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We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees., Comment: arXiv admin note: text overlap with arXiv:1611.07373

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let G be an algebraic group of classical type with defining characteristic p > 0 , μ a dominant weight and W the Weyl group of G. Let G = G ( q ) be a finite classical group, where q is a p-power. For a weight μ of G the sum s μ of distinct weights w ( μ ) with w ∈ W viewed as a function on the semisimple elements of G is known to be a generalized Brauer character of G called an orbit character of G. We compute, for certain orbit characters and every maximal torus T of G, the multiplicity of the trivial character 1 T of T in s μ . The main case is where μ = ( q − 1 ) ω and ω is a fundamental weight of G. Let St denote the Steinberg character of G. Then we determine the unipotent characters occurring as constituents of s μ ⋅ S t defined to be 0 at the p-singular elements of G. Let β μ denote the Brauer character of a representation of S L n ( q ) arising from an irreducible representation of G with highest weight μ. Then we determine the unipotent constituents of the characters β μ ⋅ S t for μ = ( q − 1 ) ω , and also for some other μ (called strongly q-restricted). In addition, for strongly restricted weights μ, we compute the multiplicity of 1 T in the restriction β μ | T for every maximal torus T of G.

In this article, we consider the actions of subgroups of the general linear group G L r ( Z n ) on Z n r , including groups of upper triangular matrices in G L r ( Z n ) , unipotent groups, Heisenberg groups and extended Heisenberg groups. By applying the Cauchy–Frobenius–Burnside lemma, we obtain several generalizations of the well known Menon's identity.

An almost nilpotent variety of linear growth is constructed in the paper for any infinite periodic word in an alphabet of two letters. A discrete series of different almost nilpotent varieties is also constructed. Only a few almost nilpotent varieties were studied previously and their existence was proved often under some additional assumptions. The existence of almost nilpotent varieties of arbitrary integer exponential growth with a fractional exponent is proved as well as the existence of a continual family of almost nilpotent varieties with not more than quadratic growth.

Let G be a split reductive group over a p-adic field F. Let B be a Borel subgroup and U the maximal unipotent subgroup of B. Let \(\psi \) be a Whittaker character of U. Let I be an Iwahori subgroup of G. We describe the Iwahori–Hecke algebra action on the Gelfand–Graev representation \((\mathrm {ind}_{U}^{G}\psi )^I\) by an explicit Hecke algebra module.

In order to find a tractable theory to substitute for the wild character theory of the group of n × n unipotent upper-triangular matrices over a finite field F q , Andre and Yan introduced the notion of supercharacter theory. In this paper, we construct a supercharacter theory from an arbitrary set S of normal subgroups of G . We call such supercharacter theory the normal supercharacter theory generated by S . It is shown that normal supercharacter theories are integral, and a recursive formula for supercharacters of the normal supercharacter theory is provided. Also, we indicate that the superclasses of the normal supercharacter theory generated by all normal subgroups of G are given by certain values on the primitive central idempotents. We study the connection between the finest normal supercharacter theory and faithful irreducible characters. Moreover, an algorithm is presented to construct the supercharacter table of the finest normal supercharacter theory from the character table. Finally, we argue that normal supercharacter theories cannot be obtained by previously known supercharacter theory constructions.

We study the simultaneously nilpotent index of a simultaneously nilpotent set of matrices over an antinegative commutative semiring S . We find an upper bound for this index and give some characterizations of the simultaneously nilpotent sets when this upper bound is met. In the special case of antinegative semirings with all zero divisors nilpotent, we also find a bound on the simultaneously nilpotent index for all nonmaximal simultaneously nilpotent sets of matrices and establish their cardinalities in case of a finite S .

In this paper, the concept of multi-fuzzy matrix (MFM), multi-fuzzy complex matrix (MFCM), generalized multi-fuzzy complex matrix (GMFCM), generalized multi-fuzzy complex nilpotent matrix (GMFCNM) are introduced and have shown that the set of GMFCMs form a distributive lattice. Some properties and characterizations for GMFCNMs are established and in particular, a necessary and sufficient condition for an n × n GMFCNM to have the nilpotent index n is given. Finally, the reduction of GMFCNMs over a distributive lattice are given with some properties.

