Unitary representations of p-adic U(5)

On the scaled trace forms and the transfer of a number field extension

For [Formula: see text] ([Formula: see text]), denote by [Formula: see text] the finite field of order [Formula: see text] and for a positive integer [Formula: see text], let [Formula: see text] be its extension field of degree [Formula: see text]. We establish a sufficient condition for existence of a primitive normal element [Formula: see text] such that [Formula: see text] is a primitive element, where [Formula: see text], with [Formula: see text] satisfying [Formula: see text] in [Formula: see text].

Over the past several years, operator algebraists have become increasingly interested in the problem of calculating the K-theory of group C∗-algebras. The focal point of research in this area ist he BaumConnesConjecture[BCH],which proposes a description of K-theory for the C∗-algebra of a group in terms of homology and the representation theory of compact subgroups. Although the main applications of the Baum-Connes Conjecture are to issues in geometry and topology, the conjecture also appears to be of interest from the point of view of harmonic analysis. Whereas for applications to topology one is concerned with discrete groups G (arising as the fundamental groups of manifolds), the conjecture’s links with harmonic analysis appear to be the strongest for reductive Lie groups and p-adic groups. The purpose of these notes is to convey to a reasonably broad audience some byproducts of the authors’ research into the C∗-algebra K-theory of the p-adic group GL(N), which culminated in a proof of the Baum-Connes Conjecture in this case [BHP2]. Along the way to the proof a number of interesting issues came to light which we feel deserve some exposure, even though our understanding of them is far from complete, and is indeed mostly very tentative. Much of what follows is focused on what we call here chamber homology, which is a type of equivariant homology associated to the action of a reductive p-adic group on its Bruhat-Tits affine building. The problem of computing chamber homology can be approached from a number of different directions. An especially interesting problem is to reconcile chamber homology with the Bernstein decomposition for representations of reductive p-adic groups [Be, BD]. This appears to be a far from trivial matter, even in comparatively simple cases. In Section 5 we formulate two very general conjectures which give a broad description of a Bernstein decomposition in chamber homology (perhaps we should call our conjectures questions, since the evidence we have gathered in their favor is not overwhelming).

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.

Weil's representation is a basic object in representation theory which plays a crucial role in many places: construction of unitary irreducible representations in the frame of the orbit method, Howe correspondence, Theta series, … The decomposition in irreducibles of the restriction of Weil's representation to maximal compact subgroups or anisotropic tori of the metaplectic group is thus an important information in representation theory. Except for $SL(2)$, this was not known in the p-adic case. In this article, we prove that the restriction of the Weil representation over a p-adic field, $p\neq2$, to maximal compact subgroups is multiplicity free and give an explicit description of the irreducibles occurring. In another paper, using our results, we describe the decomposition of the restriction of the Weil representation to maximal elliptic tori.

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.

Nondegenerate σ-additive measures with ranges in ℝ and ℚq (q≠p are prime numbers) that are quasi-invariant and pseudodifferentiable with respect to dense subgroups G′ are constructed on diffeomorphism and homeomorphism groups G for separable non-Archimedean Banach manifolds M over a local fieldK,K ⊃ ℚq, where ℚq is the field of p-adic numbers. These measures and the associated irreducible representations are used in the non-Archimedean gravitation theory.

Let G be a finite group. If there is a set containing n proper subgroups in G whose union comprises all elements of G, then G is a coverable group. The set of n proper subgroups is called an n-covering for G and the minimum number of n covers G is 3, so this means. If the n-covering has no proper subset that also covers G then it is called an irredundant n-covering for G and it is called a maximal n-covering if it consists of only maximal subgroups. A maximal irredundant n-covering with a core-free intersection is known as a Cn-covering. This study characterizes 5-groups having a C11-covering. It was found that a 5-group has a C11-covering if and only if it is isomorphic to some elementary abelian groups of a certain order.

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ℚ p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ℚ p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T p in the field ℚ p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.

Level Zero Types and Hecke Algebras for Local Central Simple Algebras

A proper subgroup [Formula: see text] of a finite group [Formula: see text] is called a second maximal subgroup of [Formula: see text] if [Formula: see text] is a maximal subgroup of every maximal subgroup [Formula: see text] in [Formula: see text] with [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] in which [Formula: see text] is a second maximal subgroup for every non-central element [Formula: see text] in [Formula: see text], and prove that either [Formula: see text] is solvable or [Formula: see text], where [Formula: see text] with [Formula: see text] a prime or [Formula: see text] with [Formula: see text] an odd prime.

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

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E7(7), they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE 7(7) in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE 7(7) leads to reducible unitary representations.

We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne–Lusztig (or Alvis–Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for $$G=GL(n)$$ and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group.

Vector decomposition problem has been proposed on some supersingular curves whose embedding degree is 2 for example. In order to apply the problem as a trapdoor for some innovative cryptosystems, it is necessary to make pairing-related calculations more efficient. Our previous work has considered an approach for Tate pairing on a supersingular curve of embedding degree 2 over extension field of extension degree 2. It improved both Miller's algorithm and final exponentiation that was required for the Tate pairing. In detail, one multiplication in the calculation of Miller's loop was eliminated by using a twist mapping. This paper shows a more improved calculation of pairing with some experimental result for the efficiency discussion.