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We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax-Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty-Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty-Kim theory to arbitrary number fields., 14 pages

We 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.

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf{K}$-types, associated varieties, and Langlands parameters of all such representations., revised exposition

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.

In this paper we describe $${\mathcal {R}}$$ -unipotent semigroups being regular extensions of a left regular band by an $$\mathcal {R}$$ -unipotent semigroup T as certain subsemigroups of a wreath product of a left regular band by T. We obtain Szendrei’s result that each E-unitary $${\mathcal {R}}$$ -unipotent semigroup is embeddable into a semidirect product of a left regular band by a group. Further, specialising the first author’s notion of $$\lambda $$ -semidirect product of a semigroup by a locally $${\mathcal {R}}$$ -unipotent semigroup, we provide an answer to an open question raised by the authors in [Extensions and covers for semigroups whose idempotents form a left regular band, Semigroup Forum 81 (2010), 51-70].

The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $Rep_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to construct level 0 decompositions when $R=\overline{\mathbb{Z}}_{\ell}$, $\ell \neq p$, by Dat for $GL_n$ and the author for a more general group. Wang proved in the case of $GL_n$ that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of $GL_n$ and a unipotent block of another group. In this paper, we generalize Wang's equivalence of category to a connected reductive group on a non-archimedean local field., 17 pages, in English

We introduce two properties: strong R-property and $C(q)$ -property, describing a special way of divergence of nearby trajectories for an abstract measure-preserving system. We show that systems satisfying the strong R-property are disjoint (in the sense of Furstenberg) with systems satisfying the $C(q)$ -property. Moreover, we show that if $u_t$ is a unipotent flow on $G/\Gamma $ with $\Gamma $ irreducible, then $u_t$ satisfies the $C(q)$ -property provided that $u_t$ is not of the form $h_t\times \operatorname {id}$ , where $h_t$ is the classical horocycle flow. Finally, we show that the strong R-property holds for all (smooth) time changes of horocycle flows and non-trivial time changes of bounded-type Heisenberg nilflows.

For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $\phi: G \to U^-\times G_{ev} \times U^+$, where $G_{ev}$ is the even subsupergroup of $G$, and $U^-$, $U^+$ are appropriate odd unipotent subsupergroups of $G$. We compute the action of odd superderivations on the images $\phi^*(x_{ij})$ of the generators of $K[G]$. We describe a specific ordering of the dominant weights $X(T)^+$ of $GL(m|n)$ for which there exists a Donkin-Koppinen filtration of the coordinate algebra $K[G]$. Let $\Gamma$ be a finitely generated ideal $\Gamma$ of $X(T)^+$ and $O_{\Gamma}(K[G])$ be the largest $\Gamma$-subsupermodule of $K[G]$ having simple composition factors of highest weights $\lambda\in \Gamma$. We apply combinatorial techniques, using generalized bideterminants, to determine a basis of $G$-superbimodules appearing in Donkin-Koppinen filtration of $O_{\Gamma}(K[G])$.

Building on the results of Deligne and Illusie on liftings to truncated Witt vectors, we give a criterion for non-liftability that involves only the dimension of certain cohomology groups of vector bundles arising from the Frobenius pushforward of the de Rham complex. Using vector bundle methods, we apply this to show that exceptional Enriques surfaces, a class introduced by Ekedahl and Shepherd-Barron, do not lift to truncated Witt vectors, yet the base of the miniversal formal deformation over the Witt vectors is regular. Using the classification of Bombieri and Mumford, we also show that bielliptic surfaces arising from a quotient by a unipotent group scheme of order p do not lift to the ring of Witt vectors. These results hinge on some observations in homological algebra that relates splittings in derived categories to Yoneda extensions and certain diagram completions.

Methods of symbolic dynamics play an essential role in the study of combinatorial properties of words, problems in number theory and the theory of dynamical systems. The paper is devoted to the problems of combinatorics on words, its applications in algebra and dynamical systems. Section 2.1 considers the one-dimensional case using the key example of Sturm’s words. The proof of the criterion for substitutionality of Sturm palindromes using the Rauzy induction is given, the case of one-dimensional facordynamics is considered. Section 2.2 discusses the shift of the torus and the Rauzy fractal that generates the word Tribonacci. The relationship between the periodicity of Rauzy’s schemes and the substitutionality of the word generated by this system is discussed. The implementation of the word Tribonacci through the rearrangement of line segments is given. An approach to the Pisot hypothesis is outlined. Section 2.3 talks about unipotent torus transformations and billiards in polygons. Chapter 3 talks about normal forms and the growth of groups and algebras. Chapter 4 is devoted to Rosie graphs, Gr¨obner bases and co-growth, and algebraic applications. Section 4.1 discusses the results in the combinatorics of multilinear words developed by V. N. Latyshev and the problems he posed. Section 4.2 talks about finitely defined objects and the problems of controlling the relationships that define them. Section 4.3 describes some monomial algebras in terms of uniformly recurrent words. Chapter 5 deals with the problem of height and normal forms.

