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Journal articles on the topic 'Unipotent automorphism'

Author: Grafiati

Published: 10 March 2023

Last updated: 31 July 2025

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1

Varea, V. R., and J. J. Varea. "On Automorphisms and Derivations of a Lie Algebra." Algebra Colloquium 13, no. 01 (2006): 119–32. http://dx.doi.org/10.1142/s1005386706000149.

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We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.

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2

Li, Sichen. "Derived length of zero entropy groups acting on projective varieties in arbitrary characteristic — A remark to a paper of Dinh-Oguiso-Zhang." International Journal of Mathematics 31, no. 08 (2020): 2050059. http://dx.doi.org/10.1142/s0129167x20500597.

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Let [Formula: see text] be a projective variety of dimension [Formula: see text] over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of [Formula: see text]. Let [Formula: see text] be a group of zero entropy automorphisms of [Formula: see text] and [Formula: see text] the set of elements in [Formula: see text] which are isotopic to the identity. We show that after replacing [Formula: see text] by a suitable finite-index subgroup, [Formula: see text] is a unipotent group of the derived length at most

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3

Lawther, Ross, Martin W. Liebeck, and Gary M. Seitz. "Outer unipotent classes in automorphism groups of simple algebraic groups." Proceedings of the London Mathematical Society 109, no. 3 (2014): 553–95. http://dx.doi.org/10.1112/plms/pdu011.

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4

REGETA, ANDRIY. "CHARACTERIZATION OF n-DIMENSIONAL NORMAL AFFINE SLn-VARIETIES." Transformation Groups 27, no. 1 (2022): 271–93. http://dx.doi.org/10.1007/s00031-022-09701-3.

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AbstractWe show that any normal irreducible affine n-dimensional SLn-variety X is determined by its automorphism group seen as an ind-group in the category of normal irreducible affine varieties. In other words, if Y is an irreducible affine normal algebraic variety such that Aut(Y) ≃ Aut(X) as an ind-group, then Y ≃ X as a variety. If we drop the condition of normality on Y , then this statement fails. In case n ≥ 3, the result above holds true if we replace Aut(X) by 𝒰(X), where 𝒰(X) is the subgroup of Aut(X) generated by all one-dimensional unipotent subgroups. In dimension 2 we have some i

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5

Waldspurger, J. L. "Le Groupe GLn Tordu, Sur un Corps Fini." Nagoya Mathematical Journal 182 (June 2006): 313–79. http://dx.doi.org/10.1017/s002776300002691x.

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AbstractLet q be a finite field, G = GLn(q), θ be the outer automorphism of G, suitably normalized. Consider the non-connected group G ⋊ {1, θ} and its connected component = Gθ. We know two ways to produce functions on , with complex values and invariant by conjugation by G: on one hand, let π be an irreducible representation of G we can and do extend to a representation π+ of G ⋊ {1, θ}, then the restriction trace to of the character of π+ is such a function; on the other hand, Lusztig define character-sheaves a, whose characteristic functions ϕ(a) are such functions too. We consider only “qu

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6

Barbari, P., and A. Kobotis. "On nilpotent filiform Lie algebras of dimension eight." International Journal of Mathematics and Mathematical Sciences 2003, no. 14 (2003): 879–94. http://dx.doi.org/10.1155/s016117120311201x.

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The aim of this paper is to determine both the Zariski constructible set of characteristically nilpotent filiform Lie algebrasgof dimension8and that of the set of nilpotent filiform Lie algebras whose group of automorphisms consists of unipotent automorphisms, in the variety of filiform Lie algebras of dimension8overC.

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7

Hanzer, Marcela, and Gordan Savin. "Eisenstein Series Arising from Jordan Algebras." Canadian Journal of Mathematics 72, no. 1 (2019): 183–201. http://dx.doi.org/10.4153/cjm-2018-033-2.

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AbstractWe describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

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8

Levchuk, V. M. "Automorphisms of unipotent subgroups of chevalley groups." Algebra and Logic 29, no. 3 (1990): 211–24. http://dx.doi.org/10.1007/bf01979936.

