On the support varieties of tilting modules

In this paper, we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$ . In particular, under certain technical conditions applicable to any root datum, we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\operatorname {\mathrm {GL}}_n$ -valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm {G}_2$ -valued representations.

Let G be a connected reductive algebraic group with a Frobenius morphism F: G → G defined over a finite field F p r . The main result of the paper is to prove that any rational G-module M which is projective when restricted to the Frobenius kernel Gr = Ker(F) is also projective over the split and twisted finite Chevalley groups. In 1987, Parshall conjectured this statement for r = 1 in the split case. The authors verified this in 1999 with the possible exclusion of primes 2 and 3 in non-simply laced cases. The converse of the main result is also discussed for split groups in this paper.

We show that a reductive group scheme over a base scheme S admits a faithful linear representation if and only if its radical torus is isotrivial, that is, it splits after a finite etale cover.

Lie algebroids arising from simple group schemes

Parabolic Subgroups in a Reductive Group Scheme

Let be a regular local ring containing a field. Let be a reductive group scheme over . We prove that a principal -bundle over is trivial if it is trivial over the field of fractions of . In other words, if is the field of fractions of , then the map of the non-Abelian cohomology pointed sets induced by the inclusion of in has trivial kernel. This result was proved in [1] for regular local rings containing an infinite field.

The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule and cominuscule nodes of the Dynkin diagram, while the infinite Coxeter elements correspond to a single node, which we call the heavy node, in all affine types other than type $A$.

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.

Representations of reductive groups over finite local rings of length two

Abstract. The computation of the cohomology for finite groups of Lie type in the describing characteristicis a challenging and difficult problem. In [BNP], the authors constructed an induction functor which takesmodules over the finite group of Lie type, G(F q ), to modules for the ambient algebraic group G. In particularthis functor when applied to the trivial module yields a natural G-filtration. This filtration was utilized in[BNP] to determine the first non-trivial cohomology class when the underlying root system is of type A n orC n . In this paper the authors extend these results toward locating the first non-trivial cohomology classesfor the remaining finite groups of Lie type (i.e., the underlying root system is of type B n , C n , D n , E 6 , E 7 ,E 8 , F 4 , and G 2 ) when the prime is larger than the Coxeter number. 1. Introduction1.1. Let Gbe a simple algebraic group scheme over a field kof prime characteristic pwhich is definedand split over the prime field F p , and F : G→ Gdenote the Frobenius map. The fixed points of therth iterate of the Frobenius map, denoted G(F

Multiplicity One Theorems for Generalized Gelfand-Graev Representations of Semisimple Lie Groups and Whittaker Models for the Discrete Series

Journal Article The space of degenerate Whittaker models for general linear groups over a finite field Get access Dipendra Prasad Dipendra Prasad Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2000, Issue 11, 2000, Pages 579–595, https://doi.org/10.1155/S1073792800000313 Published: 01 January 2000 Article history Received: 20 October 1999 Published: 01 January 2000 Revision received: 25 January 2000

We prove the multiplicity one case of Lusztig's conjecture on the irreducible characters of reductive algebraic groups for all fields with characteristic above the Coxeter number

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x0. Let (M,x0/ denote the corresponding fundamental group-scheme introduced by Nori. Let EG be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M. We prove that the following three statements are equivalent:1. The principal G-bundle EG over M is given by a homomorphism (M,x0)→G.2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ.In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.

The classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.