2021/06/03 by U. K. Anandavardhanan, Anandavardhanan, U. K., Arindam Jana +1 · 1 citation
Mathematics · Computer Science · #Finite Group Theory Research #Coding theory and cryptography #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2106.01929
Let G be a finite group with given subgroups H and K. Let π be an irreducible complex representation of G such that its space of H-invariant vectors as well as the space of K-invariant vectors are both one dimensional. Let vH (resp. vK) denote an H-invariant (resp. K-invariant) vector of unit norm in the standard G-invariant inner product ⟨ ~,~ ⟩π on π. Our interest is in computing the square of the absolute value of ⟨ vH,vK ⟩π. This is the correlation constant c(π;H,K) defined by Gross. In this paper, we give a sufficient condition for ⟨ vH, vK ⟩π to be zero and a sufficient condition for it to be non-zero (i.e., H and K are correlated with respect to π), when G=\rm GL2(\mathbb Fq), where \mathbb Fq is the finite field of q=pf elements of odd characteristic p, H is its split torus and K is a non-split torus. The key idea in our proof is to analyse the mod p reduction of π. We give an explicit formula for |⟨ vH,vK ⟩π|2 modulo p. Finally, we study the behaviour of ⟨ vH,vK ⟩π under the Shintani base change and give a sufficient condition for ⟨ vH,vK ⟩π to vanish for an irreducible representation π=\rm BC(τ) of \rm PGL2(\mathbb E), in terms of the epsilon factor of the base changing representation τ of \rm PGL2(\mathbb F), where \mathbb E/\mathbb F is a finite extension of finite fields. This is reminiscent of the vanishing of L(1/2, \rm BC(τ)), in the theory of automorphic forms, when the global root number of τ is -1.