Split Torus Research Articles

Regular elements in bases of Hecke algebras

Let G˜=GL(n,F¯q) where q is a power of a prime. Let F be the standard Frobenius map of G˜. Let B˜ denote the Borel subgroup of G˜ of upper triangular matrices in G˜ and let T˜ denote the maximal split torus of G˜ contained in B˜. Let W denote the Weyl group of the B˜,N˜ pair where N˜=NG˜(T˜). Let w∈W and let w˙ denote an element in the pre-image of the natural surjection map from NG˜(T˜) to W. The double coset B˜w˙B˜ contains only regular elements if and only if w is a Coxeter element of minimal length. In the following we consider an element w∈W that is not of minimal length and thus B˜wB˜ contains nonregular elements. We explicitly describe certain cosets that are subsets of B˜wB˜ and contain only regular elements. These cosets are chosen for their importance in the construction of a basis of a Hecke algebra associated with G=G˜F.

Expanding Cone and Applications to Homogeneous Dynamics

Abstract Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on a finite volume homogeneous space $G/\Gamma $ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma $ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In this paper, we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Rühr, we prove a multiple ergodic theorem with an error rate.

The value of the global intertwining operators on spherical vectors

Let F be a global field, G an unramified quasi-split reductive group over F and chi an everywhere unramified automorphic character of a maximal maximally split torus of G. Using Langlands-Shahidi theory, we compute the meromorphic function defined by the action of a global standard intertwining operator associated to chi on a spherical vector and show that the ratio of its poles in the positive Weyl chamber is well behaved.

Relative trace formulas for unitary hyperbolic spaces

We elaborate an explicit version of the relative trace formula on $\PGL(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution result in the level aspect for Satake parameters of holomorphic Hilbert cusp forms weighted by central $L$-values, and (ii) a bound of quadratic base change $L$-functions for Hilbert cusp forms with a subconvex exponent in the weight aspect.

Toric codes and lattice ideals

Let X be a complete simplicial toric variety over a finite field Fq with homogeneous coordinate ring S=Fq[x1,…,xr] and split torus TX≅(Fq⁎)n. We prove that submonoids of TX are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determining this parametrization if the submonoid is the zero locus of a lattice ideal in the torus. We also show that vanishing ideals of submonoids of TX are radical homogeneous lattice ideals of dimension r−n. We identify the lattice corresponding to a degenerate torus in X and completely characterize when its lattice ideal is a complete intersection. We compute dimension and length of some generalized toric codes defined on these degenerate tori.

Heights and representations of split tori

Let 𝔾 m d denote the d-dimensional split torus defined over a number field k. To each 𝔾 m d -module E we associate a height function h E defined by means of the spectral height on GL(E). This gives rise to a height pairing between the monoid of irreducible 𝔾 m d -modules of 𝔾 m d and the group 𝔾 m d (k ¯). Our main results are a characterization of those 𝔾 m d -modules E for which h E satisfeis Northcott’s finiteness theorem, the determination of the kernels of the height pairing, as well as, for a few special classes of 𝔾 m d -modules, of the group of automorphisms that preserve h E .

A new look at the decomposition of unipotents and the normal structure of Chevalley groups

The current article continues a series of papers on decomposition of unipotents and its applications. Let $G(\Phi,R)$ be a Chevalley group with a reduced irreducible root system $\Phi$ over a commutative ring $R$. Fix $h\in G(\Phi,R)$. Call an element $a\in G(\Phi,R)$ good, if it lies in the unipotent radical of a parabolic subgroup whereas the conjugate to $a$ by $h$ belongs to another proper parabolic subgroup (here we assume that all parabolics contain a given split maximal torus). Decomposition of unipotents is a representation of a root unipotent element as a product of good elements. Existence of such a decomposition implies a simple proof of the normality of the elementary subgroup and description of the normal structure of $G(\Phi,R)$. However, the decomposition is available not for all root systems. In the current article we show that for the proof of the standard normal structure it suffices to construct only one good element for the generic element of the scheme $G(\Phi,_-)$, and construct such a good element. The question whether good elements generate the whole elementary group will be addressed in a separate paper (the first version of the article was in Russian).

Characters of odd degree

We prove the McKay conjecture on characters of odd degree. A major step in the proof is the verication of the inductive McKay condition for groups of Lie type and primes ‘ such that a Sylow ‘-subgroup or its maximal normal abelian subgroup is contained in a maximally split torus by means of a new equivariant version of Harish-Chandra induction. Specics of characters of odd degree, namely, that most of them lie in the principal Harish-Chandra series, then allow us to deduce from it the McKay conjecture for the prime 2, hence for characters of odd degree.