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the n -dimensional torus. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on the n -dimensional torus; we completely describe this set, for different families containing infinitely many Morse–Smale diffeomorphisms. Moreover we show that for any given odd integer, there are Morse–Smale diffeomorphisms such that the corresponding minimal set of Lefschetz periods consists of all square free divisors of the given number. The results of the present article generalize the previous results of Berrizbeitia–Sirvent [7] . The methods used are based on the arithmetical properties of the number n .

Let k be an algebraically closed field and L = k ( ( ϵ ) ) the field of Laurent series over k. Let G be a connected reductive group over k such that the characteristic of k does not divide the order of the Weyl group of G. For q ∈ k × , the automorphism of L, defined by ∑ a i ϵ i ↦ ∑ a i ( q ϵ ) i , induces an automorphism σ q of the loop group G ( L ) . We show that, when q is not a root of unity, the classification of σ q -conjugacy classes of G ( L ) can be reduced to the classification of unipotent classes of (non-connected) reductive groups. As an application, we recover the classification (of σ q -conjugacy classes) for G = G L n .

Let F be an algebraically closed field of positive characteristic p > 3 , and A the divided power algebra in one indeterminate, which, as a vector space, coincides with the truncated polynomial ring of F [ T ] by T p n . Let g be the special derivation algebra over A which is a simple Lie algebra, and additionally non-restricted as long as n > 1 . Let N be the nilpotent cone of g , and G = Aut ( g ) , the automorphism group of g . In contrast with only finitely many nilpotent orbits in a classical simple Lie algebra, there are infinitely many nilpotent orbits in g . In this paper, we parameterize all nilpotent orbits, and obtain their dimensions. Furthermore, the nilpotent cone N is proven to be reducible and not normal. There are two irreducible components in N . The dimension of N is determined.

We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis.

In this paper, we begin to develop a theory of character sheaves on an affine algebraic group $G$ defined over an algebraically closed field $k$ of characteristic $p>0$ using the approach developed by Boyarchenko and Drinfeld for unipotent groups. Let $l$ be a prime different from $p$. Following Boyarchenko and Drinfeld, we define the notion of an admissible pair on $G$ and the corresponding idempotent in the $\overline{\mathbb{Q}_l}$-linear triangulated braided monoidal category $\mathscr{D}_G(G)$ of conjugation equivariant $\overline{\mathbb{Q}_l}$-complexes (under convolution with compact support) and study their properties. We aim to break up the braided monoidal category $\mathscr{D}_G(G)$ into smaller and more manageable pieces corresponding to these idempotents in $\mathscr{D}_G(G)$. Drinfeld has conjectured that the idempotent in $\mathscr{D}_G(G)$ obtained from an admissible pair is in fact a minimal idempotent and that any minimal idempotent in $\mathscr{D}_G(G)$ can be obtained from some admissible pair on $G$. We will prove this conjecture in the case when the neutral connected component $G^\circ \subset G$ is a solvable group. For general groups, we prove that this conjecture is in fact equivalent to an a priori weaker conjecture. Using these results, we reduce the problem of defining character sheaves on general algebraic groups to a special case which we call the "Heisenberg case"., 45 pages, added some missing details

We study Galois descents for categories of mixed Tate motives over O N [ 1 / N ] , for N ∈ { 2 , 3 , 4 , 8 } or O N for N = 6 , with O N the ring of integers of the Nth cyclotomic field, and construct families of motivic iterated integrals with prescribed properties. In particular this gives a basis of multiple zeta values via multiple zeta values at roots of unity μ N . It also gives a new proof, via Goncharov's coproduct, of Deligne's results [9] : the category of mixed Tate motives over O k N [ 1 / N ] , for N ∈ { 2 , 3 , 4 , 8 } is spanned by the motivic fundamental groupoid of P 1 ∖ { 0 , μ N , ∞ } with an explicit basis. By applying the period map, we obtain a generating family for multiple zeta values relative to μ N .