To a one-parameter family of Calabi-Yau threefolds, we can associate the extended period map by the log Hodge theory of Kato and Usui. In the present paper, we study the image of a maximally unipotent monodromy point under the extended period map. As an application, we prove the generic Torelli theorem for a large class of one-parameter families of Calabi-Yau threefolds.

We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776 ) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070 ). In particular, we note that morphisms of motivic dga’s can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga’s is a certain limit of such spectra; we give an explicit formula for this limit—a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call “motivic Chow coalgebras”. We discuss the relationship between motivic Chow coalgebras and motivic dga’s of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite etale over a number field, morphisms of associated motivic dga’s are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim’s relative unipotent section conjecture, hence as an ounce of evidence for the latter.

Following a suggestion of Peter Scholze, we construct an action of G m ^ \widehat {\mathbb {G}_m} on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p p -adic modular curve whose ring of functions is Serre’s space of p p -adic modular functions. This action is a local, p p -adic analog of a global, archimedean action of the circle group S 1 S^1 on the lattice-unstable locus of the modular curve over C \mathbb {C} . To construct the G m ^ \widehat {\mathbb {G}_m} -action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q q ; along the way we also prove a natural generalization of Dwork’s equation τ = log ⁡ q \tau =\log q for extensions of Q p / Z p \mathbb {Q}_p/\mathbb {Z}_p by μ p ∞ \mu _{p^\infty } valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of G m ^ \widehat {\mathbb {G}_m} integrates the differential operator θ \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p p -adic L L -functions.

Let $$\mathbb{G}$$ denote a higher-rank ℝ-split simple Lie group of the following type: SL(n, ℝ), SOo(m, m), E6(6), E7(7) and E8(8), where m ≥ 4 and n ≥ 3. We study the cohomological equation for discrete, abelian parabolic actions on $$\mathbb{G}$$ via representation theory. Specifically, we characterize the obstructions to solving the cohomological equation and construct smooth solutions with Sobolev estimates. We prove that global estimates of the solution are generally not tame, and our non-tame estimates in the case $$\mathbb{G}$$ = SL(n, ℝ) are sharp up to finite loss of regularity. Moreover, we prove that for general $$\mathbb{G}$$ the estimates are tame in all but one direction, and as an application, we obtain tame estimates for the common solution of the cocycle equations. We also give a sufficient condition for which the first cohomology with coefficients in smooth vector fields is trivial. In the case that $$\mathbb{G}$$ = SL(n, ℝ), we show this condition is also necessary. A new method is developed to prove tame directions involving computations within maximal unipotent subgroups of the unitary duals of SL(2, ℝ) ⋉ ℝ2 and SL(2, ℝ) ⋉ ℝ4. A new technique is also developed to prove non-tameness for solutions of the cohomological equation.

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-super\-group schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause. We also completely determine the cohomology ring in the smallest cases, namely $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb G_{a(1)}$ and $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb Z/p$. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes., 19 pages

Generalizing Lusztig's work, Malle as associated to any imprimitive complex reflection $W$ group a set of "unipotent characters", which are in bijection of the usual unipotent characters of the associated finite reductive group if $W$ is a Weyl group. He also obtained a partition of these characters into families and associated to each family a $\mathbb{Z}$-modular datum. We construct a categorification of some of these data, by studying the category of tilting modules of the Drinfeld double of the quantum enveloping algebra of the Borel of a simple complex Lie algebra.

We give a short combinatorial proof of Asai's decomposition formula for Lusztig induction of unipotent characters in groups of classical type, relying solely on the Mackey formula.

S.-Y. Pan decomposes the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. In this paper we use Pan's decomposition to study the theta correspondence for this kind of dual pairs, following the approach of Adams-Moy and Aubert-Michel-Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations., Comment: Revised version, accepted by Pacific J. Math

In this paper we will give a short proof of a special case of Zariski’s result about finite generation in connection with Hilbert’s 14th problem using a new idea. Our result is useful for invariant subrings of unipotent or connected semisimple groups. We will also prove an analogue of Miyanishi’s result for the ring of invariants of a $$ {\mathbbm{G}}_a $$ -action on R[X, Y, Z] for an affine Dedekind domain R using topological methods.

