In this paper we prove that a recent condition of Lyons--Martínez--Navarro--Tiep, regarding the field of values of extensions of characters in principal blocks, is satisfied for all finite simple groups, which when combined with their results gives a new characterization of finite groups with a normal $\ell$-complement for a prime $\ell$. This leads us to study the distribution of characters in unipotent blocks of disconnected reductive groups and show that this is well-behaved under a generalization of $d$-Harish-Chandra theory. We go on to study the blockwise Galois--McKay (also known as the Alperin--McKay--Navarro) conjecture for the blocks of almost (quasi-)simple groups above unipotent blocks.
We prove a strengthening of Brauer's height zero conjecture for principal 2-blocks with Galois automorphisms. This requires a new extension of the Itô--Michler theorem for the prime~2, again with Galois automorphisms. We close, this time for odd primes $p$, with a new characterisation of $p$-closed groups via the decomposition numbers of certain characters.
In this paper, we construct twist automorphisms on quantum unipotent cells, which are quantum analogues of the Berenstein-Fomin-Zelevinsky twist automorphisms on unipotent cells. We show that those quantum twist automorphisms preserve the dual canonical bases of quantum unipotent cells. Moreover, we prove that quantum twist automorphisms are described by the syzygy functors for representations of preprojective algebras in the symmetric case. This is the quantum analogue of Geiß-Leclerc-Schröer's description, and Geiß-Leclerc-Schröer's results are essential in our proof. As a consequence, we sh…
Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of every involutorial automorphism (involution) of G is a spherical subgroup of G.
In this note, we prove that if every character of a finite group $G$ fixed by an order-2 Galois automorphism has odd degree, then $G$ has a normal Sylow $2$-subgroup. On the way, we study extensions of characters of $GL_n(q)$, $q$ odd, to the group extended by the transpose-inverse automorphism and prove that unipotent characters of $PSL_n(q)$ extend to rational characters of its automorphism group.
G. Navarro has conjectured a necessary and sufficient condition for a finite group $G$ to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of $G$. In a previous article, the author has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the require…
When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the values of irreducible characters of the extended groups on these elements. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.
In this paper, we prove quantum analogues of the Chamber Ansatz formulae for unipotent cells. These formulae imply that the quantum twist automorphisms, constructed by Kimura and the author, are generalizations of Berenstein-Rupel's quantum twist automorphisms for unipotent cells associated with the squares of acyclic Coxeter elements. This conclusion implies that the known compatibility between quantum twist automorphisms and dual canonical bases corresponds to the property conjectured by Berenstein and Rupel.
We generalize the theory of ordering character triples, developed by Navarro and Späth, by taking into account the action of Galois automorphisms on characters. This new technique, together with previous results of Ladisch and Turull, allows us to reduce the Galois--McKay conjecture to a question about simple groups.
If A is a finite dimensional nilpotent associative algebra over a finite field k, the set G=1+A of all formal expressions of the form 1+a, where a is an element of A, has a natural group structure, given by (1+a)(1+b)=1+(a+b+ab). A finite group arising in this way is called an algebra group. One can also consider G as a unipotent algebraic group over k. We study representations of G from the point of view of ``geometric character theory'' for algebraic groups over finite fields (cf. G. Lusztig, ``Character sheaves and generalizations'', math.RT/0309134). The main result of this paper is a cons…
Let $G\subset\GL(\BC^r)$ be a finite complex reflection group. We show that when $G$ is irreducible, apart from the exception $G=\Sgot_6$, as well as for a large class of non-irreducible groups, any automorphism of $G$ is the product of a central automorphism and of an automorphism which preserves the reflections. We show further that an automorphism which preserves the reflections is the product of an element of $N_{\GL(\BC^r)}(G)$ and of a "Galois" automorphism: we show that $\Gal(K/\BQ)$, where $K$ is the field of definition of $G$, injects into the group of outer automorphisms of $G$, and …
Let $W$ be a finite Weyl group and $\sg$ be a non-trivial graph automorphism of $W$. We show a remarkable relation between the $\sg$-twisted involution module for $W$ and the Frobenius--Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. This extends earlier results of Lusztig-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 study locally transitive actions of the commutative unipotent group $\Gan$ on a nondegenerate quadric in the projective space $¶^{n+1}$. It is shown that for each $n$ such an action is unique up to isomorphism.
We prove that for most groups of Lie type, the bijections used by Malle and Spaeth in the proof of Isaacs-Malle-Navarro's inductive McKay conditions for the prime 2 and odd primes dividing q - 1 are also equivariant with respect to certain Galois automorphisms. In particular, this shows that these bijections are candidates for proving Navarro-Spaeth-Vallejo's recently-posited inductive Galois-McKay conditions. On the way, we show that several simple groups of Lie type satisfy the McKay--Navarro conjecture for the prime 2.
