Overgroups of Levi subgroups I. The case of abelian unipotent radical

The Saito–Kurokawa lifting of automorphic representations from PGL(2) to the projective symplectic group of similitudes PGSp(4) of genus 2 is studied using the Fourier summation formula (an instance of the "relative trace formula"), thus characterizing the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F) associated to a quadratic extension E of the global base field F, and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup. The image is nongeneric and almost everywhere nontempered, violating a naive generalization of the Ramanujan conjecture. Technical advances here concern the development of the summation formula and matching of relative orbital integrals.

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Calculating Galois groups of third-order linear differential equations with parameters

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500389 ◽

2018 ◽

Vol 20 (04) ◽

pp. 1750038

Author(s):

Andrei Minchenko ◽

Alexey Ovchinnikov

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Galois Group ◽

Galois Groups ◽

Linear Differential Equations ◽

Unipotent Radical ◽

Third Order ◽

Differential Galois Group ◽

Bessel Differential Equation ◽

Linear Differential

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

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THE MINIMAL MODULAR FORM ON QUATERNIONIC

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748020000213 ◽

2020 ◽

pp. 1-34

Author(s):

Aaron Pollack

Keyword(s):

Modular Form ◽

Eisenstein Series ◽

Explicit Form ◽

Modular Forms ◽

Fourier Expansion ◽

Reductive Group ◽

Automorphic Function ◽

Unipotent Radical ◽

Exceptional Groups ◽

Dynkin Type

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$ -valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$ , which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$ . We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$ , which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$ .

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The automorphisms of the unipotent radical of certain parabolic subgroups of GL(1 + l, K)

Journal of Algebra ◽

10.1016/0021-8693(85)90039-0 ◽

1985 ◽

Vol 96 (1) ◽

pp. 54-77

Author(s):

Hoe Peng Khor

Keyword(s):

Unipotent Radical ◽

Parabolic Subgroups

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Growth of linear semigroups

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700037368 ◽

1996 ◽

Vol 60 (1) ◽

pp. 18-30 ◽

Cited By ~ 2

Author(s):

Jan Okniński

Keyword(s):

Finite Rank ◽

Polynomial Growth ◽

Growth Function ◽

Unipotent Radical ◽

Maximal Degree ◽

Finitely Generated ◽

Group Of Fractions

AbstractWe show that the growth function of a finitely generated linear semigroup S ⊆ Mn(K) is controlled by its behaviour on finitely many cancellative subsemigroups of S. If the growth of S is polynomially bounded, then every cancellative subsemigroup T of S has a group of fractions G ⊆ Mn (K) which is nilpotent-by-finite and of finite rank. We prove that the latter condition, strengthened by the hypothesis that every such G has a finite unipotent radical, is sufficient for S to have a polynomial growth. Moreover, the degree of growth of S is then bounded by a polynomial f(n, r) in n and the maximal degree r of growth of finitely generated cancellative T ⊆ S.

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Extensions panachées autoduales

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is013001030jkt213 ◽

2013 ◽

Vol 11 (2) ◽

pp. 393-411 ◽

Cited By ~ 1

Author(s):

Daniel Bertrand

Keyword(s):

Equivalence Classes ◽

Unipotent Radical ◽

Self Duality ◽

Tannakian Category ◽

Monodromy Groups

AbstractWe study self-duality of Grothendieck's blended extensions in the context of a tannakian category. The set of equivalence classes of symmetric, resp. antisymmetric, blended extensions is naturally endowed with a torsor structure, which enables us to compute the unipotent radical of the associated monodromy groups in various situations.

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