In this paper we study the Gan–Gross–Prasad problem for unitary groups over finite fields. Our results provide complete answers for unipotent representations, and we obtain the explicit branching of these representations.

Let G(F_q) be the group of rational points of a simple algebraic group defined and split over a finite field F_q. In this paper we define a new basis for the Grothendieck group of unipotent representations of G(F_q)., 36 pages. arXiv admin note: text overlap with arXiv:1805.03770

Let G be a connected reductive algebraic group defined over a finite field with q elements. In the 1980’s, Kawanaka introduced generalised Gelfand–Graev representations of the finite group $$ G\left({\mathbbm{F}}_q\right) $$ G F q , assuming that q is a power of a good prime for G. These representations have turned out to be extremely useful in various contexts. Here we investigate to what extent Kawanaka’s construction can be carried out when we drop the assumptions on q. As a curious by-product, we obtain a new, conjectural characterisation of Lusztig’s concept of special unipotent classes of G in terms of weighted Dynkin diagrams.

We study the local zeta integrals attached to a pair of generic representations ( π , τ ) of GL n × GL m , n > m , over a p-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of π and τ. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin–Selberg (local) L-function.

Given a Speh representation π of GL n ( F ) for a non-archimedean local field F, we obtain necessary and sufficient conditions on standard parabolic subgroups for the Jacquet module of π with respect to the parabolic subgroup to be generic. Using this we describe the precise set of characters Θ of the maximal unipotent radical U n of GL n ( F ) such that Hom U n ( π , Θ ) is non-zero. We then describe the behavior of this set under the base change map (with respect to a finite cyclic extension of F of prime degree).

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.

Falbel, Koseleff and Rouillier computed a large number of boundary unipotent CR representations of fundamental groups of non compact three-manifolds. Those representations are not always discrete. By experimentally computing their limit set, one can determine that those with fractal limit sets are discrete. Many of those discrete representations can be related to (3,3,n) complex hyperbolic triangle groups. By exact computations, we verify the existence of those triangle representations, which have boundary unipotent holonomy. We also show that many representations are redundant: for n fixed, all the (3,3,n) representations encountered are conjugate and only one among them is uniformizable.

В статье изучаются p--расширения полных дискретно нормированных полей смешанной характеристики, где p-характеристика поля вычетов рассматриваемого поля. Известно, что любое вполне разветвленное расширение Галуа степени p-с немаксимальным скачком ветвления может быть задано уравнением Артина-Шрайера; при этом ограничение сверху на скачок ветвления соответствует ограничению снизу на нормирование правой части уравнения. Задача построения расширений с заданной группой Галуа произвольного конечного порядка не решена. В работах Инабы рассматривались p-расширения полей характеристики p, заданные матричным уравнением $$X^{(p)}=AX$$, которое мы здесь называем уравнением Инабы. В этом уравнении $$X^{(p)}$$ обозначает матрицу, полученную возведением каждого элемента квадратной матрицы X в степень p, а - некоторая унипотентная матрица A над данным полем. Такое уравнение задает последовательность расширений полей, каждое из которых задано уравнением Артина-Шрайера. Было доказано, что любое уравнение Инабы задает расширение Галуа, и обратно, любое конечное p-расширение Галуа задается уравнением такого вида. В настоящей работе для полей смешанной характеристики доказано, что расширение, задаваемое уравнением Инабы, является расширением Галуа, если нормирования элементов матрицы удовлетворяют некоторым оценкам снизу, т.е. если скачки промежуточных расширений степени p достаточно малы. Данная конструкция может применяться при решении задачи погружения расширений полей. Уравнение Инабы задает последовательность расширений полей, полученную последовательным присоединением элементов диагоналей матрицы. Это означает, что, если расширение L/K задано уравнением Инабы, и матрица A выбрана так, что на диагоналях с большими номерами записаны нули, то можно получать расширения, содержащие L/K, заменяя нули другими элементами. В работе доказано, что любое нециклическое расширение степени $$p^2$$ с достаточно маленькими скачками можно погрузить в расширение с группой Галуа, изоморфнной группе унипотентных матриц $$3\times 3$$ над полем из p элементов. В конце статьи сформулирован ряд открытых вопросов, при исследовании которых, возможно, окажется полезной данная конструкция.

The dimensions of the Jordan blocks in the images of regular unipotent elements from subsystem subgroups of type C2 in p-restricted irreducible representations of groups of type Cn in characteristic p ≥ 11 with locally small highest weights are found. These results can be applied for investigating the behavior of unipotent elements in modular representations of simple algebraic groups and recognizing representations and linear groups. The article consists of 3 parts. In the first one, preliminary lemmas that are necessary for proving the principal results, are contained and the case where all weights of the restriction of a representation considered to a subgroup of type A1 containing a relevant unipotent element are less than p, is investigated.

