Orthogonality of invariant vectors

Let G be a finite group with given subgroups H and K. π an irreducible complex representation of such that its space H-invariant vectors as well the K-invariant are both one dimensional. vH (resp. vK) denote K-invariant) vector unit norm in G-invariant inner product 〈,〉π on π. Our interest is computing square absolute value 〈vH,vK〉π. This correlation constant c(π;H,K) defined by Gross [3]. In this paper, we study question for G=GL2(Fq), where Fq field q=pm elements odd characteristic p, split torus K non-split torus. The first main theorem paper gives explicit formula |〈vH,vK〉π|2 modulo p. key idea here to analyse mod p reduction second relates behaviour 〈vH,vK〉π under Shintani base change sufficient condition vanish π=BC(τ) PGL2(E), terms epsilon factor changing τ PGL2(F), E/F extension fields.