Matrix coefficients of depth-zero supercuspidal representations of \(\mathrm{GL}(2)\) (Q399467)

The paper under review explicitly computes the matrix coefficients of depth-zero supercuspidal representations of \(\mathrm{GL}(2)\) over a nonarchimedean local field \(F\). The authors reduce the computation to that of the matrix coefficients of the cuspidal representations of \(\mathrm{GL}_2(k)\), where \(k\) is the residue field of \(F\). Then they make use of the explicit model by Piatetski-Shapiro.NEWLINENEWLINENext the authors focus on the case where the test vector in the supercuspidal matrix coefficient is a unit new vector. The resulting function, which is quite interesting, may have some global applications towards the automorphic forms of \(\mathrm{GL}(2)\).NEWLINENEWLINEAlso, the authors group together depth-zero supercuspidal representations according to central character, and give formulas for the sums of matrix coefficients over all such representations in a given class, i.e. a class with a given central character.