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9

Bavula, V. V., and T. H. Lenagan. "Quadratic and cubic invariants of unipotent affine automorphisms." Journal of Algebra 320, no. 12 (2008): 4132–55. http://dx.doi.org/10.1016/j.jalgebra.2008.07.029.

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10

Gourevitch, Dmitry, Henrik P. A. Gustafsson, Axel Kleinschmidt, Daniel Persson, and Siddhartha Sahi. "A reduction principle for Fourier coefficients of automorphic forms." Mathematische Zeitschrift 300, no. 3 (2021): 2679–717. http://dx.doi.org/10.1007/s00209-021-02784-w.

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AbstractWe consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent

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11

Ginzburg, David. "Certain conjectures relating unipotent orbits to automorphic representations." Israel Journal of Mathematics 151, no. 1 (2006): 323–55. http://dx.doi.org/10.1007/bf02777366.

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12

Feng, Zhicheng, and Britta Späth. "Unitriangular basic sets, Brauer characters and coprime actions." Representation Theory of the American Mathematical Society 27, no. 6 (2023): 115–48. http://dx.doi.org/10.1090/ert/635.

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We show that the decomposition matrix of a given group G G is unitriangular, whenever G G has a normal subgroup N N such that the decomposition matrix of N N is unitriangular, G / N G/N is abelian and certain characters of N N extend to their stabilizer in G G . Using the recent result by Brunat–Dudas–Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix whenever they are related via Bonnafé–Dat–Rouquier’s equivalence to a unipotent block. Th

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13

Khukhro, E. I., and P. Shumyatsky. "Length-type parameters of finite groups with almost unipotent automorphisms." Doklady Mathematics 95, no. 1 (2017): 43–45. http://dx.doi.org/10.1134/s1064562417010124.

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14

Rohde, Jan Christian. "Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy." manuscripta mathematica 131, no. 3-4 (2010): 459–74. http://dx.doi.org/10.1007/s00229-009-0329-5.

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15

Salmon, Andrew. "Unipotent nearby cycles and the cohomology of shtukas." Compositio Mathematica 159, no. 3 (2023): 590–615. http://dx.doi.org/10.1112/s0010437x23007017.

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We give cases in which nearby cycles commute with pushforward from sheaves on the moduli stack of shtukas to a product of curves over a finite field. The proof systematically uses the property that taking nearby cycles of Satake sheaves on the Beilinson–Drinfeld Grassmannian with parahoric reduction is a central functor together with a ‘Zorro's lemma’ argument similar to that of Xue [Smoothness of cohomology sheaves of stacks of shtukas, Preprint (2020), arXiv:2012.12833]. As an application, for automorphic forms at the parahoric level, we characterize the image of tame inertia under the Langl

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16

Lifschitz, Lucy, and Dave Witte. "ON AUTOMORPHISMS OF ARITHMETIC SUBGROUPS OF UNIPOTENT GROUPS IN POSITIVE CHARACTERISTIC." Communications in Algebra 30, no. 6 (2002): 2715–43. http://dx.doi.org/10.1081/agb-120003985.

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17

Levchuk, V. M., and G. S. Suleimanova. "Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups." Proceedings of the Steklov Institute of Mathematics 267, S1 (2009): 118–27. http://dx.doi.org/10.1134/s0081543809070128.

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18

Levchuk, V. M. "Automorphisms of unipotent subgroups of lie type groups of small ranks." Algebra and Logic 29, no. 2 (1990): 97–112. http://dx.doi.org/10.1007/bf02001355.

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19

Voll, Christopher. "IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS." Quarterly Journal of Mathematics 71, no. 3 (2020): 959–80. http://dx.doi.org/10.1093/qmathj/haaa010.

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Abstract We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in M. N. Berman, B. Klopsch and U. Onn (A family of class-2 nilpotent groups, their automorphisms and pro-isomorphic zeta functions, Math. Z. 290 (2018), 909935), in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced and graded ideal zeta functions, as well as representation zeta functions for the unipotent group sc

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20

Gan, Wee Teck, Nadya Gurevich, and Dihua Jiang. "Cubic unipotent Arthur parameters and multiplicities of square integrable automorphic forms." Inventiones Mathematicae 149, no. 2 (2002): 225–65. http://dx.doi.org/10.1007/s002220200210.