Let G be a special orthogonal group SO(2n+1) defined over a p-adic field F. Let $\pi$ be an admissible irreducible representation of G(F) which is tempered and of unipotent reduction. We prove that $\pi$ has a wave front set. In some particular cases, for instance if $\pi$ is of the discrete series, we give a method to compute this wave front set., Comment: in French

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the $p$-adic de Rham period map $j^{dr}_n$ on elliptic and hyperelliptic curves over number fields via their universal unipotent connections $\mathcal{U}$. Several algorithms forming part of the computation of finite level versions $j^{dr}_n$ of the unipotent Albanese maps are presented. The computation of the logarithmic extension of $\mathcal{U}$ in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on $\mathcal{U}$ over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated $p$-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well., Comment: 60 pages

It is proved that all maximal unipotent subgroups of a special linear group over commutative ring with identity (such that the factor ring of its modulo primitive radical is a finite direct sum of Bezout domains) are pairwise conjugated and describe one maximal unipotent subgroup of the general linear group (and of a special linear group) over an arbitrary commutative ring with identity.

We compute the K-automorphism group of A K = K ⊗ F A f of a weak crossed product algebra A f for a weak 2-cocycle f over a Galois extension K / F with Galois group G. The K-automorphism group of A K decomposes into its unipotent part and the reductive part H ˆ . Automorphisms in H ˆ are computed via their restriction to semi-simple part of A K . There is a strong relationship between H ˆ and the lower-subtractive relation (≤) induced by f on G. We introduce a subgroup of H ˆ , namely Λ, which contains interesting combinatorial information of ≤. We also present a duality on lower subtractive relations which simplifies the computation of the automorphism group. For the Weak Bruhat order of a Coxeter system, which is an important example of a lower-subtractive relation, it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph.

We construct two supercharacter theories (in the sense of P. Diaconis and I.M. Isaacs) for the parabolic subgroups in orthogonal and symplectic groups. For each supercharacter theory, we obtain a supercharacter analog of the A.A. Kirillov formula for irreducible characters of finite unipotent groups.

In this article, it is proved that every symplectic matrix can be decomposed into a product of three symplectic unipotent matrices of index 2, i.e., every complex matrix A satisfying ATJA = J with J = [0

We establish an effective version of the classical Lie–Kolchin Theorem. Namely, let A , B ∈ GL m ( C ) be quasi-unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k ⩾ 0 the matrix A B k is also quasi-unipotent. Then A and B have a common eigenvector. In particular, 〈 A , B 〉 GL m ( C ) is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.

In this paper we start the study of configurations of flags in closed orbits of real forms using mainly tools of GIT. As an application, using cross ratio coordinates for generic configurations, we identify boundary unipotent representations of the fundamental group of the figure eight knot complement into real forms of $${{\,\mathrm{PGL}\,}}(4,{\mathbb {C}})$$ .

Let $G$ be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of $G$ by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on $G$. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where $G$ is the (twisted or non-twisted) group of type $E_6$ over the finite field with $q$ elements, for $q$ a power of the bad prime $p=3$, by exploiting known facts about the representation theory of the Hecke algebra associated with $G$., Comment: 12 pages

We study some special types of bands of E-inversive unipotent semigroups. It has been proved that in any R-semigroup S, which is a band of E-inversive unipotent semigroups, the set of its regular elements is a retract of S. Also, some characterizations of E-inversive rectangular bands of unipotent semigroups are given. This theorem extends nearly 40-old results from the theory of epigroups. In fact, a more general result is valid in some special subclass of the class of E-inversive semigroups; this result seems to be (partially) new for all epigroups.

We study representations of the general and the special linear groups over a non-archimedean local field that admit a non-zero invariant linear form with respect to the symplectic group (henceforth distinguished representations). For the class of ladder representations that are distinguished we show that applying the highest derivative operation twice results in a distinguished (ladder) representation. Furthermore, we characterize the unique distinguished component of the associated L-packet of representations of the special linear group in terms of the associated maximal unipotent orbit. We also obtain a sufficient condition for distinction of standard modules.

