Automorphic Integral Transforms for Classical Groups II: Twisted Descents

The classification of irreducible square integrable representations of classical p p -adic groups is completed in this paper, under a natural local assumption. Further, this classification gives a parameterization of irreducible tempered representations of these groups. Therefore, it implies a classification of the non-unitary duals of these groups (modulo cuspidal data). The classification of irreducible square integrable representations is directly related to the parameterization of irreducible square integrable representations in terms of dual objects, which is predicted by Langlands program.

The reduced determinantal forms for the character of all the irreducible representations of the classical Lie groups are derived from Weyl's character formulae. The derivations use a lemma due to Frobenius and the results are expressed in terms of characters of representations specified by hook diagrams. The results are appropriate to all irreducible representations whether specified by ordinary or composite Young diagrams, and whether tensor or spinor in nature.

Let $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $ -approximable points and for the set of inhomogeneous multiplicative $\psi $ -approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math.740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$ , to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

This paper deals with a new analytic type of vector- and Clifford algebra valued\nautomorphic forms in one and two vector variables. For hypercomplex\ngeneralizations of the classical modular group and their arithmetic congruence\nsubgroups Eisenstein- and Poincaré type series that are annihilated by Dirac\noperators, and more generally, by iterated Dirac operators on the upper\nhalf-space of $\\mathbb{R}^n$ are discussed. In particular we introduce\n(poly-)monogenic modular forms on hypercomplex generalizations of the classical\ntheta group.

In this paper we exhibit a new method of proving non-unitarity of representations, based on semi simplicity of unitarizable representations. Non-unitarity is proved for a half of all irreducible representations of classical p-adic groups whose infinitesimal character is the same as the infinitesimal character of a generalized Steinberg representation (as defined in Tadic, Am J Math 120:159–210, 1998). Only the Steinberg representation and its Aubert dual are expected to be unitary here. In this way we partially generalize a result of Casselman to the case of classical groups. Our argument is completely different from Casselman’s argument (which is hard to extend to this case). It requires a very limited knowledge of the inducing cuspidal representation.

Almost unramified automorphic representations for split groups over Fq(t)

Structure Arising from Induction and Jacquet Modules of Representations of Classical p-Adic Groups

Automorphic forms and automorphic representations

Automorphic Forms with Anisotropic Periods on a Unitary Group

For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.

It is the main objective of this paper to show a vanishing result for cuspidal cohomology of arithmetic groups in classical groups~$G$ defined over some number field~$k$. Our approach relies on the fact that cuspidal automorphic forms are nonsingular in the sense of Howe. This result puts a strong constraint on the (archimedean components of) irreducible cuspidal automorphic representations of $G$ that can possibly contribute to the cuspidal cohomology. Combining this with the classification of Vogan-Zuckerman of unitary representations with nonzero cohomology provides a constant $r_0(G/k)$, only depending on $G/k$, below which the cuspidal cohomology vanishes. We will give a formula for this constant for each classical group of type (I) in the classification scheme due to Weil. We conclude with making this result explicit for some split classical groups over a totally real algebraic number field.

Each irreducible n-dimensional complex representation of an arbitrary group G is shown to be associated with a unique classical group. Such representations are thereby classified into 'ten classical types' associated with the respective classical groups GL(n;C), O(n;C), Sp(n;C), U(r,s), GL(n;R), GL(n/2;H), O(r,s), Sp(n;R), O(n/2;H), Sp(r/2,s/2), where r+s=n.

Our aim is to describe the image and fibres of the base change lift constructed by Langlands, for solvable extensions of number fields. A particular instance of the principle of functoriality enunciated by Lang- lands, is the base change map relating automorphic forms on GL2(Ak) to forms on GL2(AK), for an extension K/k of number fields. Using the trace formula and following earlier workby Saito and Shintani, Langlands in (L), established the existence of the base change lift BCK/k sending cuspidal automorphic rep- resentations of GL2(Ak) to automorphic representations of GL2(AK), provided K/k is a cyclic extension of prime degree. Langlands further characterized the image and the fibres of the base change map BCK/k. Given an automorphism σ of K, and an automorphic representation Π of GL2(AK), let Π σ denote the automorphic representation of GL2(AK) defined by Π σ (g )=Π (σ −1 (g)). The descent property for invariant, automorphic representations established by Langlands is that if Π is a cuspidal automorphic representation of GL2(AK), which is invariant with respect to the action of the Galois group G(K/k )o f a cyclic extension K/k of prime degree, then Π lies in the image of the base change map BCK/k from automorphic representations on GL2(Ak) to automor- phic representations on GL2(AK). Further if π1 and π2 are distinct, cuspidal automorphic representations of GLn(Ak) which base change to Π, then

A realization of the quantum Lorentz group

ABSTRACTThe unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.