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21

FLICKER, YUVAL Z. "CUSP FORMS ON GSp(4) WITH SO(4)-PERIODS." International Journal of Number Theory 07, no. 04 (2011): 855–919. http://dx.doi.org/10.1142/s1793042111004186.

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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, violat

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22

Jiang, Dihua, and Baiying Liu. "On special unipotent orbits and Fourier coefficients for automorphic forms on symplectic groups." Journal of Number Theory 146 (January 2015): 343–89. http://dx.doi.org/10.1016/j.jnt.2014.03.002.

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23

Khor, Hoe Peng. "The automorphisms of the unipotent radical of certain parabolic subgroups of GL(1 + l, K)." Journal of Algebra 96, no. 1 (1985): 54–77. http://dx.doi.org/10.1016/0021-8693(85)90039-0.

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24

Sun, Heng. "Remarks on Certain Metaplectic Groups." Canadian Mathematical Bulletin 41, no. 4 (1998): 488–96. http://dx.doi.org/10.4153/cmb-1998-064-1.

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AbstractWe study metaplectic coverings of the adelized group of a split connected reductive group G over a number field F. Assume its derived group G′ is a simply connected simple Chevalley group. The purpose is to provide some naturally defined sections for the coverings with good properties which might be helpful when we carry some explicit calculations in the theory of automorphic forms on metaplectic groups. Specifically, we1.construct metaplectic coverings of G(A) from those of G′(A);2.for any non-archimedean place v, show the section for a covering of G(Fv) constructed from a Steinberg s

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25

Booher, Jeremy. "Minimally ramified deformations when." Compositio Mathematica 155, no. 1 (2018): 1–37. http://dx.doi.org/10.1112/s0010437x18007546.

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Let $p$ and $\ell$ be distinct primes, and let $\overline{\unicode[STIX]{x1D70C}}$ be an orthogonal or symplectic representation of the absolute Galois group of an $\ell$-adic field over a finite field of characteristic $p$. We define and study a liftable deformation condition of lifts of $\overline{\unicode[STIX]{x1D70C}}$ ‘ramified no worse than $\overline{\unicode[STIX]{x1D70C}}$’, generalizing the minimally ramified deformation condition for $\operatorname{GL}_{n}$ studied in Clozel et al. [Automorphy for some$l$-adic lifts of automorphic mod$l$Galois representations, Publ. Math. Inst. Hau

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26

Hanzer, Marcela. "An explicit construction of automorphic representations of the symplectic group with a given quadratic unipotent Arthur parameter." Monatshefte für Mathematik 177, no. 2 (2014): 235–73. http://dx.doi.org/10.1007/s00605-014-0686-3.

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27

RICHARD, RODOLPHE. "RÉPARTITION GALOISIENNE D'UNE CLASSE D'ISOGÉNIE DE COURBES ELLIPTIQUES." International Journal of Number Theory 09, no. 02 (2012): 517–43. http://dx.doi.org/10.1142/s1793042112501199.

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Dans cet article, on montre que les orbites sous Galois des invariants modulaires associés à des courbes elliptiques complexes sans multiplication complexe variant dans une même classe d'isogénie s'équidistribuent dans la courbe modulaire vers la probabilité hyperbolique. La démonstration repose sur des arguments de théorie ergodique, notamment le théorème de Ratner (cf. [A. Eskin et H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26(1) (2006) 163–167]), ainsi que sur le théorème de l'image ouverte de Serre [J.-P. Serre, Abelian l-Adic Represent

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28

Regeta, Andriy, and Immanuel van Santen. "Maximal commutative unipotent subgroups and a characterization of affine spherical varieties." Journal of the European Mathematical Society, May 8, 2025. https://doi.org/10.4171/jems/1651.

