Abstract
We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a$p$-adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet, Piatetski-Shapiro and Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky’s degenerate Whittaker model.
Highlights
We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a p-adic field
We do not know whether the restriction of Sp(π, m) to a parabolic subgroup of type ((n − 1)m, m) is of finite length. Another aspect of the paper is to provide an explicit, manifestly positive, unitary structure for the Speh representation in its Shalika model. (By this we mean that the positive-definiteness is ‘clear and obvious’ from the definition.) Once again, this is modeled on the case of generic unitarizable representations, in which Bernstein gives a unitary structure for their Whittaker models by taking the L2-inner product of Whittaker functions restricted to the mirabolic subgroup [Ber84]
We introduce the class of mhomogeneous representations, which includes the usual Speh representations and which is the main focus of the paper
Summary
2.1 Notation Throughout the paper, fix a non-archimedean local field F with ring of integers O and absolute value |·|. We denote the set of irreducible representations of GLn(F ) (up to equivalence) by Irr GLn and set Irr = n 0 Irr GLn. We write Irr GL0 = {1}. (In contrast, the one-dimensional trivial character of GL1 will be denoted by 1F ∗.) The subset of supercuspidal (respectively, squareintegrable, essentially square-integrable, tempered, generic) representations will be denoted by Irrcusp (respectively, Irrsqr, Irresqr, Irrtmp, Irrgen). For any character ω of F ∗ (i.e. ω ∈ Irr GL1) we denote by πω the representation obtained from π by twisting by the character ω ◦ det. For any τ ∈ Irresqr, let e(τ ) be the unique real number s such that the twisted representation τ |·|−s is unitarizable (i.e. has a unitary central character). (This follows from the classification of the unitary dual of GLn(F ) by Tadic [Tad86].) We denote by Irr(AT ) the set of (AT) representations. An element of M(A) (a multiset of A) is a finite (possibly empty) formal sum of element of A
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