We prove that every finite simple group G of Lie type satisfies G = U U - U U -, where U is a unipotent Sylow subgroup of G and U - is its opposite. We also characterize the cases for which G = U U - U. These results are best possible in terms of the number of conjugates of U in the above factorizations.

We introduce the notion of spectral transfer morphisms between normalized affine Hecke algebras, and show that such morphisms induce spectral measure preserving correspondences on the level of the tempered spectra of the affine Hecke algebras involved. We define a partial ordering on the set of isomorphism classes of normalized affine Hecke algebras, which plays an important role for the Langlands parameters of Lusztig's unipotent representations., Comment: 38 pages; The ordering of the material has been improved in this version

We introduce the notion of a lowering–raising (or LR) triple of linear transformations on a nonzero finite-dimensional vector space. We show how to normalize an LR triple, and classify up to isomorphism the normalized LR triples. We describe the LR triples using various maps, such as the reflectors, the inverters, the unipotent maps, and the rotators. We relate the LR triples to the equitable presentation of the quantum algebra U q ( sl 2 ) and Lie algebra sl 2 .

Let G(F_q) be the group of rational points of a split connected reductive group G defined over the finite field F_q. In this paper we show that the category of representations of G(F_q) which are finite direct sums of unipotent representations in a fixed two-sided cell is equivalent to the centre of a certain monoidal category of sheaves on the product of two copies of the flag manifold of G. We also consider a version of this for nonsplit groups., 26 pages. This version considers also nonsplit groups. arXiv admin note: text overlap with arXiv:1308.1082

For the finite simple groups of twisted Lie types 2Al and 2Dl, we specify the description for the chief factors of a maximal parabolic subgroup which are involved in its unipotent radical. We prove a theorem in which, for every maximal parabolic subgroup of the groups 2Al(q2) and 2Dl(q2), we give the fragments of the chief series involving in the unipotent radical of this parabolic subgroup. The generators of the corresponding chief factors are presented in tables.

The nilpotent graph of a group G is a simple graph whose vertex set is G∖nil(G), where nil(G) = {y ∈ G | ⟨ x, y ⟩ is nilpotent ∀ x ∈ G}, and two distinct vertices x and y are adjacent if ⟨ x, y ⟩ is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.

For G a connected, reductive group over an algebraically closed field k of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of g : = Lie ( G ) and the unipotent variety of G to study the projective variety of elementary subalgebras of g of rank r, denoted E ( r , g ) . In the case that G is defined over F p , we define the category of F q -expressible subalgebras of g for q = p d , and prove that this category is isomorphic to a subcategory of Quillen's category of elementary abelian subgroups of the finite Chevalley group G ( F q ) . This isomorphism of categories leads to a correspondence between G-orbits of E ( r , g ) defined over F q and G-conjugacy classes of certain elementary abelian subgroups of rank rd in G ( F q ) which satisfy a closure property characterized by the Springer isomorphism. We use Magma to compute examples for G = GL n , n ≤ 5 .

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \u* of \u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E8. When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q), \u*(q)) of coadjoint orbits of U(q) on \u*(q). Since k(U(q), \u*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in our earlier paper. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q)., Comment: 14 pages; v2 23 pages; to appear in Transformation Groups

Let $$\Pi L^1$$ denote a direct power of $$L^1$$ , the two-element left zero semigroup with identity adjoined. We prove that the semigroups in the title are embeddable into transformation semigroups which naturally generalize the Vagner one-point completion of the symmetric inverse semigroup. We give some examples and show that our representation provides a convenient way to construct E-unitary covers and embeddings of E-unitary such semigroups into semidirect products of $$\Pi L^1$$ -embeddable bands by groups.

Let H be a quadratic homogeneous polynomial map of dimension n over an infinite field in which 2 is invertible such that its Jacobian JH is nilpotent. Meisters and Olech have shown that JH is strongly nilpotent if n ≤ 4 . They also proved that it is not true when n = 5 . We show that if rank J H ≤ 2 and n arbitrary, then JH is strongly nilpotent. We also give examples to show that this is no longer true for any rank and dimension as long as the rank is greater than 2.

