Let π1 be a cuspidal Saito-Kurokawa representation of SO(3,2) and π2 a cuspidal or trivial representation of SO(3,1) or SO(2,2). We present period formulas that show how π1 restricts to π2.

Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight g-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuper-algebra. Then we classify the cuspidal Levi sub...

Let ∧ : GLn(C) −→ GLN (C), where N = n(n−1) 2 , be the map given by the exterior square. Then Langlands’ functoriality predicts that there is a map from cuspidal representations of GLn to automorphic representations of GLN , which satisfies certain canonical properties. To explain, let F be a number field, and let A be its ring of adeles. Let π = ⊗ v πv be a cuspidal (automorphic) representatio...

We present a unified parametrisation of `-blocks of quasi-simple finite groups of Lie type in non-defining characteristic via Lusztig’s induction functor in terms of eJordan-cuspidal pairs and e-Jordan quasi-central cuspidal pairs.

ior of small cuspidal eigenpairs of ΔS. In Theorem 1.7, we describe limiting behavior of these eigenpairs on surfaces Sm ∈Mg,n when (Sm) converges to a point in M̄g,n. Then, we consider the ith cuspidal eigenvalue, λc i (S), of S∈Mg,n. Since noncuspidal eigenfunctions (residual eigenfunctions or generalized eigenfunctions) may converge to cuspidal eigenfunctions, it is not known if λc i (S) is a...

Let F be a number field, G the general linear group of degree n defined over F. Let π be an irreducible cuspidal automorphic representation of G(A). In [1–3], a Rankin-Selbergtype integral is constructed to represent the L function of π. That the integrals of Jacquet, Piatetski-Shapiro, and Shalika are Eulerian follows from the uniqueness of Whittaker models and the fact that cuspidal represent...

Let O be the ring of integers of a p-adic field and p its maximal ideal. We compute the Jordan-Hölder decomposition of the reduction modulo p of the cuspidal representations of GL2(O/p) for e ≥ 1. We also provide an alternative formulation of Serre’s conjecture for Hilbert modular forms. 1. Cuspidal representations and weights 1.1. Cuspidal representations. Let K/Qp be a local field, where p is...

This note pursues the techniques of modified psi classes on the stack of stable maps (cf. [Graber-Kock-Pandharipande]) to give concise solutions to the characteristic number problem of rational curves in P2 or P1×P1 with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via...

A strong impetus for this paper came, at least for the first author, from a question of Avner Ash, asking whether one can construct non-selfdual, nonmonomial cuspidal cohomology classes for suitable congruence subgroups Γ of SL(n,Z), say for n = 6. Such a construction, in special examples, has been known for some time for n = 3 ([AGG1984], [vGT1994], [vGKTV1997], [vGT2000]); it is of course not...

Horseshoes play a central role in dynamical systems and are observed in many chaotic systems. However most points in a neighborhood of the horseshoe escape after finite iterations. In this work we construct a model that possesses an attracting set that contains a cuspidal horseshoe with positive entropy. This model is obtained by reinjecting the points that escape the horseshoe and can be reali...