Non-stable K1-functors of Multiloop Groups

AbstractLet k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials . Assume that G contains amaximal R-torus, and that every semisimple normal subgroup of G contains a two-dimensional split torus G2m. We show that the natural map of non-stable K1-functors, also called Whitehead groups, KG1(R) → KG1 ( k((x1)) … ((xn))) is injective, and an isomorphism if G is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii–Neher) and the subgroup generated by exponential automorphisms.

From pro-p Iwahori–Hecke modules to (φ,Γ)-modules, I

Let ${\mathfrak o}$ be the ring of integers in a finite extension $K$ of ${\mathbb Q}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb Q}_p$, let $T$ be a maximal split torus in $G$. Let ${\mathcal H}(G,I_0)$ be the pro-$p$-Iwahori Hecke ${\mathfrak o}$-algebra. Given a semiinfinite reduced chamber gallery (alcove walk) $C^{({\bullet})}$ in the $T$-stable apartment, a period $\phi\in N(T)$ of $C^{({\bullet})}$ of length $r$ and a homomorphism $\tau:{\mathbb Z}_p^{\times}\to T$ compatible with $\phi$, we construct a functor from the category ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ of finite length ${\mathcal H}(G,I_0)$-modules to \'{e}tale $(\varphi^r,\Gamma)$-modules over Fontaine's ring ${\mathcal O}_{\mathcal E}$. If $G={\rm GL}_{d+1}({\mathbb Q}_p)$ there are essentially two choices of ($C^{({\bullet})}$, $\phi$, $\tau$) with $r=1$, both leading to a functor from ${\rm Mod}^{\rm fin}({\mathcal H}(G,I_0))$ to \'{e}tale $(\varphi,\Gamma)$-modules and hence to ${\rm Gal}_{{\mathbb Q}_p}$-representations. Both induce a bijection between the set of absolutely simple supersingular ${\mathcal H}(G,I_0)\otimes_{\mathfrak o} k$-modules of dimension $d+1$ and the set of irreducible representations of ${\rm Gal}_{{\mathbb Q}_p}$ over $k$ of dimension $d+1$. We also compute these functors on modular reductions of tamely ramified locally unitary principal series representations of $G$ over $K$. For $d=1$ we recover Colmez' functor (when restricted to ${\mathfrak o}$-torsion ${\rm GL}_{2}({\mathbb Q}_p)$-representations generated by their pro-$p$-Iwahori invariants)

Covers of tori over local and global fields

Langlands has described the irreducible admissible representations of $T$, when $T$ is the group of points of an algebraic torus over a local field. Also, Langlands described the automorphic representations of $T_{\Bbb{A}}$ when $T_{\Bbb{A}}$ is the group of adelic points of an algebraic torus over a global field $F$. We describe irreducible (in the local setting) and automorphic (in the global setting) $\epsilon$-genuine representations for ``covers'' of tori, also known as ``metaplectic tori,'' which arise from a framework of Brylinski and Deligne. In particular, our results include a description of spherical Hecke algebras in the local unramified setting, and a global multiplicity estimate for automorphic representations of covers of split tori. For automorphic representations of covers of split tori, we prove a multiplicity-one theorem.

Additive group actions on affine [formula omitted]-varieties of complexity one in arbitrary characteristic

Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T=Gmn stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero.With the assumption that the characteristic of k is positive, we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine T-varieties normalized up to a Frobenius map. As a preliminary result, we provide a complete description of these Ga-actions in the toric situation.

Torus quotients as global quotients by finite groups

This article is motivated by the following local-to-global question: is every variety with tame quotient singularities globally the quotient of a smooth variety by a finite group? We show that this question has a positive answer for all quasi-projective varieties which are expressible as a quotient of a smooth variety by a split torus (e.g. simplicial toric varieties). Although simplicial toric varieties are rarely toric quotients of smooth varieties by finite groups, we give an explicit procedure for constructing the quotient structure using toric techniques. This result follow from a characterization of varieties which are expressible as the quotient of a smooth variety by a split torus. As an additional application of this characterization, we show that a variety with abelian quotient singularities may fail to be a quotient of a smooth variety by a finite abelian group. Concretely, we show that $\mathbb{P}^2/A_5$ is not expressible as a quotient of a smooth variety by a finite abelian group.

