Algebraic groups over a 2-dimensional local field: Irreducibility of certain induced representations

Let X X be a smooth curve over a finite field of characteristic p p , let E E be a number field, and let L = { L λ } \mathbf {L} = \{\mathcal {L}_\lambda \} be an E E -compatible system of lisse sheaves on the curve X X . For each place λ \lambda of E E not lying over p p , the λ \lambda -component of the system L \mathbf {L} is a lisse E λ E_\lambda -sheaf L λ \mathcal {L}_\lambda on X X , whose associated arithmetic monodromy group is an algebraic group over the local field E λ E_\lambda . We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the E E -compatible system L \mathbf {L} is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of λ \lambda ”. More precisely, after replacing E E by a finite extension, there exists a connected split reductive algebraic group G 0 G_0 over the number field E E such that for every place λ \lambda of E E not lying over p p , the identity component of the arithmetic monodromy group of L λ \mathcal {L}_\lambda is isomorphic to the group G 0 G_0 with coefficients extended to the local field E λ E_\lambda .

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space \({\cal S}(X)\) (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of \( {\cal S}(X)^I \), which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of \( {\cal S}(X) \), which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.

A note on Galois representations valued in reductive groups with open image

Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be identified with the dual of a coalgebra defined using exclusively the root datum of $(G,T)$, a set of simple roots defining $P$ and the formal group law of $h$. In two papers [Push-pull operators on the formal affine Demazure algebra and its dual, arXiv:1312.0019] and [A coproduct structure on the formal affine Demazure algebra, arXiv:1209.1676], we studied the properties of this dual and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper can be viewed as a companion paper, that justifies all the definitions of the algebraic objects and operators by explaining how to match them to equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. Our main tool is the pull-back to the $T$-fixed points of $G/P$ which injects the cohomology ring in question into a direct product of a finite number of copies of the $T$-equivariant oriented cohomology of a point.

Let ${\mathfrak o}$ be the ring of integers in a finite extension field of ${\mathbb Q}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb Q}_p$, let ${\mathcal H}(G,I_0)$ be its pro-$p$-Iwahori Hecke ${\mathfrak o}$-algebra. In \cite{dfun} we introduced a general principle how to assign to a certain additionally chosen datum $(C^{(\bullet)},\phi,\tau)$ an exact functor $M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$ from finite length ${\mathcal H}(G,I_0)$-modules to $(\varphi^r,\Gamma)$-modules. In the present paper we concretely work out such data $(C^{(\bullet)},\phi,\tau)$ for the classical matrix groups. We show that the corresponding functor identifies the set of (standard) supersingular ${\mathcal H}(G,I_0)\otimes_{{\mathfrak o}}k$-modules with the set of $(\varphi^r,\Gamma)$-modules satisfying a certain symmetry condition.

In this work we extend the first part of the previous paper [F4] to higher dimensional local fields. We introduce a nontrivial translation invariant measure on the additive group of higher dimensional local fields, and then develop elements of integration and harmonic analysis. We also discuss its relation with several other measure theories. For an n-dimensional local field F a translation invariant measure μ is defined on a certain ring A of measurable sets and takes values in R ((X1)) . . . ((Xn−1)) (which itself is an n-dimensional local field whose last residue field is the archimedean field R). The algebra A in the case of higher dimensional fields with finite last residue field is the algebra generated by characteristic functions of shifts of fractional ideals, i.e. a + bO with a, b ∈ F and O the ring of integers of F with respect to the n-dimensional structure. The measure is countably additive in a refined sense (see sections 7 and 8). Elements of integration theory are introduced in sections 9–13. The additive group of a higher dimensional field has a certain topology on it with respect to which it is not locally compact for n > 1. This topology plays a key role in higher class field theory [F1–F3]. The additive group of F is self dual, which together with the measure and integration leads to analogs of many classical results in Fourier analysis (section 14). In particular, for functions in certain space we introduce their transform

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

This book is, on the one hand, a pedagogical introduction to the formalism of slopes, of semi-stability and of related concepts in the simplest possible context. It is therefore accessible to any graduate student with a basic knowledge in algebraic geometry and algebraic groups. On the other hand, the book also provides a thorough introduction to the basics of period domains, as they appear in the geometric approach to local Langlands correspondences and in the recent conjectural p-adic local Langlands program. The authors provide numerous worked examples and establish many connections to topics in the general area of algebraic groups over finite and local fields. In addition, the end of each section includes remarks on open questions, historical context and references to the literature.

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

Objectives: The purpose of this work is to study 𝑝-Bratteli diagrams arising from a class of group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 for every odd prime 𝑝, to define the irreducible representations using mutually orthogonal primitive idempotents and to index the complete set of irreducible representations for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟. Method: We use primitive idempotents to define irreducible representations and systematically index the irreducible representations to establish vertices for the 𝑝-Bratteli diagram. Also, we compute the matrix units using the orthogonal primitive idempotents of the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟. Findings: The complete set of irreducible representations for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 has been successfully indexed, and matrix units for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 were determined using the orthogonal primitive idempotents. Novelty: This work provides a structured approach to constructing 𝑝-Bratteli diagrams for the group algebras related to odd primes, contributing new insights into their irreducible representations and matrix unit’s structures. Keywords: Primitive idempotent; Irreducible representation; Matrix units; Young diagram; Bratteli diagram

Let $K$ denote an $n$-dimensional local field. The aim of this expository paper is to survey the basic arithmetic theory of the $n$-dimensional local field $K$ together with its Milnor $K$-theory and Parshin topological $K$-theory; to review Kato's ramification theory for finite abelian extensions of the $n$-dimensional local field $K$, and to state the local abelian higher-dimensional $K$-theoretic generalization of local abelian class field theory of Hasse, which is developed by Kato and Parshin. The paper is geared toward non-abelian generalization of this theory.

Let k be a finite field, a global field, or a local non-archimedean field, and let H 1 and H 2 be split, connected, semisimple algebraic groups over k. We prove that if H 1 and H 2 share the same set of maximal k-tori, up to k-isomorphism, then the Weyl groups W(H 1 ) and W(H 2 ) are isomorphic, and hence the algebraic groups modulo their centers are isomorphic except for a switch of a certain number of factors of type B n and C n . (Due to a recent result of Philippe Gille, this result also holds for fields which admit arbitrary cyclic extensions.).

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren, and the author.

A linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and $\dim G=\dim V$ . A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions.

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,