Correspondances de Howe sur un corps p-adique

A Simple Proof of Rationality of Siegel-Weil Eisenstein Series

“…Assertion (b) is standard (cf. [MVW,p. 36 (2)]), while (c) follows from properties of splittings proved in [K] and recalled in [HKS], specifically [HKS,(1.8)].…”

The conservation relation for cuspidal representations

“…The representation theory of O(V ) follows from its connected component SO(V ), see for example [MVW,3.II.5]. Let sgn be the non trivial character of O(V )/SO(V ), and ε n an element of O(V ) not in SO(V ).…”

“…We say that a genuine irreducible representation π of G occurs in the theta correspondence for the dual pair (G, G ) if π occurs as a quotient of ω. The maximal quotient of ω which is π-isotypic has the form π ⊗Θ(π, W m ), where Θ(π, W m ) is a smooth representation of G [MVW,2.III.5]. The Howe duality principle asserts that Θ(π, W m ) has a unique irreducible quotient, denoted by θ(π, W m ).…”

“…The Howe duality principle asserts that Θ(π, W m ) has a unique irreducible quotient, denoted by θ(π, W m ). This principle is known to hold if the residue characteristic of F is not 2 [Wa1] or when π is a cuspidal representation [MVW,page 69]. In this article, we will just be interested in cuspidal representations so we will not need to suppose that the residue characteristic of F is different from 2.…”

“…Denote by P 1 the parabolic subgroup of U(V n+2 ) associated to this decomposition. We will write [MVW,page 69], the Jacquet functor…”