The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence for G L ( n ) GL(n) over a nonarchimedean local field. In this article we make progress toward a purely local proof of this fact. To wit, we find a family of formal schemes V \mathcal {V} such that the generic fiber of V \mathcal {V} is isomorphic to an open subset of Lubin-Tate space at infinite level, and such that the middle cohomology of the special fiber of V \mathcal {V} realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degree n n extension). The special fiber of V \mathcal {V} is related to an interesting variety X X , defined over a finite field, which is “maximal” in the sense that the number of rational points of X X is the largest possible among varieties with the same Betti numbers as X X . The variety X X is derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups.

Let G be a simple algebraic group defined over an algebraically closed field K and let G = G ( K ) . We consider here the problem when an element g ∈ G can be presented in the form g = e n ( g 1 , g 2 ) : = [ g 1 , [ g 1 , ⋯ [ g 1 ︸ n -times , g 2 ] ⋯ ] ] for some g 1 , g 2 ∈ G . This can always be done if g is a semisimple or a unipotent element. Also, it is known that this holds for G = SL 2 ( K ) . Here we prove that g = e n ( g 1 , g 2 ) for every n if G is a group of type B 2 or G 2 and we prove g = e 2 ( g 1 , g 2 ) = [ g 1 , [ g 1 , g 2 ] ] if G = PGL 3 ( K ) . We also show g = e 2 ( g 1 , g 2 ) = [ g 1 , [ g 1 , g 2 ] ] if g is a regular element of G and G is a group of rank 3. For any simple group G we give a criterion for some “general” regular elements of G to be presented in the form [ g 1 , [ g 1 , g 2 ] ] .

Let k be a finitely generated field of characteristic 0 embedded into \({\mathbb{C},\,X}\) a smooth, separated and geometrically connected scheme over k with generic point \({\eta}\) and \({f : Y \rightarrow X}\) a smooth proper morphism. Let \({f^{an}_{\mathbb{C}} : Y^{an}_{\mathbb{C}} \rightarrow X^{an}_{\mathbb{C}}}\) denote the associated morphism of complex analytic spaces. For \({x \in X(\mathbb{C})}\), write H for the Betti cohomology of \({Y^{an}_{\mathbb{C}, x}}\) with coefficients in \({\mathbb{Q}}\) and for \({x \in X}\), write \({{\rm H}_{\ell}}\) for the l-adic cohomology of \({Y_{\overline{x}}}\) with coefficients in \({\mathbb{Q}_{\ell}}\) (Under our assumptions on \({f : Y \rightarrow X,\,{\rm H}}\) and \({{\rm H}_{\ell}}\) are independent of x). For every prime l, let \({X^{ex}_{\ell}}\) be the set of all \({x \in X}\) where the Zariski closure \({G_{\ell, x}}\) of the image of the Galois representation \({\Gamma_{k(x)} \rightarrow {\rm GL}({\rm H}_{\ell})}\) has dimension strictly smaller than the dimension of \({G_{\ell,\eta}}\). By previous works of A. Tamagawa and the author, \({X^{ex}_{\ell}}\) is ‘small’ in the sense that if X is a curve then for every integer \({\delta \geq 1}\) the set of all \({x \in X^{ex}_{\ell}}\) with \({[k(x) : k] \leq \delta}\) is finite. Set \({X^{ex} := {\bigcap}_{\ell} X^{ex}_{\ell}}\). The Tate conjectures predict that for every \({x \in X}\) the \({G_{\ell, x}}\) are defined over \({\mathbb{Q}}\), reductive and independent of l hence, in particular, that the sets \({X^{ex}_{\ell}}\) are independent of l. Let \({\overline{G}}\) denote the Zariski closure of the image of the monodromy representation \({\pi_{1} (X_{\mathbb{C}}^{an}; \, x) \rightarrow {\rm GL}({\rm H})}\). Then \({\overline{G}}\) is a semi-simple algebraic group of rank—say—r. The main result of this note is that for \({x \notin X^{ex},\,G_{\ell, x} \cap \overline{G}_{\mathbb{Q}_{\ell}}}\) is a semi-simple algebraic group of rank r. This implies in particular that: (1) If \({\overline{G}_{\overline{\mathbb{Q}}}}\) has only simple factors of type \({A_{n}}\) then \({X^{ex}_{\ell}}\) is independent of l; (2) For every prime l and \({x \notin X^{ex}}\) the unipotent radical of \({G_{\ell, x}}\) coincides with the unipotent radical of \({G_{\ell, \eta}}\) and, in particular, is independent of \({x \notin X^{ex}}\); (3) For every prime l, if there exists \({x_{\ell} \in X}\) such that \({G_{\ell, x_{\ell}}}\) is reductive then for every \({x \notin X^{ex},\,G_{\ell, x}}\) is reductive. (3) applies in particular when \({{\rm H}}\) is a geometrically irreducible \({\overline{G}}\)-module. This implies, for instance, that apart from a few exceptional cases, for every r-tuple \({\underline{d}=(d_{1}, \dots, d_{r})}\) of integers \({\geq 2}\) there exists a non-singular complete intersection in \({\mathbb{P}^{n+r}_{\mathbb{Q}}}\) with multi-degree \({\underline{d}}\) for which the Tate semi-simplicity conjecture holds (for every prime l).

