Nonsplit Torus Research Articles

Orthogonality of invariant vectors

Let G be a finite group with given subgroups H and K. Let π be an irreducible complex representation of G such that its space of H-invariant vectors as well as the space of K-invariant vectors are both one dimensional. Let vH (resp. vK) denote an H-invariant (resp. K-invariant) vector of unit norm in a given G-invariant inner product 〈,〉π on π. Our interest is in computing the square of the absolute value of 〈vH,vK〉π. This is the correlation constant c(π;H,K) defined by Gross [3]. In this paper, we study this question for G=GL2(Fq), where Fq is the finite field of q=pm elements of odd characteristic p, H is its split torus and K is a non-split torus. The first main theorem of this paper gives an explicit formula for |〈vH,vK〉π|2 modulo p. The key idea here is to analyse the mod p reduction of π. The second main theorem relates the behaviour of 〈vH,vK〉π under Shintani base change and gives a sufficient condition for 〈vH,vK〉π to vanish for an irreducible representation π=BC(τ) of PGL2(E), in terms of the epsilon factor of the base changing representation τ of PGL2(F), where E/F is a finite extension of finite fields.

Mass equidistribution on the torus in the depth aspect

In this paper we prove the equidistribution of the restriction of the mass of automorphic newforms to a nonsplit torus in the depth aspect. This result is better than the current known results on the similar problem in the eigenvalue aspect. The method is relatively elementary and makes use of the known effective QUE result in the depth aspect.

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).

On the Construction of Finite Oscillator Dictionary

A finite oscillator dictionary which has important applications in sequences designs and the compressive sensing was introduced by Gurevich, Hadani, and Sochen in [3]. In this paper, closed formulae of the finite split oscillator dictionary 𝔖 s are revisited by a simple proof, the structure of nonsplit tori of the group SL(2, 𝔽 p ) is studied, and an explicit algorithm for computing the finite nonsplit oscillator dictionary 𝔖 ns is described.

Toroidal automorphic forms for function fields

The space of toroidal automorphic forms was introduced by Zagier in 1979. Let F be a global field. An automorphic form on GL(2) is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems from the fact (amongst others) that an Eisenstein series of weight s is toroidal if s is a non-trivial zero of the zeta function, and thus a connection with the Riemann hypothesis is established. In this paper, we concentrate on the function field case. We show the following results. The (n −1)-th derivative of a non-trivial Eisenstein series of weight s and Hecke character x is toroidal if and only if L(x, s+1/2) vanishes in s to order at least n (for the “only if” part we assume that the characteristic of F is odd). There are no non-trivial toroidal residues of Eisenstein series. The dimension of the space of derivatives of unramified Eisenstein series equals h(g −1)+1 if the characteristic is not 2; in characteristic 2, the dimension is bounded from below by this number. Here g is the genus and h is the class number of F. The space of toroidal automorphic forms is an admissible representation and every irreducible subquotient is tempered.

Toric varieties and spherical embeddings over an arbitrary field

We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We characterize those combinatorial objects corresponding to varieties defined over an arbitrary field k. Then we provide some situations where toric varieties over k are classified by Galois-stable fans, and spherical embeddings over k by Galois-stable colored fans. Moreover, we construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over k whose fan is Galois-stable but which admits no k-form. In the spherical setting, we offer an example of a spherical homogeneous space X 0 over R of rank 2 under the action of SU ( 2 , 1 ) and a smooth embedding of X 0 whose fan is Galois-stable but which admits no R -form.

Remarks on $\aone$-homotopy groups of smooth toric models

We extend previous results on A 1 -homotopy groups of smooth proper toric varieties to the case of smooth proper toric models, i.e., smooth proper equivariant compactifications of possibly non-split tori, in characteristic 0.

On subgroups of the general linear group that contain a maximal nonsplit torus

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let $$ K = k\left( {\sqrt[n]{d}} \right) $$ be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σR denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σR denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σR) be the subgroup generated by all transvections from the net group G(σR). In the paper it is proved that the product TE(σR) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σR,then TE(σR) ≤ H ≤ N(σR),where N(σR) is the normalizer of the elementary net group E(σR) in G. For the normalizer N(σR),the formula N(σR)= TG(σR) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.

Toroidal automorphic forms for some function fields

Zagier introduced toroidal automorphic forms to study the zeros of zeta functions: an automorphic form on GL 2 is toroidal if all its right translates integrate to zero over all non-split tori in GL 2 , and an Eisenstein series is toroidal if its weight is a zero of the zeta function of the corresponding field. We compute the space of such forms for the global function fields of class number one and genus g ⩽ 1 , and with a rational place. The space has dimension g and is spanned by the expected Eisenstein series. We deduce an “automorphic” proof for the Riemann hypothesis for the zeta function of those curves.

Ramanujan graphs on cosets of [formula omitted

In this paper, we study Cayley graphs on PGL 2 ( F q ) mod the unipotent subgroup, the split and nonsplit tori, respectively. Using the Kirillov models of the representations of PGL 2 ( F q ) of degree greater than one, we obtain explicit eigenvalues of these graphs and the corresponding eigenfunctions. Character sum estimates are then used to conclude that two types of the graphs are Ramanujan, while the third is almost Ramanujan. The graphs arising from the nonsplit torus were previously studied by Terras et al. We give a different approach here.

The Lower Garland of Subgroup Lattices in Linear Groups

The lower garland of some lattices of intermediate subgroups in linear groups is described. The results are applied to the case of subgroup lattices in general and special linear groups over a class of rings that contain the group of rational points T of a maximal nonsplit torus in the corresponding algebraic group. It turns out that the lower garlands of these lattices coincide with their intervals consisting of subgroups lying between T and its normalizer. Bibliography: 10 titles.

The subgroups of the group gl(2,k) that contain a nonsplit maximal torus

In the group GL(2,k), the lattice of the intermediate subgroups that contain the maximal nonsplit torus is studied for a field k of characteristic different from 2. In a number of cases, when formulating the results some additional restrictions are imposed. Bibliography: 3 titles.

Lattices of subgroups inGL(2,Q) containing a non-split torus

In a general linear group of degree 2 over field of rational numbersQ we consider subgroups containing a maximal non-split torus T=T(d) (that is, the image in G=GL(2,Q) of a multiplicative group of quadratic fieldQ(√d) under a regular imbedding). We investigate lattices Lat(d) depending on d of intermediate subgroups as well as structure of these subgroups. For any subgroup H from Lat(d) the factor-group Ng(H)/H is an Abelian group of exponent 2. The chain of successive normalizes H ≤NG(H) ≤NG(NG(H))≤... is stabilized in the final step. The dual descending chains H ≥TH ≥ T(tH) ≥...of successive normal closuresof torus T do not always terminate. A termination of such descending chains for all H from Lat(d) holds ifand only if d ≡ 1(rood 4). All the connected components of the graph of the normality relation on Lat(d)(garlands) are in a bijective correspondence to all the self-normalizing intermediate subgroups. We obtain adescription of all the self-normalizing and all the complete intermediate subgroups (F is complete if 7 FH = F).The proofs of the results are not given.