Hiraga, Ichino and Ikeda have conjectured an explicit expression for the Plancherel density of the group of points of a reductive group defined over a local field F, in terms of local Langlands parameters. In these lectures we shall present a proof of these conjectures for Lusztig’s class of representations of unipotent reduction if F is p-adic and G is of adjoint type and splits over an unramified extension of F. This is based on the author’s paper [Spectral] transfer morphisms for unipotent affine Hecke algebras, Selecta Math. (N.S.) 22 (2016), no. 4, 2143–2207]. More generally for G connected reductive (still assumed to be split over an unramified extension of F), we shall show that the requirement of compatibility with the conjectures of Hiraga, Ichino and Ikeda essentially determines the Langlands parameterisation for tempered representations of unipotent reduction. We shall show that there exist parameterisations for which the conjectures of Hiraga, Ichino and Ikeda hold up to rational constant factors. The main technical tool is that of spectral transfer maps between normalised affine Hecke algebras used in op. cit

We consider a family of smooth perturbations of unipotent flows on compact quotients of SL $${(3,\mathbb{R})}$$ which are not time-changes. More precisely, given a unipotent vector field, we perturb it by adding a non-constant component in a commuting direction. We prove that, if the resulting flow preserves a measure equivalent to Haar, then it is parabolic and mixing. The proof is based on a geometric shearing mechanism together with a non-homogeneous version of Mautner Phenomenon for homogeneous flows. Moreover, we characterize smoothly trivial perturbations and we relate the existence of non-trivial perturbations to the failure of cocycle rigidity of parabolic actions in SL $${(3,\mathbb{R})}$$ .

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (resp. subalgebra), or a natural generalisation if $G$ is of type $D_n$. From these representatives, it is straightforward to generate representatives for any given class. Along the way we also prove recognition theorems for identifying both the unipotent classes and nilpotent orbits of exceptional algebraic groups., 26 pages

This work concerns the representation theory and cohomology of a finite unipotent supergroup scheme $G$ over a perfect field $k$ of positive characteristic $p\ge 3$. It is proved that an element $x$ in the cohomology of $G$ is nilpotent if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $x_K$ to $E$ is nilpotent. It is also shown that a $kG$-module $M$ is projective if and only if for every extension field $K$ of $k$ and every elementary sub-supergroup scheme $E\subseteq G_K$, the restriction of $M_K$ to $E$ is projective. The statements are motivated by, and are analogues of, similar results for finite groups and finite group schemes, but the structure of elementary supergroups schemes necessary for detection is more complicated than in either of these cases. One application is a detection theorem for the nilpotence of cohomology, and projectivity of modules, over finite dimensional Hopf subalgebras of the Steenrod algebra., 46 pages; Sections 12 on Z-graded group schemes and the Steenrod algebra is revised compared to the previous version

We prove a general theorem for constructing integral quantum cluster algebras over Z[q±1/2], namely, that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over Z[q±1/2]. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that, for every symmetrizable Kac–Moody algebra g and Weyl group element w, the dual canonical form Aq(n+(w))Z[q±1] of the corresponding quantum unipotent cell has the property that Aq(n+(w))Z[q±1]⊗Z[q±1]Z[q±1/2] is isomorphic to a quantum cluster algebra over Z[q±1/2] and to the corresponding upper quantum cluster algebra over Z[q±1/2].

In this paper, we look at the notion of cohomological triviality of fibrations of homogeneous spaces of affine algebraic groups defined over $${\mathbb {C}}$$ and use topological methods, primarily the theory of covering spaces. This is made possible because of the structure theory of affine algebraic groups. Further, we generalize our results for arbitrary connected algebraic groups and their homogeneous spaces. As an application of our methods, we give a structure result for quasi- reductive algebraic groups (i.e., algebraic groups whose unipotent radical is trivial), up to isogeny.

We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

We give a new realization of the prefundamental representations $L^\pm_{r,a}$ introduced by Hernandez and Jimbo, when the quantum loop algebra $U_q(\mathfrak{g})$ is of types $A_n^{(1)}$ and $D_n^{(1)}$, and the $r$-th fundamental weight $\varpi_r$ for types $A_n$ and $D_n$ is minuscule. We define an action of the Borel subalgebra $U_q(\mathfrak{b})$ of $U_q(\mathfrak{g})$ on the unipotent quantum coordinate ring associated to the translation by $-\varpi_r$, and show that it is isomorphic to $L^\pm_{r,a}$. We then give a combinatorial realization of $L^+_{r,a}$ in terms of the Lusztig data of the dual PBW vectors., Comment: 41 pages, introduction is revised, Lemma 3.2 is revised, the proofs of Lemma 3.8 and Corollary 4.8 are revised, added Example 3.9, Example 4.4 and Example 4.19, comments added on Remark 3.11, Remark 4.9, Remark 4.21 and Remark 4.30, typos corrected, minor corrections

Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold., Comment: 20 pages. A new section has been added