Deligne–Lusztig duality and wonderful compactification

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.

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).

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.

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.

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

We discuss some connections between the closure of a Steinberg fiber in the wonderful compactification of an adjoint group and the affine Deligne–Lusztig varieties Xw(1) in the affine flag variety. Among other things, we describe the emptiness/nonemptiness pattern of Xw(1) if the translation part of w is quasi-regular. As a by-product, we give a new proof of the explicit description of ⁠, first obtained in He [“Unipotent variety in the group compactification.” Advances in Mathematics 203 (2006): 109–31].

Complete group rings as Hecke algebras

MacMahon's Master Theorem, Double Tableau Polynomials, and Representations of Groups

We establish the stability of Rankin–Selberg local γ-factors attached to generic representations of symplectic groups and general linear groups over p-adic fields. Our approach uses the Langlands–Shahidi method and provides a new proof that does not rely on previously known stability results. The key innovations involve handling the geometry of orbit spaces with non-trivial stabilizers and developing new techniques for analyzing partial Bessel integrals in this setting. The results have important implications for the Langlands program, particularly for local converse theorems and the functoriality conjecture.

This paper is concerned with Higher Algebraic K-theory and actions of algebraic groups G on such ‘nice’ categories as the category of algebraic vector bundles on a scheme X. Such ‘nice’ categories are examples of ‘exact’ categories with the observation that the category of actions on G on such exact categories also form an exact category called equivariant exact categories on which one can do higher Algebraic K-theory (of Quillen) called equivariant higher Algebraic K-theory—the higher dimensional generalizations of classical equivariant K-theory which belongs to the field of representation theory. Thus, for an Algebraic group G over a number field or p-adic field F, we present constructions and computations of equivariant higher K-groups as well as ‘profinite’ or ‘continuous’ higher K-groups for some G-Scheme X. In particular, we present explicit l-completeness (l a rational prime) and finiteness computations for higher K-groups and profinite higher K-groups for twisted flag varieties.

We study the parabolically induced complex representations of the unitary group in 5 variables,U(5), defined over a p-adic field.Let F be a p-adic field. Let E:F be a field extension of degree two. U(5) has three proper standard Levi subgroups, the minimal Levi subgroup M 0 ≅E * ×E * ×E 1 and the two maximal Levi subgroups M 1 ≅GL(2,E)×E 1 and M 2 ≅E * ×U(3).We consider representations induced from M 0 , representations induced from non-cuspidal, not fully-induced representations of M 1 and M 2 and representations induced from cuspidal representations of M 1 .We determine the points and lines of reducibility and the irreducible subquotients of these representations. Further we describe - except several particular cases - the unitary dual in terms of Langlands quotients.

Introduction. Let G1 and G be connected semisimple algebraic groups defined over a field K of characteristic zero and assume that there is an isomorphism f of G1 onto G which is defined over R, the algebraic closure of K. If p: G -? GL(V) is an absolutely irreducible (finite-dimensional) representation of G defined over K, then p of is an absolutely irreducible representation of G1 defined over K. Satake [7, p. 230] has shown that there is a field K1 which is a finite extension of K, a (unique) central simple division algebra K# defined over K1, a finite-dimensional right vector space V1 over K#, and a K1-homomorphism pi: G1 -GL(V1/K#) (the group of all nonsingular K#-linear endomorphisms of V1) such that (p of)(g) = 01(p1(g)) for all g E G1 where 01 is a unique absolutely irreducible representation of End (V1/K#) (the algebra of all K#-linear endomorphisms of V1) onto End (V). In this paper we are interested in the case where K= K1 and where there are invariant forms on V and V1. More precisely, we state the following two problems. PROBLEM 1. Assume that K#=K and that there are invariant bilinear forms B on V and B1 on V1 which are defined over K. What is the relationship between these two forms over K? Of course, if B is alternating, so is B1 and both are determined by dim V=dim V1. Hence, we shall always take B and B1 to be symmetric. PROBLEM 2. Assume that K# is a nontrivial division algebra over K (i.e., K# AK) and that there is an invariant bilinear form B on V and an invariant ?hermitian form F (c = + 1 or 1) on V1 both of which are defined over K. What is the relationship between these two forms over K? We are especially interested in the case K= Q,, a p-adic field. (In a future paper, we shall discuss the case K=R.) Here, some simplifications are immediately available. In Problem 2, it can be shown [7, p. 232] that K# has an involution of the first kind; but over Q,, it is known that the only such division algebra is the quaternion division algebra. Furthermore, it is known that a hermitian form on a finite-dimensional vector space over a quaternion division algebra defined over Q, is determined only by the dimension of the vector space. Therefore, in Problem 2 we shall always take F to be skew-hermitian; in the case where K# is a quaterion division algebra, this means that the form B is symmetric [7, p. 233]. If W is a vector space defined over K and if S is a symmetric form on W which is also defined over K, then three invariants can be associated with the pair (W, S),

Minimal representations of exceptional 𝑝-adic groups

Simple subquotients of big parabolically induced representations of p-adic groups

Jiang and Rallis (1997) defined a family of local integrals attached to a cubic polynomial and proved explicit evaluations of them over a non-archimedean local field F, when either F contains three third roots of unity, or the defining polynomial is reducible. The restriction on F allowed them, among other things, to reduce the case of irreducible polynomials of the form x3 - a. Pleso (2009) began the work of removing the restriction on F by expressing the integral as a sum of 16 integrals for the cubic polynomial x3 -bx -c, with b,c in F, and computing nine of them. In this work, we compute 15 of Pleso's integrals explicitly, and reduce the last to a conjecture about the number of points on a surface over a finite field, in the special case when F is the p-adic numbers (F = Qp) and p is equivalent to 5 mod 6. The proof of this conjecture is provided in the appendix section. Our computations essentially complete Pleso's work in that special case. In the interim, Xiong (2020) has computed the integrals for an arbitrary non-archimedean local field by a totally different approach. Our direct approach might be more extendable to analogous integrals defined using quintic polynomials in a higher-rank setting.