Statistics of Hecke eigenvalues for GL(𝑛)

A two-dimensional central limit theorem for the eigenvalues of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>GL</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {GL(n)} Hecke–Maass cusp forms is newly derived. The covariance matrix is diagonal and hence verifies the statistical independence between the real and imaginary parts of the eigenvalues. We also prove a central limit theorem for the number of weighted eigenvalues in a compact region of the complex plane, and evaluate some moments of eigenvalues for the Hecke operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {Tₚ} which reveal interesting interferences.