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An Isaac Newton Institute Workshop

Matrix Ensembles and L-Functions

Symmetry Beyond Root Numbers: a GL(6) Example

Authors: Eduardo Dueñez (University of Texas), Steven J. Miller (Ohio State University)

Abstract

Going against the "folklore" belief that even orthogonal families arise splitting a full orthogonal family by sign, we show that the lone-standing family {L(s,g x Sym2(f)} (where g is a fixed Hecke-Maass form and f varies over holomorphic modular forms of level 1) has SO(even) symmetry. Thus, the theory of symmetry types is not merely about root numbers (sign of the functional equation). The family above is connected with the relation between classical and quantum fluctuations of observables in the modular surface by work of Luo and Sarnak.