We show that over any field $k$, a smooth unipotent algebraic $k$-group is special if and only if it is $k$-split., Comment: arXiv admin note: text overlap with arXiv:1012.2984

We consider unipotent, weakly cancellative, left cancellative, right cancellative, cancellative monoids and groups with the unary operation of mapping onto the identity element and bands with the identity mapping as unary operations. Then form Malcev products of each of the former with varieties of rectangular bands, semilattices, normal bands and all bands. Regarding the resulting semigroups with the induced unary operations, we characterize these classes, some of them we provide with a construction, and determine a basis for their implications, for they all turn out to be quasivarieties. We also determine their relationship which we clarify by an inclusion diagram.

Let$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$be a semigroup. Elements$a,b$of$S$are$\widetilde{\mathscr{R}}$-related if they have the same idempotent left identities. Then$S$is weakly left ample if (1) idempotents of$S$commute, (2) $\widetilde{\mathscr{R}}$is a left congruence, (3) for any$a \in S$,$a$is$\widetilde{\mathscr{R}}$-related to a (unique) idempotent, say$a^+$, and (4) for any element$a$and idempotent$e$of$S$,$ae=(ae)^+a$. Elements$a,b$of$S$are$\mathscr{R}^*$-related if, for any$x,y \in S^1$,$xa=ya$if and only if$xb=yb$. Then$S$is left ample if it satisfies (1), (3) and (4) relative to$\mathscr{R}^*$instead of$\widetilde{\mathscr{R}}$. Further,$S$is (weakly) ample if it is both (weakly) left and right ample. We establish several characterizations of these classes of semigroups. For weakly left ample ones we provide a construction of all such semigroups with zero all of whose nonzero idempotents are primitive. Among characterizations of weakly ample semigroups figure (strong) semilattices of unipotent monoids, and among those for ample semigroups, (strong) semilattices of cancellative monoids. This describes the structure of these two classes of semigroups in an optimal way, while, for the ‘one-sided’ case, the problem of structure remains open.

In this paper we are interested in computing representations of the fundamental group of SL 3-manifold into \(\mathrm{PGL}(3,\mathbb {C})\) (in particular in \(\mathrm{PGL}(2,\mathbb {C}), \mathrm{PGL}(3,\mathbb {R})\) and \(\mathrm{PU}(2,1)\)). The representations are obtained by gluing decorated tetrahedra of flags as in Falbel (J Differ Geom 79:69–110, 2008), Bergeron et al. (Tetrahedra of flags, volume and homology of SL(3), 2011). We list complete computations (giving 0-dimensional or 1-dimensional solution sets (for unipotent boundary holonomy) for the first complete hyperbolic non-compact manifolds with finite volume which are obtained gluing less than three tetrahedra with a description of the computer methods used to find them. The methods we use work for non-unipotent boundary holonomy as shown in some examples.

