Levi Factor Research Articles

Systems of second-order linear ODE’s with constant coefficients and their symmetries II. The case of non-diagonal coefficient matrices

We complete the analysis of the symmetry algebra L for systems of n second-order linear ODEs with constant real coefficients, by studying the case of coefficient matrices having a non-diagonal Jordan canonical form. We also classify the Levi factor (maximal semisimple subalgebra) of L , showing that it is completely determined by the Jordan form. A universal formula for the dimension of the symmetry algebra of such systems is given. As application, the case n = 5 is analyzed.

Slices for biparabolics of index 1

Let \( \mathfrak{a} \) be an algebraic Lie subalgebra of a simple Lie algebra \( \mathfrak{g} \) with index \( \mathfrak{a} \) ≤ rank \( \mathfrak{g} \). Let \( Y\left( \mathfrak{a} \right) \) denote the algebra of \( \mathfrak{a} \) invariant polynomial functions on \( {\mathfrak{a}^*} \). An algebraic slice for \( \mathfrak{a} \) is an affine subspace η + V with \( \eta \in {\mathfrak{a}^*} \) and \( V \subset {\mathfrak{a}^*} \) subspace of dimension index \( \mathfrak{a} \) such that restriction of function induces an isomorphism of \( Y\left( \mathfrak{a} \right) \) onto the algebra R[η + V] of regular functions on η + V. Slices have been obtained in a number of cases through the construction of an adapted pair (h, η) in which \( h \in \mathfrak{a} \) is ad-semisimple, η is a regular element of \( {\mathfrak{a}^*} \) which is an eigenvector for h of eigenvalue minus one and V is an h stable complement to \( \left( {{\text{ad}}\;\mathfrak{a}} \right)\eta \) in \( {\mathfrak{a}^*} \). The classical case is for \( \mathfrak{g} \) semisimple [16], [17]. Yet rather recently many other cases have been provided; for example, if \( \mathfrak{g} \) is of type A and \( \mathfrak{a} \) is a “truncated biparabolic” [12] or a centralizer [13]. In some of these cases (in particular when the biparabolic is a Borel subalgebra) it was found [13], [14], that η could be taken to be the restriction of a regular nilpotent element in \( \mathfrak{g} \). Moreover, this calculation suggested [13] how to construct slices outside type A when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically, let \( \mathfrak{a} \) be a truncated biparabolic of index one. (This only arises if \( \mathfrak{g} \) is of type A and \( \mathfrak{a} \) is the derived algebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes are coprime.) In this case it is shown that the second member of an adapted pair (h, η) for \( \mathfrak{a} \) is the restriction of a particularly carefully chosen regular nilpotent element of \( \mathfrak{g} \). A by-product of our analysis is the construction of a map from the set of pairs of coprime integers to the set of all finite ordered sequences of ±1.

Emerton's Jacquet functors for non-Borel parabolic subgroups

This paper studies Emerton's Jacquet module functor for locally analytic representations of p -adic reductive groups, introduced in citeemerton-jacquet. When P is a parabolic subgroup whose Levi factor M is not commutative, we show that passing to an isotypical subspace for the derived subgroup of M gives rise to essentially admissible locally analytic representations of the torus Z(M) , which have a natural interpretation in terms of rigid geometry. We use this to extend the construction in of eigenvarieties in citeemerton-interpolation by constructing eigenvarieties interpolating automorphic representations whose local components at p are not necessarily principal series.

On the structure of maximal solvable extensions and of Levi extensions of nilpotent Lie algebras

We establish an improved upper estimate on the dimension of any solvable Lie algebra with its nilradical isomorphic to a given nilpotent Lie algebra . Next we consider Levi decomposable algebras with a given nilradical and investigate restrictions on possible Levi factors originating from the structure of characteristic ideals of . We present a new perspective on Turkowski's classification of Levi decomposable algebras up to dimension 9.

Systems of second-order linear ODE’s with constant coefficients and their symmetries

Starting from the study of the symmetries of systems of 4 second-order linear ODEs with constant real coefficients, we determine the dimension and generators of the symmetry algebra for systems of n equations described by a diagonal Jordan canonical form. We further prove that some dimensions between the lower and upper bounds cannot be attained in the diagonal case, and classify the Levi factors of the symmetry algebras.

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class. In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.

Levi decompositions of a linear algebraic group

If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. In this paper we give some sufficient conditions for the existence and conjugacy of the Levi factors of G. Let $$ \mathcal{A} $$ be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth $$ \mathcal{A} $$ -group schemes $$ \mathcal{P} $$ whose generic fibers $$ {{\mathcal{P}} \left/ {K} \right.} $$ coincide with G; these are known as parahoric group schemes. The special fiber $$ {{\mathcal{P}} \left/ {K} \right.} $$ of a parahoric group scheme is a linear algebraic group over k. If G splits over an unramified extension of K, we show that $$ {{\mathcal{P}} \left/ {K} \right.} $$ has a Levi factor, and that any two Levi factors of $$ {{\mathcal{P}} \left/ {K} \right.} $$ are geometrically conjugate.

