Split Semisimple Group Research Articles

Cyclic base change of cuspidal automorphic representations over function fields

Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

On Lie algebra modules which are modules over semisimple group schemes

Let p be a prime. Given a split semisimple group scheme G over a normal integral domain R which is a faithfully flat $${\mathbb {Z}}_{(p)}$$ -algebra, we classify all finite dimensional representations V of the fiber $$G_K$$ of G over $$K:=Frac (R)$$ with the property that the set of lattices of V with respect to R which are G-modules is as well the set of lattices of V with respect to R which are $$Lie (G)$$ -modules. We apply this classification to get a general criterion of extensions of homomorphisms between reductive group schemes over $$Spec \,K$$ to homomorphisms between reductive group schemes over $$Spec \,R$$ . We also show that for a simply connected semisimple group scheme over a reduced $${\mathbb {Q}}$$ –algebra, the category of its representations is equivalent to the category of representations of its Lie algebra.

Support theory for Drinfeld doubles of some infinitesimal group schemes

Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representation category of the Drinfeld double of kG. We show that thick ideals in the corresponding stable category are classified by cohomological support, and calculate the Balmer spectrum of the stable category of Z(rep(G)). We also construct a $\pi$-point style rank variety for the Drinfeld double, identify $\pi$-point support with cohomological support, and show that both support theories satisfy the tensor product property. Our results hold, more generally, for Drinfeld doubles of Frobenius kernels in any smooth algebraic group which admits a quasi-logarithm, such as a Borel subgroup in a split semisimple group in very good characteristic.

Coherence of augmented Iwasawa algebras

The augmented Iwasawa algebra of a p-adic Lie group is a generalisation of the Iwasawa algebra of a compact p-adic Lie group. We prove that a split-semisimple group over a p-adic field has a coherent augmented Iwasawa algebra if and only if its root system is of rank one. We deduce that the general linear group of degree n has a coherent augmented Iwasawa algebra precisely when n is at most two. We also characterise when certain solvable p-adic Lie groups have a coherent augmented Iwasawa algebra.

A parametrization of unipotent representations

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cells representations of the Weyl group in the same two-sided cell. We use this map to parametrize the unipotent representations.

On the Ramanujan conjecture for automorphic forms over function fields I. Geometry

Let G G be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of G G , subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of Bun G \operatorname {Bun}_G . It is independent of the work of Lafforgue on the global Langlands correspondence.

Degree 3 unramified cohomology of classifying spaces for exceptional groups

Let G be a reductive group defined over an algebraically closed field of characteristic 0 such that the Dynkin diagram of G is the disjoint union of diagrams of types G 2 , F 4 , E 6 , E 7 , E 8 . We show that the degree 3 unramified cohomology of the classifying space of G is trivial. In particular, combined with articles by Merkurjev [11] and the author [1] , this completes the computations of degree 3 unramified cohomology and reductive invariants for all split semisimple groups of a homogeneous Dynkin type.

Triviality Properties of Principal Bundles on Singular Curves. II

AbstractFor $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\rightarrow S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D\subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected, but it can be made to vanish by a faithfully flat base change, if $S$ is the spectrum of a dvr (and some other hypotheses). The vanishing of this obstruction is shown to be a sufficient condition for étale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

Degree three invariants for semisimple groups of types B, C, and D

We determine the group of reductive cohomological degree 3 invariants of all split semisimple groups of types B, C, and D. We also present a complete description of such cohomological invariants. As an application, we show that the group of degree 3 unramified cohomology of the classifying space BG is trivial for all split semisimple groups G of types B, C, and D.