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We describe maximal commutative unipotent subgroups of the automorphism group \mathrm{Aut}(X) of an irreducible affine variety X . Furthermore, we show that a group isomorphism \mathrm{Aut}(X) \allowbreak\to \mathrm{Aut}(Y) maps unipotent elements to unipotent elements, where Y is irreducible and affine. Using this result, we show that the automorphism group detects sphericity and the weight monoid.As an application, we show that an affine toric variety different from an algebraic torus is determined by its automorphism group among normal irreducible affine varieties, and we show that a smooth

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29

Frati, Marco. "Unipotent automorphisms of soluble groups with finite Prüfer rank." Journal of Group Theory 17, no. 3 (2014). http://dx.doi.org/10.1515/jgt-2013-0041.

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30

Parkinson, James, and Hendrik Van Maldeghem. "Automorphisms and opposition in spherical buildings of exceptional type, I." Canadian Journal of Mathematics, July 5, 2021, 1–62. http://dx.doi.org/10.4153/s0008414x21000341.

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Abstract To each automorphism of a spherical building, there is a naturally associated opposition diagram, which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber, then the automorphism is called domestic. In this paper, we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf {E}_6$ , $\mathsf {F}_4$ , and $\mathsf {G}_2$ . Moreover, for all split spherical buildings of exceptional type, we classify (i) the

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31

BONNAFÉ, CÉDRIC. "AUTOMORPHISMS AND SYMPLECTIC LEAVES OF CALOGERO–MOSER SPACES." Journal of the Australian Mathematical Society, October 17, 2022, 1–32. http://dx.doi.org/10.1017/s1446788722000180.

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Abstract We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero–Moser space induced by an element of finite order of the normalizer of the associated complex reflection group. We give a parametrization à la Harish-Chandra of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero–Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper, C. Bonnafé [‘Calogero–Moser spaces vs unipotent representations’,

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32

Frati, Marco. "Unipotent automorphisms of soluble groups with finite Prüfer rank." September 20, 2013. https://doi.org/10.1515/jgt-2013-0041.

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Abstract. In this paper we prove that a unipotent automorphism group H acting upon a soluble group with finite Prüfer rank is nilpotent. Furthermore, if H is finitely generated and the cardinality of a minimal set of generators is t, it is then possible to find an upper bound for the nilpotency class of H which depends only on n and t.

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33

Hu, Fei. "Polynomial Volume Growth of Quasi-Unipotent Automorphisms of Abelian Varieties (with an Appendix in Collaboration with Chen Jiang)." International Mathematics Research Notices, July 25, 2023. http://dx.doi.org/10.1093/imrn/rnad170.

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Abstract Let $X$ be an abelian variety over an algebraically closed field $\textbf{k}$ and $f$ a quasi-unipotent automorphism of $X$. When $\textbf{k}$ is the field of complex numbers, Lin, Oguiso, and D.-Q. Zhang provide an explicit formula for the polynomial volume growth of (or equivalently, for the Gelfand–Kirillov dimension of the twisted homogeneous coordinate ring associated with) the pair $(X, f)$, by an analytic argument. We give an algebraic proof of this formula that works in arbitrary characteristic. In the course of the proof, we obtain the following: (1) a new description of the

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34

Kraft, Hanspeter, and Mikhail Zaidenberg. "Algebraically generated groups and their Lie algebras." Journal of the London Mathematical Society 109, no. 2 (2024). http://dx.doi.org/10.1112/jlms.12866.

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AbstractThe automorphism group of an affine variety is an ind‐group. Its Lie algebra is canonically embedded into the Lie algebra of vector fields on . We study the relations between subgroups of and Lie subalgebras of . We show that a subgroup generated by a family of connected algebraic subgroups of is algebraic if and only if the Lie algebras generate a finite‐dimensional Lie subalgebra of . Extending a result by Cohen–Draisma (Transform. Groups 8 (2003), no. 1, 51–68), we prove that a locally finite Lie algebra generated by locally nilpotent vector fields is algebraic, that is, for an alge

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35

Etingof, Pavel, and Shlomo Gelaki. "Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof." International Mathematics Research Notices, May 28, 2019. http://dx.doi.org/10.1093/imrn/rnz093.