Jordan–Hölder reductions for principal Higgs bundles on curves

It is known that semistable sheaves V admit a filtration whose quotients are stable and have the same slope of V , named the Jordan–Hölder filtration. We give the analogous result for principal Higgs bundles on curves. Let G be a reductive algebraic group over C , if E = ( E , ϕ ) is a semistable principal Higgs G -bundle, there exists a parabolic subgroup P of G and an admissible reduction of the structure group of E to that parabolic such that the principal Higgs bundle obtained by extending the structure group to the Levi factor L ( P ) of P is a stable principal Higgs bundle. The extension of the structure group L ( P ) to G of the latter stable principal bundle is the graded module gr ( E ) .

On Eisenstein series and the cohomology of arithmetic groups

The automorphic cohomology of a reductive Q -group G captures essential analytic aspects of the arithmetic subgroups of G. The subspace spanned by all possible residues and principal values of derivatives of Eisenstein series, attached to cuspidal automorphic forms π on the Levi factor of proper parabolic Q -subgroups of G, forms the Eisenstein cohomology. We show that non-trivial classes can only arise if the point of evaluation features a “half-integral” property. Consequently, only the analytic behavior of the automorphic L-functions at half-integral arguments matters whether an Eisenstein series attached to a globally generic π gives rise to a residual class or not.

On the residual spectrum of split classical groups supported in the Siegel maximal parabolic subgroup

For the split symplectic and special orthogonal groups over a number field, we decompose the part of the residual spectrum supported in the maximal parabolic subgroup with the Levi factor isomorphic to GL n . The decomposition depends on the analytic properties of the symmetric and exterior square automorphic L-functions, but seems sufficient for the computation of the corresponding part of the Eisenstein cohomology. We also prove that if one assumed Arthur’s conjectural description of the discrete spectrum for the considered groups, then one would be able to find the poles of the L-functions in question, and would make the decomposition more precise.

Poisson structures on affine spaces and flag varieties. II

The standard Poisson structures on the flag varieties $G/P$ of a complex reductive algebraic group $G$ are investigated. It is shown that the orbits of symplectic leaves in $G/P$ under a fixed maximal torus of $G$ are smooth irreducible locally closed subvarieties of $G/P$, isomorphic to intersections of dual Schubert cells in the full flag variety $G/B$ of $G$, and their Zariski closures are explicitly computed. Two different proofs of the former result are presented. The first is in the framework of Poisson homogeneous spaces, and the second one uses an idea of weak splittings of surjective Poisson submersions, based on the notion of Poisson–Dirac submanifolds. For a parabolic subgroup $P$ with abelian unipotent radical (in which case $G/P$ is a Hermitian symmetric space of compact type), it is shown that all orbits of the standard Levi factor $L$ of $P$ on $G/P$ are complete Poisson subvarieties which are quotients of $L$, equipped with the standard Poisson structure. Moreover, it is proved that the Poisson structure on $G/P$ vanishes at all special base points for the $L$-orbits on $G/P$ constructed by Richardson, Röhrle, and Steinberg.

Induced Modules for Affine Lie Algebras

We study induced modules of nonzero central charge with arbitrary multiplicities over affine Lie algebras. For a given pseudo parabolic subalgebra P of an affine Lie algeb- ra G, our main result establishes the equivalence between a certain category of P-induced G-modules and the category of weight P-modules with injective action of the central element of G. In particular, the induction functor preserves irreducible modules. If P is a parabolic subalgebra with a finite-dimensional Levi factor then it defines a unique pseudo parabolic subalgebra P ps , P P ps . The structure of P-induced modules in this case is fully deter- mined by the structure of P ps -induced modules. These results generalize similar reductions in particular cases previously considered by V. Futorny, S. Konig, V. Mazorchuk (Forum Math. 13 (2001), 641-661), B. Cox (Pacific J. Math. 165 (1994), 269-294) and I. Dimitrov, V. Futorny, I. Penkov (Comm. Math. Phys. 250 (2004), 47-63).

Weight elements of Chevalley groups

The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field — the weight elements. These are the conjugates of certain semisimple elements hω(e) of extended Chevalley groups G = G(Φ ,K ), where ω is a weight of the dual root system Φ ∨ and e ∈ K ∗ . In the adjoint case the hω(e)'s were defined by Chevalley himself and in the simply connected case they were constructed by Berman and Moody. The conjugates of hω(e) are called weight elements of type ω .V arious constructions of weight elements are discussed in the paper, in particular, their action in irreducible rational representations and weight elements induced on a regularly embedded Chevalley subgroup by the conjugation action of a larger Chevalley group. It is proved that for a given x ∈ G all elements x(e )= xhω(e)x −1 , e ∈ K ∗ ,a part maybe from a finite number of them, lie in the same Bruhat coset BwB ,w herew is an involution of the Weyl group W = W (Φ). The elements hω(e) are particularly important when ω = � i is a microweight of Φ ∨ . The main result of the paper is a calculation of the factors of the Bruhat decomposition of microweight elements x(e )f or the case whereω = � i. It turns out that all nontrivial x(e)'s lie in the same Bruhat coset BwB ,w herew is a product of reflections in pairwise strictly orthogonal roots γ1 ,...,γ r+s. Moreover, if among these roots r are long and s are short, then r +2 s does not exceed the width of the unipotent radical of the ith maximal parabolic subgroup in G. A version of this result was first announced in a paper by the author in Soviet Mathematics: Doklady in 1988. From a technical viewpoint, this amounts to the determination of Borel orbits of a Levi factor of a parabolic subgroup with Abelian unipotent radical and generalizes some results of Richardson, Rohrle, and Steinberg. These results are instrumental in the description of overgroups of a split maximal torus and in the recent papers by the author and V. Nesterov on the geometry of tori.

