Heredity of Whittaker Models on the Metaplectic Group

and the Whittaker models of the inducing representation π M (cf. Theorem 2 of [4]). In this paper, Rodier’s theorem on the “heredity” of Whittaker models is extended to the non-algebraic setting of the n-fold metaplectic cover G̃ of G, where n is a positive integer such that F contains all of the n-th roots of unity. The main result is stated as a theorem in §2. In order to illustrate the situation, consider the example of a representation of G̃ induced from the metaplectic preimage B̃ of the standard Borel subgroup B of G. Since the Levi component T of B is a (maximal) torus in G, its metaplectic preimage T̃ is a Heisenberg group. Consequently, the dimension of any irreducible representation π T of T̃ is equal to the index [T̃ : T̃∗], where T̃∗ is an arbitrary maximal abelian subgroup of T̃ . In this example, every linear functional on the space of π T is a Whittaker functional, hence the inducing representation π T has precisely [T̃ : T̃∗] distinct Whittaker models. Now extend πT to a representation π B of B̃ (see §2 below), and let π G be the normalized, fullinduced representation Ind(B̃, G̃;π B ) of G̃. By Lemma I.3.2 of [3], it follows that π G also has [T̃ : T̃∗] distinct Whittaker models, thus Rodier’s theorem evidently extends to this example.