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Abstract We prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p&gt;2$ admits a symmetric fiber functor to the category of supervector spaces. This proves Ostrik’s conjecture [25, Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme $\mathcal{G}$ over $k$ and a grouplike element $\epsilon \in k\mathcal{G}$ of order $\le 2$, whose action by conjugation on $\mathcal{G}$ coincides with the parity automorphism of $\mathcal{G}$, such tha

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36

SHAH, RIDDHI, and ALOK KUMAR YADAV. "Distal Actions of Automorphisms of Lie Groups G on Sub_G." Mathematical Proceedings of the Cambridge Philosophical Society, December 21, 2021, 1–22. http://dx.doi.org/10.1017/s0305004121000694.

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Abstract For a locally compact metrisable group G, we study the action of ${\rm Aut}(G)$ on ${\rm Sub}_G$ , the set of closed subgroups of G endowed with the Chabauty topology. Given an automorphism T of G, we relate the distality of the T-action on ${\rm Sub}_G$ with that of the T-action on G under a certain condition. If G is a connected Lie group, we characterise the distality of the T-action on ${\rm Sub}_G$ in terms of compactness of the closed subgroup generated by T in ${\rm Aut}(G)$ under certain conditions on the center of G or on T as follows: G has no compact central subgroup of pos

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37

Schempp, Walter J. "A Cubic Pure Half-Spinor Approach to High Performance Photon-Counting Computerized X-ray Tomography – Transcendence of the Brauer Mirror Symmetry Functor –." Complex Analysis and Operator Theory 18, no. 7 (2024). http://dx.doi.org/10.1007/s11785-024-01605-2.

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AbstractThe goal of this paper is to present a spectral theory of advanced photon-counting computerized X-ray tomography, which is a promising microelectronical semiconductor wafer X-ray technique on the verge of becoming clinically feasible and has the potential to dramatically alter the clinical application of computer-aided medical diagnosis in the upcoming decades. In analogy to the deep fact that any real division algebra has dimension 1, 2, 4 or 8, the isogeneous third Galois cohomological, mirror symmetrical approach to high spatial resolution spectral photon-counting computerized tomog

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38

Kimura, Yoshiyuki, Fan Qin, and Qiaoling Wei. "Twist Automorphisms and Poisson Structures." Symmetry, Integrability and Geometry: Methods and Applications, December 23, 2023. http://dx.doi.org/10.3842/sigma.2023.105.

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We introduce (quantum) twist automorphisms for upper cluster algebras and cluster Poisson algebras with coefficients. Our constructions generalize the twist automorphisms for quantum unipotent cells. We study their existence and their compatibility with Poisson structures and quantization. The twist automorphisms always permute well-behaved bases for cluster algebras. We explicitly construct (quantum) twist automorphisms of Donaldson-Thomas type and for principal coefficients.

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39

Puglisi, Orazio, and Gunnar Traustason. "Unipotent automorphisms of solvable groups." Journal of Group Theory 20, no. 3 (2017). http://dx.doi.org/10.1515/jgth-2016-0049.

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40

Hultgren, Jakob, and Erlend F. Wold. "Unipotent Factorization of Vector Bundle Automorphisms." International Journal of Mathematics, January 4, 2021. http://dx.doi.org/10.1142/s0129167x21500130.

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41

Pioline, Boris. "R4 couplings and automorphic unipotent representations." Journal of High Energy Physics 2010, no. 3 (2010). http://dx.doi.org/10.1007/jhep03(2010)116.

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42

MATRINGE, Nadir. "Symmetric periods for automorphic forms on unipotent groups." Journal of the Mathematical Society of Japan -1, no. -1 (2024). http://dx.doi.org/10.2969/jmsj/91279127.

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43

Andrist, Rafael B. "Integrable Generators of Lie Algebras of Vector Fields on $$\textrm{SL}_2(\mathbb {C})$$ and on $$xy = z^2$$." Journal of Geometric Analysis 33, no. 8 (2023). http://dx.doi.org/10.1007/s12220-023-01294-x.

