Weighted Sato–Tate Vertical Distribution of the Satake Parameter of Maass Forms on PGL(N)

Abstract

We formulate a conjectured orthogonality relation between the Fourier coefficients of Maass forms on PGL(N). Based on the works of Goldfeld-Kontorovich and Blomer for N=3, and on our conjecture for N≥4, we prove a weighted vertical equidistribution theorem (with respect to the generalized Sato-Tate measure) for the Satake parameter of Maass forms at a finite prime. For N=3, the rate of convergence for the equidistribution theorem is obtained.

The Voronoi formula and double Dirichlet series

“…. , p kN−1 ) by the Casselman-Shalika formula (Proposition 5.1 of [Zho14]) and they are compatible with the Hecke relations. More explicitly, for a prime number p, we define A(p k1 , .…”

Fourier coefficients of GL(N) automorphic forms in arithmetic progressions

“…To conclude this section, we have shown that, for X ą 2p N´1 , we have 27) in the case of a form f which is not self-dual, and of odd N .…”

An orthogonality relation for a thin family of GL(3) Maass forms

“…A well known fact about Dirichlet characters is the following orthogonality relation for integers m, n coprime to q, where the sum on the left is over all characters (mod q). Since Dirichlet characters can be viewed as automorphic representations of GL(1, A Q ), this result can be interpreted as the simplest case of the orthogonality relation conjectured by Zhou [19] concerning Fourier-Whittaker coefficients of Maass forms on the space SL n (Z)\SL n (R)/SO n (R), n ≥ 2. This orthogonality relation conjectured by Zhou was proved by Bruggeman [2] in the case n = 2 and by Goldfeld-Kontorovich [6] and Blomer [1] in the case n = 3.…”