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Journal articles on the topic 'Unipotent automorphism'

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Published: 10 March 2023

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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 (March 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 (June 23, 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 [Formula: see text]. This result was first proved by Dinh et al. for compact Kähler manifolds.

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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 (March 25, 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 (March 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 interesting exceptions.

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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 “quadratic-unipotent” representations. For all such representation π, we define a suitable extension π+, a character-sheave f(π) and we prove an identity trace = γ(π)ϕ(f(π)) with an explicit complex number γ(π).

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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 (January 9, 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 (May 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 (December 2008): 4132–55. http://dx.doi.org/10.1016/j.jalgebra.2008.07.029.

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10

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

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11

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 (October 15, 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 radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

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12

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

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13

Rohde, Jan Christian. "Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy." manuscripta mathematica 131, no. 3-4 (January 22, 2010): 459–74. http://dx.doi.org/10.1007/s00229-009-0329-5.

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14

Lifschitz, Lucy, and Dave Witte. "ON AUTOMORPHISMS OF ARITHMETIC SUBGROUPS OF UNIPOTENT GROUPS IN POSITIVE CHARACTERISTIC." Communications in Algebra 30, no. 6 (June 19, 2002): 2715–43. http://dx.doi.org/10.1081/agb-120003985.

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15

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 (December 2009): 118–27. http://dx.doi.org/10.1134/s0081543809070128.

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16

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

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17

Voll, Christopher. "IDEAL ZETA FUNCTIONS ASSOCIATED TO A FAMILY OF CLASS-2-NILPOTENT LIE RINGS." Quarterly Journal of Mathematics 71, no. 3 (June 17, 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 schemes associated to the Lie rings in question.

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18

Gan, Wee Teck, Nadya Gurevich, and Dihua Jiang. "Cubic unipotent Arthur parameters and multiplicities of square integrable automorphic forms." Inventiones Mathematicae 149, no. 2 (August 1, 2002): 225–65. http://dx.doi.org/10.1007/s002220200210.

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19

FLICKER, YUVAL Z. "CUSP FORMS ON GSp(4) WITH SO(4)-PERIODS." International Journal of Number Theory 07, no. 04 (June 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, 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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20

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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21

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

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22

Sun, Heng. "Remarks on Certain Metaplectic Groups." Canadian Mathematical Bulletin 41, no. 4 (December 1, 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 section is an isomorphism, both algebraically and topologically in an open subgroup of G(Fv);3.define a global section which is a product of local sections on a maximal torus, a unipotent subgroup and a set of representatives for the Weyl group.

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23

Booher, Jeremy. "Minimally ramified deformations when." Compositio Mathematica 155, no. 1 (November 12, 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. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.

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24

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 (September 20, 2014): 235–73. http://dx.doi.org/10.1007/s00605-014-0686-3.

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25

RICHARD, RODOLPHE. "RÉPARTITION GALOISIENNE D'UNE CLASSE D'ISOGÉNIE DE COURBES ELLIPTIQUES." International Journal of Number Theory 09, no. 02 (December 5, 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 Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math.15(4) (1972) 259–331] dans le cas où les invariants modulaires considérés sont algébriques sur Q, et des résultats de G. Shimura dans le cas transcendant [Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)]. In this article, it is shown that Galois orbits of invariants associated with non-CM and pairwise isogeneous complex elliptic curves equidistribute in the classical modular curve towards the hyperbolic probability measure. The proof is based on arguments from ergodic theory, especially Ratner's theorem on unipotent flows (cf. [A. Eskin and H. Oh, Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26(1) (2006) 163–167]), as well as on Serre's open image theorem [J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves (W. A. Benjamin, New York, 1968); Propriétés Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math.15(4) (1972) 259–331] in case of algebraic invariants, and on G. Shimura's work in the transcendant case [Introduction to the Arithmetic Theory of Automorphic Functions, Publications of the Mathematical Society of Japan (Princeton University Press, Princeton, NJ, 1994)].

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26

Frati, Marco. "Unipotent automorphisms of soluble groups with finite Prüfer rank." Journal of Group Theory 17, no. 3 (May 1, 2014). http://dx.doi.org/10.1515/jgt-2013-0041.

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27

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 domestic homologies, (ii) the opposition diagrams arising from elements of the standard unipotent subgroup of the Chevalley group, and (iii) the automorphisms with opposition diagrams with at most two distinguished orbits encircled. Our results provide unexpected characterizations of long root elations and products of perpendicular long root elations in long root geometries, and analogues of the density theorem for connected linear algebraic groups in the setting of Chevalley groups over arbitrary fields.

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28

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’, Pure Appl. Math. Q., to appear, Preprint, 2021, arXiv:2112.13684].

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29

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 $p>2$ 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 that $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(\mathcal{G},\epsilon )$. In particular, when $\mathcal{C}$ is unipotent, the functor lands in $\textrm{Vec}$, so $\mathcal{C}$ is symmetric tensor equivalent to $\textrm{Rep}(U)$ for a unique finite unipotent group scheme $U$ over $k$. We apply our result and the results of [17] to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over $k$ (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field $k$ of characteristic $p>0$, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field $k$ of characteristic $p>0$. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper [4], and more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic $\ne 2$ is always a Serre subcategory.

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30

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 positive dimension or T is unipotent or T is contained in the connected component of the identity in ${\rm Aut}(G)$ . Moreover, we also show that a connected Lie group G acts distally on ${\rm Sub}_G$ if and only if G is either compact or it is isomorphic to a direct product of a compact group and a vector group. All the results on the Lie groups mentioned above hold for the action on ${\rm Sub}^a_G$ , a subset of ${\rm Sub}_G$ consisting of closed abelian subgroups of G.

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31

Puglisi, Orazio, and Gunnar Traustason. "Unipotent automorphisms of solvable groups." Journal of Group Theory 20, no. 3 (January 1, 2017). http://dx.doi.org/10.1515/jgth-2016-0049.

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32

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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33

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

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34

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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35

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 all morphisms and isomorphisms of algebraic varieties between G and H. We also characterize all connected commutative algebraic groups over K whose only variety automorphisms are compositions of automorphisms of algebraic groups with translations.

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36

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 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.

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37

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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38

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 $E_{8}$ , 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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39

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 nonzero ${\mathfrak {h}}$ -invariant functional. We show that the maximal ${\mathbf {G}}$ -orbit in the annihilator variety of any irreducible ${\mathfrak {h}}$ -distinguished ${\mathfrak {g}}$ -module intersects ${\mathfrak {h}}^{\bot }$ . This generalises a result of Vogan [Vog91]. We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G. Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup $\bf H$ , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20]. Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models.

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