scholarly journals Bijective PC-maps of the unipotent radical of the Borel subgroup of the classical symplectic group

2019 ◽  
pp. 1-36
Author(s):  
Alexander Shchegolev
Keyword(s):  
Symplectic Group ◽  
Borel Subgroup ◽  
Unipotent Radical

scholarly journals Commuting Varieties for Nilpotent Radicals

2018 ◽  
Vol 62 (2) ◽  
pp. 559-594
Author(s):  
Rolf Farnsteiner
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Unipotent Radical ◽  
Closed Field ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Commuting Variety ◽  
Good For

AbstractLetUbe the unipotent radical of a Borel subgroup of a connected reductive algebraic groupG, which is defined over an algebraically closed fieldk. In this paper, we extend work by Goodwin and Röhrle concerning the commuting variety of Lie(U) for Char(k) = 0 to fields whose characteristic is good forG.

scholarly journals Coordinate rings and birational charts

2022 ◽  
Vol 26 (1) ◽  
pp. 1-16
Author(s):  
Sergey Fomin ◽  
George Lusztig
Keyword(s):  
Algebraic Group ◽  
Borel Subgroup ◽  
Total Positivity ◽  
Unipotent Radical ◽  
Reductive Groups ◽  
Content Type ◽  
Coordinate Rings ◽  
Simply Connected

Let G G be a semisimple simply connected complex algebraic group. Let U U be the unipotent radical of a Borel subgroup in  G G . We describe the coordinate rings of U U (resp., G / U G/U , G G ) in terms of two (resp., four, eight) birational charts introduced by Lusztig [Total positivity in reductive groups, Birkhäuser Boston, Boston, MA, 1994; Bull. Inst. Math. Sin. (N.S.) 14 (2019), pp. 403–459] in connection with the study of total positivity.

scholarly journals Residual automorphic representations of Sp4

1992 ◽  
Vol 127 ◽  
pp. 15-47 ◽  
Author(s):  
Takao Watanabe
Keyword(s):  
Hilbert Space ◽  
Symplectic Group ◽  
Maximal Compact Subgroup ◽  
Borel Subgroup ◽  
Compact Subgroup ◽  
Algebraic Number Field ◽  
Orthogonal Decomposition ◽  
Parabolic Subgroups ◽  
Automorphic Representations ◽  
Adele Group

Let G = Sp4 be the symplectic group of degree two defined over an algebraic number field F and K the standard maximal compact subgroup of the adele group G (A). By the general theory of Eisenstein series ([14]), one knows that the Hilbert space L2(G(F)\G(A)) has an orthogonal decomposition of the formL2(G(F)\G(A)) = L2(G) ⊕ L2(B) ⊕ L2(P1) ⊕ L2(P1),where B is a Borel subgroup and Pi are standard maximal parabolic subgroups in G for i = 1,2. The purpose of this paper is to study the space L2d(B) associated to discrete spectrurns in L2(B).

The stabilizer of two-dimensional vector space of 27-dimensional module of type E6 over a field of characteristic two

2020 ◽  
pp. 2150151
Author(s):  
Yousuf Alkhezi ◽  
Mshhour Bani-Ata
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Borel Subgroup ◽  
Unipotent Radical ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Type Formula ◽  
Characteristic Two ◽  
Dimensional Module

The purpose of this paper is to use the notion of [Formula: see text]-sets (cocliques) introduced by the second author in [S. Aldhafeeri and M. Bani-Ata, On the construction of Lie-algebras of type [Formula: see text] for fields of characteristic two, Beitrag Zur Algebra und Geometry 58 (2017) 529–534.] and using Levi components and unipotent radical subgroups of [Formula: see text] to give an elementary and self-contained construction of the stabilizer of two dimensional vector space of 27-dimensional module of type [Formula: see text] over a field of characteristic two. This stabilizer is in fact the maximal parabolic subgroup [Formula: see text] of [Formula: see text] or a Borel subgroup. This construction is elementary on the account that we use not more than little naive linear algebra notions. For more information one can see [M. E. Aschbacher, The 27-dimensional module for [Formula: see text], 1, Invent. Math. 89 (1987) 159–195; M. E. Aschbacher, The 27-dimensional module for [Formula: see text], II, J. London Math. Soc. 37 (1988) 275–293; M. Bani-Ata, On Lie algebras of type [Formula: see text] and [Formula: see text] over finite fields of characteristic two, Preprint; B. Cooperstein, Subgroups of the group [Formula: see text] which are generated by root-subgroups, J. Algebra 46 (1977) 355–388.].

scholarly journals Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups

2020 ◽  
pp. 1-48
Author(s):  
Dmitry Gourevitch ◽  
Henrik P. A. Gustafsson ◽  
Axel Kleinschmidt ◽  
Daniel Persson ◽  
Siddhartha Sahi
Keyword(s):  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Borel Subgroup ◽  
Automorphic Representation ◽  
Cusp Forms ◽  
Unipotent Radical ◽  
Finite Cover ◽  
Explicit Formulas ◽  
Automorphic Representations ◽  
Minimal Automorphic

Abstract In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let $\pi $ be a minimal or next-to-minimal automorphic representation of G. We prove that any $\eta \in \pi $ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on $\operatorname {GL}_n$ . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type $D_5$ and $E_8$ with a view toward applications to scattering amplitudes in string theory.

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

-ADIC -FUNCTIONS FOR ORDINARY FAMILIES ON SYMPLECTIC GROUPS

2019 ◽  
Vol 19 (4) ◽  
pp. 1287-1347 ◽  
Author(s):  
Zheng Liu
Keyword(s):  
Eisenstein Series ◽  
Symplectic Group ◽  
Symplectic Groups ◽  
Simple Strategy ◽  
Doubling Method

We construct the $p$-adic standard $L$-functions for ordinary families of Hecke eigensystems of the symplectic group $\operatorname{Sp}(2n)_{/\mathbb{Q}}$ using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group $\operatorname{Sp}(4n)_{/\mathbb{Q}}$, which guarantees the nonvanishing of local zeta integrals and allows us to $p$-adically interpolate the restrictions of the Siegel Eisenstein series to $\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$.

scholarly journals Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

2009 ◽  
Vol 146 (1) ◽  
pp. 21-57 ◽  
Author(s):  
Harald Grobner
Keyword(s):  
Algebraic Group ◽  
Eisenstein Series ◽  
Symplectic Group ◽  
Automorphic Forms ◽  
General Theorem ◽  
Simple Algebraic Group ◽  
Ramanujan Conjecture ◽  
Eisenstein Cohomology ◽  
Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

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