficients of unitary representations of G, not containing the trivial representation. We utilize these estimates to find an error term in the problem of counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in G. We choose to focus on the special linear groups of rank at least 2, where we give a detailed proof of error estimates which constitute the first improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991, and are nearly twice as good in some cases. Another application of quantitative estimates on matrix coefficients was in establishing new examples of tempered subgroups of special linear groups. These subgroups are of great interest as they were found to give raise to best possible Diophantine approximation of lattice orbits in an associated homogenous space.