We characterize finite groups having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block for p=2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.
Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel's equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In the present paper, we give the first known algorithm that calculates the differential Galois group of a third order parameterized linear differential equation.
Let K be a cyclic Galois extension of degree f over k and T a generator of the Galois group. For any v=(v_1,... , v_m)\in K^m such that v is linearly independent over k, and any 0< d < m the Gabidulin-like code C(v, T , d) is a maximum rank distance code in the space of f times m matrices over k of dimension fd. This construction unifies the ones available in the literature. We characterise the K-linear codes that are Gabidulin-like codes and determine their rank-metric automorphism group.
Let $F$ be a $p$-adic field and $\mathbf{U}$ be a unipotent group defined over $F$, and set $U=\mathbf{U}(F)$. Let $σ$ be an involution of $\mathbf{U}$ defined over $F$. Adapting the arguments of Yves Benoist in the real case, we prove the following result: an irreducible representation $π$ of $U$ is $U^σ$-distinguished if and only if it is $σ$-self-dual and in this case $\mathrm{Hom}_{U^σ}(π,\mathbb{C})$ has dimension one. When $σ$ is a Galois involution these results imply a bijective correspondence between the set $\mathrm{Irr}(U^σ)$ of isomorphism classes of irreducible representations of …
Lusztig's classification of unipotent representations of finite reductive groups depends only on the associated Weyl group $W$ (endowed with its Frobenius automorphism). All the structural questions (families, Harish-Chandra series, partition into blocks...) have an answer in a combinatorics that can be entirely built directly from $W$. Over the years, we have noticed that the same combinatorics seems to be encoded in the Poisson geometry of a Calogero-Moser space associated with $W$ (roughly speaking, families correspond to ${\mathbb{C}}^\times$-fixed points, Harish-Chandra series correspond …
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 $W$. We give a parametrization {\it 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 will be the theme of a forthcoming paper.
The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without par…
Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$. Let ${\mathcal U}_n(A)$ be the subgroup of $\mathrm{GL}_n(A)$ of all upper triangular matrices with 1's along the main diagonal. Let $P=H_n(A)\rtimes {\mathcal U}_n(A)$, where ${\mathcal U}_n(A)$ acts on $H_n(A)$ by $*$-congruence transformations. We may view $P$ as a unipotent subgroup of either a symplectic group $\mathrm{Sp}_{2n}(A)$, if $*=1_A$ (in which c…
We study the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group $W$. We determine some of (and conjecturally all) the ${\mathbb{C}}^\times$-fixed points of its unique irreducible component of maximal dimension in terms of the character table of $W$. This is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.
We introduce the concept of a Galois covering of a pointed coalgebra. The theory developed shows that Galois coverings of pointed coalgebras can be concretely expressed by smash coproducts using the coaction of the automorphism group of the covering. Thus the theory of Galois coverings is seen to be equivalent to group gradings of coalgebras. An advantageous feature of the coalgebra theory is that neither the grading group nor the quiver is assumed finite in order to obtain a smash product coalgebra.
We prove that finite groups have the same complex character tables iff the group algebras are twisted forms of each other as Drinfel'd quasi-bialgebras or iff there is non-associative bi-Galois algebra over these groups. The interpretations of class-preserving automorphisms and permutation representations with the same character in terms of Drinfel'd algebras are also given.
We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space.
The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup of G. Let W be a subgroup of G such that Ad(W) is contained in the Zariski closure (in the group of automorphisms of the Lie algebra of G) of the subgroup generated by the unipotent elements of Ad(W). Then any finite ergodic invariant measure for the action of W on G/Gamma is a h…
Let $G$ be a finite group isomorphic to $SL_n(q)$ or $SU_n(q)$ for some prime power $q$. In this paper, we give an explicit description of the action of automorphisms of $G$ on the set of its irreducible complex characters. This is done by showing that irreducible constituents of restrictions of irreducible characters of $GL_n(q)$ (resp. $GU_n(q)$) to $SL_n(q)$ (resp. $SU_n(q)$) can be distinguished by the rational classes of their unipotent support which are equivariant under the action of automorphisms. Meanwhile, we give a criterion to explicitly determine whether an irreducible character i…
Given the spherical subalgebra $B$ of a rational Cherednik algebra, we aim to classify all finite groups $Γ$ for which there exists a domain $R$ on which $Γ$ acts by ring automorphisms, such that $B=R^Γ.$ We describe such groups in terms of geometry of the center of the reduction of $B$ modulo a large prime.
This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for functors on Artin local rings. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmeti…
To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than those coming from deformations of "abelian" Galois representations induced by the Tate module of associated Jacobian variety. We develop an arithmetic deformation theory of graded Lie algebras with finite…