Let KG be the group ring of a group G over a field K. Let * be an involution of a group G extended linearly to the group ring KG. Suppose that G is a torsion group without 2-elements and K is a field with characteristic different from 2. We prove that KG is Lie *-nilpotent if and only if KG is Lie nilpotent.

We compute the Lefschetz zeta function for quasi-unipotent maps on the n-dimensional torus, using arithmetical properties of the number n. In particular we compute the Lefschetz zeta function for quasi-unipotent maps, such that the characteristic polynomial of the induced map on the first homology group is the th cyclotomic polynomial, when is an odd prime. These computations involve fine combinatorial properties of roots of unity. We also show that the Lefchetz zeta functions for quasi-unipotent maps on are rational functions of total degree zero. We use these results in order to characterized the minimal set of Lefschetz periods for quasi-unipotent maps on , having finitely many periodic points all of them hyperbolic. Among this class of maps are the Morse–Smale diffeomorphisms of .

Let k be an arbitrary field, let G be a (smooth) linear algebraic group over k, and let U be a vector group over k on which G acts by automorphisms of algebraic groups. The action of G on U is said to be linear if there is a G-equivariant isomorphism of algebraic groups U ≃ Lie ( U ) . Suppose that G is connected and that the unipotent radical of G is defined over k. If the G-module Lie ( U ) is simple, we show that the action of G on U is linear. If G acts by automorphisms on a connected, split unipotent group U, we deduce that U has a filtration by G-invariant closed subgroups for which the successive factors are vector groups with a linear action of G. When G is connected and the unipotent radical of G is defined and split over k, this verifies an assumption made in earlier work of the author on the existence of Levi factors. On the other hand, for any field k of positive characteristic we show that if the category of representations of G is not semisimple, there is an action of G on a suitable vector group U which is not linear.

We show that a conjecture of Groves for modules over nilpotent groups of class 2 holds for the codimension 2 case with certain assumptions.

Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \emph{of finite type} if there are only finitely many $ K $-orbits on it. In this paper, we give a parametrization of $ K $-orbits on $ G/P \times K/Q $ in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of $ P \subset G $ or $ Q \subset K $ is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of $ K $-spherical flag varieties $ G/P $ and $ G $-spherical homogeneous spaces $ G/Q $., 47 pages, 3 tables; add all the details of the classification

Let \(X\) be a smooth variety over an algebraically closed field of characteristic \(p > 0, Z\) a smooth divisor, and \(j: U=X {\setminus } Z \rightarrow X\) the natural inclusion. We introduce in an axiomatic way the notion of a \(V\)-filtration on unit \(F\)-crystals and prove such axioms determine a unique filtration. It is shown that if \(\mathcal M \) is a tame unit \(F\)-crystal on \(U\), then such a \(V\)-filtration along \(Z\) exists on \(j_*\mathcal M \). The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton–Kisin Riemann–Hilbert correspondence. A few applications to \(\mathbb A ^1\) and gluing are then discussed.

Unitriangular factorization is a presentation of a linear group as a product of unipotent radicals of a Borel subgroup and its opposite. Whether this decomposition is known for Chevalley groups over rings of stable rank 1 and some Dedekind rings of arithmetic type, the case of twisted groups has been studied only over finite fields. In the present paper we give a much simpler proof for twisted groups over finite fields and the field of complex numbers.

The concept of the derived π-length for finite π-soluble groups is introduced and its elementary properties are described. The dependence between the π-length, nilpotent π-length, and derived π-length, and also between the derived and nilpotent lengths of a π-Hall subgroup, is determined.

Let G G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K K and let V V be a unipotent representation of G ( K ) G(K) (for example, an Iwahori-spherical representation). We calculate the character of V V at compact very regular elements of G ( K ) G(K) .

