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Memorial Volume for Professor Hiroshi Saito

Volume 208 - December 2012

Contents

Lefschetz operator and local Langlands mod ℓ: The regular case
Jean-François Dat
Published online by Cambridge University Press: 11 January 2016, pp. 1-38
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Let p and ℓ be two distinct primes. The aim of this paper is to show how, under a certain congruence hypothesis, the mod ℓ cohomology complex of the Lubin-Tate tower, together with a natural Lefschetz operator, provides a geometric interpretation of Vignéras’s local Langlands correspondence modulo ℓ for unipotent representations.

Fonctions zêta ℓ-modulaires
Alberto Mínguez
Published online by Cambridge University Press: 11 January 2016, pp. 39-65
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Let F be a non-Archimedean locally compact field, of residual characteristic p, and let D be a finite-dimensional central division F-algebra. Let ℓ be a prime number different from p. In this article, generalizing the results of [GJ], we associate, to each ℓ-modular smooth irreducible representation π of GL m (D), two invariants L(T,π), ε(T,π,ψ), where T is an indeterminate and ψ is a nontrivial character of F.

Doubling zeta integrals and local factors for metaplectic groups
Wee Teck Gan
Published online by Cambridge University Press: 11 January 2016, pp. 67-95
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We develop the theory of the doubling zeta integral of Piatetski-Shapiro and Rallis for metaplectic groups Mp2n , and we use it to give precise definitions of the local γ-factors, L-factors, and ε-factors for irreducible representations of Mp2n × GL1, following the footsteps of Lapid and Rallis.

On the Kottwitz-Shelstad normalization of transfer factors for automorphic induction for GL n
Kaoru Hiraga, Atsushi Ichino
Published online by Cambridge University Press: 11 January 2016, pp. 97-144
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Automorphic induction for GL n is a case of endoscopic transfer, and its character identity was established by Henniart and Herb, up to a constant of proportionality. We determine this constant in terms of the Kottwitz-Shelstad normalization of transfer factors, which involves certain ε-factors.

Some conjectures on endoscopic representations in odd orthogonal groups
David Ginzburg, Dihua Jiang
Published online by Cambridge University Press: 11 January 2016, pp. 145-170
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In this paper, we introduce two conjectures on characterizations of endoscopy structures of irreducible generic cuspidal automorphic representations of odd special orthogonal groups in terms of nonvanishing of certain period of automorphic forms. We discuss a relation between the two conjectures and prove that a special case of Conjecture 1 (and hence Conjecture 2) is true.

Extensions of representations of p-adic groups
Jeffrey D. Adler, Dipendra Prasad
Published online by Cambridge University Press: 11 January 2016, pp. 171-199
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We calculate extensions between certain irreducible admissible representations of p-adic groups.

Matrix coefficients of the large discrete series representations of Sp(2; R) as hypergeometric series of two variables
Takayuki Oda
Published online by Cambridge University Press: 11 January 2016, pp. 201-263
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We investigate the radial part of the matrix coefficients with minimal K-types of the large discrete series representations of Sp(2; R). They satisfy certain difference-differential equations derived from Schmid operators. This system is reduced to a holonomic system of rank 4, which is finally found to be equivalent to higher-order hypergeometric series in the sense of Appell and Kampé de Fériet.

On zeta functions associated to symmetric matrices, II: Functional equations and special values
Tomoyoshi Ibukiyama, Hiroshi Saito
Published online by Cambridge University Press: 11 January 2016, pp. 265-316
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New simple functional equations of zeta functions of the prehomogeneous vector spaces consisting of symmetric matrices are obtained, using explicit forms of zeta functions in the previous paper, Part I, and real analytic Eisenstein series of half-integral weight. When the matrix size is 2, our functional equations are identical with the ones by Shintani, but we give here an alternative proof. The special values of the zeta functions at nonpositive integers and the residues are also explicitly obtained. These special values, written by products of Bernoulli numbers, are used to give the contribution of “central” unipotent elements in the dimension formula of Siegel cusp forms of any degree. These results lead us to a conjecture on explicit values of dimensions of Siegel cusp forms of any torsion-free principal congruence subgroups of the symplectic groups of general degree.