The Normalizer of the Elementary Net Group Associated with a Nonsplit Torus in the General Linear Group Over a Field

UDC 512.5 In this paper, the normalizer N (σ) of the elementary net group E(σ) associated with a nonsplit maximal torus T (d) in the general linear group GL(n, k) over a field k of odd characteristic is computed. The nonsplit maximal torus T = T (d) is determined by the radical extension k( n √ d) of degree n of the ground field k (minisotropic torus). Bibliography: 18 titles. The problem of describing the overgroups of a split maximal torus in linear groups and in Chevalley groups is virtually solved. The fundamental contribution to the solution of this problem was made by the Leningrad–Petersburg algebraic school (Z. I. Borevich, N. A. Vavilov, and their pupils; for example, see [1, 2, 4–6]). The description of overgroups of a nonsplit torus is a significantly less investigated area. At the present time, a complete description of overgroups of a nonsplit torus has been obtained only for some special fields such as finite or local fields. For finite fields, this was carried out in papers by W. Kantor and G. Seitz (see [15, 17, 18]). Important results on overgroups of a nonsplit torus for local and global fields have been obtained by V. P. Platonov [16]. The case of the field of real numbers has been considered in [14]. For arbitrary fields, the study of overgroups of a nonsplit torus has been conducted in papers [3, 7–13]. The present paper is devoted to an investigation of intermediate subgroups of the general linear group that contain the nonsplit maximal torus related to a radical extension K = k( n √ d) of the ground field k, d ∈ k. The elementary net groups E(σ) associated with the torus T = T (d) and their normalizer N (σ) (which is an overgroup of the nonsplit torus) in the general linear group G =G L( n, k) are of primary importance in studies of the intermediate subgroups mentioned above (see [7–11]). In the present paper, we calculate the normalizer N (σ) of the elementary net group E(σ) associated with a nonsplit maximal torus T (d) in the general linear group GL(n, k )o ver afi eld k of odd characteristic. Moreover, the nonsplit maximal torus T = T (d) is determined by the radical degree-n extension k( n √ d) of the ground field k (minisotropic torus).

Asymptotic behaviors of means of central values of automorphic L-functions for [formula omitted

Let A be the adele ring of a totally real algebraic number field F. We push forward an explicit computation of a relative trace formula for periods of automorphic forms along a split torus in GL(2) from a square-free level case done by Masao Tsuzuki [20], to an arbitrary level case. By using a relative trace formula, we study central values of automorphic L-functions for cuspidal automorphic representations of GL(2,A) corresponding to Maass forms with arbitrary level.

Smoothness of limit functors

Let S be a scheme. Assume that we are given an action of the one dimensional split torus \(\mathbb {G}_{m,S}\) on a smooth affine S-scheme \(\mathfrak {X}\). We consider the limit (also called attractor) subfunctor \(\mathfrak {X}_{\lambda }\) consisting of points whose orbit under the given action ‘admits a limit at 0’. We show that \(\mathfrak {X}_{\lambda }\) is representable by a smooth closed subscheme of \(\mathfrak {X}\). This result generalizes a theorem of Conrad et al. (Pseudo-reductive groups (2010) Cambridge Univ. Press) where the case when \(\mathfrak {X}\) is an affine smooth group and \(\mathbb {G}_{m,S}\) acts as a group automorphisms of \(\mathfrak {X}\) is considered. It also occurs as a special case of a recent result by Drinfeld on the action of \(\mathbb {G}_{m,S}\) on algebraic spaces (Proposition 1.4.20 of Drinfeld V, On algebraic spaces with an action of \(\mathfrak {G}_{m}\), preprint 2013) in case S is of finite type over a field.

On measures invariant under tori on quotients of semisimple groups

We classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple S-algebraic groups invariant under a maximal split torus in at least one simple local factor and show that the algebraic support of such a measure splits into the product of four homogeneous spaces: a torus, a homogeneous space on which the measure is (up to nite index) the Haar measure, a product of homogeneous spaces on each of which the action degenerates to a rank one action, and a homogeneous space in which every element of the action acts with zero entropy.

Twisted Geometric Langlands Correspondence for a Torus

Let T be a split torus over local or global function field. The theory of Brylinski-Deligne gives rise to the metaplectic central extensions of T by a finite cyclic group. The representation theory of these metaplectic tori has been developped to some extent in the works of M. Weissman, G. Savin, W. T. Gan, P. McNamara and others. In this paper we propose a geometrization of this theory in the framework of the geometric Langlands program (in the everywhere nonramified case).

Polyhedral divisors and torus actions of complexity one over arbitrary fields

We show that the presentation of affine T-varieties of complexity one in terms of polyhedral divisors holds over an arbitrary field. We also describe a class of multigraded algebras over Dedekind domains. We study how the algebra associated with a polyhedral divisor changes when we extend the scalars. As another application, we provide a combinatorial description of affine G-varieties of complexity one over a field, where G is a (not necessarily split) torus, by using elementary facts on Galois descent. This class of affine G-varieties is described via a new combinatorial object, which we call (Galois) invariant polyhedral divisor.

Holomorphic jets in symplectic manifolds

We define pointwise partial differential relations for holomorphic discs. Given a relative homotopy class, a relation, and a generic almost complex structure we provide the moduli space of discs which have an injective point with the structure of a smooth manifold. Applications to the local behaviour are given and an adjunction inequality for singularities is derived. Moreover, we show that for a coordinate class of a monotone Lagrangian split torus generically the number of non-immersed holomorphic discs is even.