In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic p>0. In particular, we show that every admissible pair for such a group G gives rise to an L-packet of character sheaves on G, and that, conversely, every L-packet of character sheaves on G arises from a (non-unique) admissible pair. In the appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck-Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third appendix proves that the "naive" definition of the equivariant constructible derived category with respect to a unipotent algebraic group is equivalent to the "correct" one., 106 pages, LaTeX; to appear in Selecta Mathematica

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p>0$, and assume $p$ is good for $G$. Let $P$ be a parabolic subgroup with unipotent radical $U$. For $r \ge 1$, denote by $\mathbb{G}_{a(r)}$ the $r$-th Frobenius kernel of $\mathbb{G}_a$. We prove that if the nilpotence class of $U$ is less than $p$, then any embedding of $\mathbb{G}_{a(r)}$ in $U$ lies inside a one-parameter subgroup of $U$, and there is a canonical way in which to choose such a subgroup. Applying this result, we prove that if $p$ is at least as big as the Coxeter number of $G$, then the cohomological variety of $G_{(r)}$ is homeomorphic to the variety of $r$-tuples of commuting elements in $\mathcal{N}_1(\mathfrak{g})$, the $[p]$-nilpotent cone of $Lie(G)$., 15 pages

We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms. A reductive symmetric space is a homogeneous space G / H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution σ on G . Let P be a parabolic subgroup such that σ ( P ) is opposite to P and let N P be the unipotent radical of P . For a compactly supported smooth function ϕ on G / H , we define R P ( ϕ ) ( g ) to be the integral of N P ∋ n ↦ ϕ ( g n ⋅ H ) over N P . The Radon transform R P thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions. For these transforms we derive support theorems in which the support of ϕ is (partially) characterized in terms of the support of R P ϕ . The proof is based on the relation between the Radon transform and the Fourier transform on G / H , and a Paley–Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.

Let u be a unipotent element of a simple algebraic group G over a field k of characteristic p. We develop a method for computing the connected component of Z(C-G(u)) in the cases where p is positive and both it and the rank of G are small enough. The method is then carried out for G of type G(2), F-4 and E-6 in bad characteristic. (C) 2013 Elsevier Inc. All rights reserved.

In this paper, we introduce the smallest equivalence relation ν⁎ on a hypergroup H such that the quotient H/ν⁎, the set of all equivalence classes, is a nilpotent group. The characterization of nilpotent groups via strongly regular relations is investigated and several results on the topic are presented.

Let W W be a finite Weyl group and σ \sigma a non-trivial graph automorphism of W W . We show a remarkable relation between the σ \sigma -twisted involution module for W W and the Frobenius–Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig and Vogan for the untwisted case and then allows us to state a general result valid for any finite group of Lie type. Inspired by recent work of Marberg, we also formally define Frobenius–Schur indicators for “unipotent characters” of twisted dihedral groups.

We show that several character correspondences for finite reductive groups $$G$$ are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to $$G$$ has connected center. The correspondences we consider are the so-called Jordan decomposition of characters introduced by Lusztig and the generalized Harish-Chandra theory of unipotent characters due to Broue–Malle–Michel. In addition we consider a correspondence giving character extensions, due to the second author, in order to verify the inductive McKay condition from Isaacs–Malle–Navarro for the non-abelian finite simple groups of Lie types $$^3\mathsf{D }_4,\mathsf{E }_8,\mathsf{F }_4,^2\mathsf{F }_4$$ , and $$\mathsf{G }_2$$ .

We show that the statement analogous to the Mumford–Tate conjecture for Abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image of the absolute Galois group with the unipotent part of the motivic fundamental group.

We study the effect of Alvis-Curtis duality on the unipotent representations of $\mathrm{GL}_n(q)$ in non-defining characteristic $\ell$. We show that the permutation induced on the simple modules can be expressed in terms of a generalization of the Mullineux involution on the set of all partitions, which involves both $\ell$ and the order of $q$ modulo $\ell$.