Mixed motives and geometric representation theory in equal characteristic

Let $\mathbb{k}$ be a field of characteristic $p$. We introduce a formalism of mixed sheaves with coefficients in $\mathbb{k}$ and showcase its use in representation theory. More precisely, we construct for all quasi-projective schemes $X$ over an algebraic closure of $\mathbb{F}_p$ a $\mathbb{k}$-linear triangulated category of motives on $X$. Using work of Ayoub (2007), Cisinski-Deglise (2012) and Geisser-Levine (2000), we show that this system of categories has a six functors formalism and computes higher Chow groups. Indeed, it behaves similarly to other categories of sheaves that one is used to. We attempt to make its construction also accessible to non-experts. We then consider the subcategory of stratified mixed Tate motives defined for affinely stratified varieties $X$, discuss perverse and parity motives and prove formality results. As an example, we combine these results and Soergel (2000) to construct a geometric and graded version of Soergel's modular category $\mathscr O(G)$, consisting of rational representations of a split semisimple group $G/\mathbb{k}$, and thereby equip it with a full six functor formalism (see Riche-Soergel-Williamson (2014) and Achar-Riche (2016) for other approaches). The main idea of using motives in geometric representation theory in this way as well as many results about stratified mixed Tate motives are directly borrowed from Soergel and Wendt, who tell the story in characteristic zero.

Triviality properties of principal bundles on singular curves

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected but it can be made to vanish, if $S$ is the spectrum of a dvr (and some other hypotheses), by a faithfully flat base change. The vanishing of this obstruction is shown to be a sufficient condition for etale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

Wonderful compactifications of Bruhat-Tits buildings

Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.

Cohomological invariants of central simple algebras

We determine the indecomposable degree cohomological invariants of tuples of central simple algebras with linear relations. Equivalently, we determine the degree reductive cohomological invariants of all split semisimple groups of type .

Chow groups of products of Severi-Brauer varieties and invariants of degree 3

We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension 2 2 Chow groups of a product of Severi-Brauer varieties. In particular, for any n ≥ 2 n\geq 2 we completely determine the degree 3 3 invariants of a split semisimple group, the quotient of ( S L 2 ) n (\mathbf {SL}_{2})^{n} by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants of a split semisimple group of type A.

Effective decay of multiple correlations in semidirect product actions

We prove effective decay of certain multiple correlation coefficients for measure preserving, mixing Weyl chamber actions of semidirect products of semisimple groups with $G$-vector spaces. These estimates provide decay for actions in split semisimple groups of higher rank.

Some Examples of Hecke Algebras for Two-Dimensional Local Fields

Let K be a local non-archimedian field, F = K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group G = G(F) and its central extension Ĝ. For instance our spherical Hecke algebra corresponds to the subgroup G (A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

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

Let G be a split semisimple group over a finite field F q , F the field F q (t) of rational functions in t with coefficients in F q and A the adèles of F . We describe the irreducible automorphic representations of G( A ) which have non-zero vectors invariant under Iwahori subgroups at two places and under maximal compact subgroups at all other places in terms of the irreducible square-integrable representations of an Iwahori–Hecke algebra associated to G and the Satake isomorphism.

Explicit realization of a metaplectic representation

F + 2. In [W] Weil explicitly con- structed a model of a genuine unitary representation 0 of the two-fold covering group Sp of the symplectic group Sp over F. In particular, the existence of the covering group Sp was first proven in [W]. It is now known (see, e.g., [M]) how to construct r-fold covering groups of split semi-simple groups over a field F ~ C containing a primitive rth root of unity. In particular, when r = 2, such F has

Fourier inversion on Borel subgroups of Chevalley groups: the symplectic group case

In recent papers, the author and J. A. Wolf have developed the Plancherel theory of parabolic subgroups of real reductive Lie groups. This includes describing the irreducible unitary representations, computing the Plancherel measure, and-since parabolic groups are nonunimodular-explicating the (unbounded) Dixmier-Pukanszky operator that appears in the Plancherel formula. The latter has been discovered to be a special kind of pseudodifferential operator. In this paper, the author considers the problem of extending this analysis to parabolic subgroups of semisimple algebraic groups over an arbitrary local field. Thus far he has restricted his attention to Borel subgroups (i.e. minimal parabolics) in Chevalley groups (i.e. split semisimple groups). The results he has obtained are described in this paper for the case of the symplectic group. The final result is (perhaps surprisingly), to a large extent, independent of the local field over which the group is defined. Another interesting feature of the work is the description of the “pseudodifferential” Dixmier-Pukanszky operator in the nonarchimedean situation.