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AbstractFor the special linear group $$\textrm{SL}_2(\mathbb {C})$$ SL 2 ( C ) and for the singular quadratic Danielewski surface $$x y = z^2$$ x y = z 2 we give explicitly a finite number of complete polynomial vector fields that generate the Lie algebra of all polynomial vector fields on them. Moreover, we give three unipotent one-parameter subgroups that generate a subgroup of algebraic automorphisms acting infinitely transitively on $$x y = z^2$$ x y = z 2 .

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44

Kimura, Yoshiyuki, and Hironori Oya. "Twist Automorphisms on Quantum Unipotent Cells and Dual Canonical Bases." International Mathematics Research Notices, March 7, 2019. http://dx.doi.org/10.1093/imrn/rnz040.

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45

Dill, Gabriel A. "On Morphisms Between Connected Commutative Algebraic Groups over a Field of Characteristic 0." Transformation Groups, July 26, 2022. http://dx.doi.org/10.1007/s00031-022-09748-2.

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AbstractLet K be a field of characteristic 0 and let G and H be connected commutative algebraic groups over K. Let Mor0(G,H) denote the set of morphisms of algebraic varieties G → H that map the neutral element to the neutral element. We construct a natural retraction from Mor0(G,H) to Hom(G,H) (for arbitrary G and H) which commutes with the composition and addition of morphisms. In particular, if G and H are isomorphic as algebraic varieties, then they are isomorphic as algebraic groups. If G has no non-trivial unipotent group as a direct factor, we give an explicit description of the sets of

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46

Gourevitch, Dmitry, Henrik P. A. Gustafsson, Axel Kleinschmidt, Daniel Persson, and Siddhartha Sahi. "Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups." Canadian Journal of Mathematics, September 21, 2020, 1–48. http://dx.doi.org/10.4153/s0008414x20000711.

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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 coe

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47

Mœglin, Colette. "Représentations unipotentes et formes automorphes de carré intégrable." Forum Mathematicum 6, no. 6 (1994). http://dx.doi.org/10.1515/form.1994.6.651.

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48

Pollack, Aaron. "THE MINIMAL MODULAR FORM ON QUATERNIONIC." Journal of the Institute of Mathematics of Jussieu, August 20, 2020, 1–34. http://dx.doi.org/10.1017/s1474748020000213.

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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 $E_{8}$

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49

Flicker, Yuval Z. "Euler-Poincaré functions and local systems." manuscripta mathematica 176, no. 4 (2025). https://doi.org/10.1007/s00229-025-01647-6.

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Abstract We develop here a new proof – using the trace formula – of a crucial reduction step done first in [7, section 5] using the geometric mass of a category technique. The result that we reprove is a count of the equivalence classes of local systems of rank n on a curve $$X_1$$ X 1 over a finite field $${\mathbb {F}}_q$$ F q , fixed by the Frobenius, with principal unipotent monodromy at least at two points; or rather the corresponding automorphic representations of $$\operatorname {GL}(n,{\mathbb {A}})$$ GL ( n , A ) over the function field F of $$X_1$$ X 1 that have unramified components

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50

Gourevitch, Dmitry, Eitan Sayag, and Ido Karshon. "Annihilator varieties of distinguished modules of reductive Lie algebras." Forum of Mathematics, Sigma 9 (2021). http://dx.doi.org/10.1017/fms.2021.42.

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Abstract We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let ${\mathbf {G}}$ be a complex algebraic reductive group and ${\mathbf {H}}\subset {\mathbf {G}}$ be a spherical algebraic subgroup. Let ${\mathfrak {g}},{\mathfrak {h}}$ denote the Lie algebras of ${\mathbf {G}}$ and ${\mathbf {H}}$ , and let ${\mathfrak {h}}^{\bot }$ denote the orthogonal complement to ${\mathfrak {h}}$ in ${\mathfrak {g}}^*$ . A ${\mathfrak {g}}$ -module is called ${\mathfrak {h}}$ -distinguished if it admits a nonz

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