Principal series representations of metaplectic groups

Abstract

We study the principal series representations of central extensions of a split reductive algebraic group by a cyclic group of order n. We compute the Plancherel measure of the representation using Eisenstein series and a comparison method. In addition, we construct genuine central characters of the metaplectic torus in the simply-laced case.

On Shahidi local coefficients matrix

“…Proof. From the Knapp-Stein dimension Theorem extended by Savin in the Appendix of [31] to a maximal parabolic induction on metaplectic groups, it follows that given that σ is unitary, I(σ) is reducible if and only if σ ∼ = σ w and µ n (σ, s) −1 is analytic at s = 0. In this case I(σ) is the sum of two non-isomorphic irreducible representations.…”