On a theorem of Schmid

We establish for which parabolic subgroups P of a simply connected and semisimple algebraic group G with unipotent radical U and Levi factor H the two rings \mathbb k[G/H]^{U} and \mathbb k[U^-] are isomorphic as H algebras. We show the relation of this problem with a Theorem of Schmid and we compare the multiplications in the rings \mathbb k[U^-] and \mathbb k[G/H] .

Residues of Intertwining Operators for Classical Groups

Let be a symplectic or even orthogonal group over a p-adic field F, and M the Levi factor of a maximal parabolic subgroup of . Suppose that M has the shape of three blocks of the same size. Let be a supercuspidal representation of M. In this paper, we give a simple explicit expression for the residue of the standard intertwining operator for the parabolic induction of from M to G.

The Centralizer of a Nilpotent Section

Let F be an algebraically closed field and let G be a semisimple F-algebraic group for which the characteristic of F is very good. If X ∈ Lie(G) = Lie(G)(F) is a nilpotent element in the Lie algebra of G, and if C is the centralizer in G of X, we show that (i) the root datum of a Levi factor of C, and (ii) the component group C/C° both depend only on the Bala-Carter label of X; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when G is (simple and) of adjoint type.The proofs are achieved by studying the centralizer of a nilpotent section X in the Lie algebra of a suitable semisimple group scheme over a Noetherian, normal, local ring . When the centralizer of X is equidimensional on Spec(), a crucial result is that locally in the étale topology there is a smooth -subgroup scheme L of such that Lt is a Levi factor of for each t ∈ Spec ().

Normalizers of subsystem subgroups in finite groups of Lie type

Finite groups of Lie type form the greater part of known finite simple groups. An important class of subgroups of finite groups of Lie type are so-called reductive subgroups of maximal rank. These arise naturally as Levi factors of parabolic groups and as centralizers of semisimple elements, and also as subgroups with maximal tori. Moreover, reductive groups of maximal rank play an important part in inductive studies of subgroup structure of finite groups of Lie type. Yet a number of vital questions dealing in the internal structure of such subgroups are still not settled. In particular, we know which quasisimple groups may appear as central multipliers in the semisimple part of any reductive group of maximal rank, but we do not know how normalizers of those quasisimple groups are structured. The present paper is devoted to tackling this problem.

Group-theoretic compactification of Bruhat–Tits buildings

Let G F denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat–Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of G F in the Chabauty topology, which enables us to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat–Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We finally prove two parametrization theorems extending the Bruhat–Tits dictionary between maximal compact subgroups and vertices of X: one is about Zariski connected amenable subgroups and the other is about subgroups with distal adjoint action.

A slice theorem for truncated parabolics of index one and the Bezout equation

Let p be a parabolic subalgebra of s l ( n ) and p E its canonical truncation. In [A. Joseph, Parabolic actions in type A and their eigenslices, 7.19] it was conjectured that the coadjoint action in p E ∗ admits a slice. This was established [A. Joseph, Parabolic actions in type A and their eigenslices, 7.18] when p is invariant under the Dynkin diagram involution. In the present work, this is proved when the Levi factor of p has just two blocks of sizes p , q with p , q coprime. The solution gives rise to an algorithm for solving the Bezout equation r p − s q = − 1 . The construction involves finding an adapted pair ( h , y ) ∈ h E × ( p E ) reg , and here the choice of y is partly motivated by an observation [P. Tauvel, R.W.T. Yu, Sur l'indice de certaines algèbres de Lie, Ann. Inst. Fourier (Grenoble) 54 (2004) 1793–1810, 3.9] of P. Tauvel and R.W.T. Yu.

Structure of isometry group of bilinear spaces

We describe the structure of the isometry group G of a finite-dimensional bilinear space over an algebraically closed field of characteristic not two. If the space has no indecomposable degenerate orthogonal summands of odd dimension, it admits a canonical orthogonal decomposition into primary components and G is isomorphic to the direct product of the isometry groups of the primary components. Each of the latter groups is shown to be isomorphic to the centralizer in some classical group of a nilpotent element in the Lie algebra of that group. In the general case, the description of G is more complicated. We show that G is a semidirect product of a normal unipotent subgroup K with another subgroup which, in its turn, is a direct product of a group of the type described in the previous paragraph and another group H which we can describe explicitly. The group H has a Levi decomposition whose Levi factor is a direct product of several general linear groups of various degrees. We obtain simple formulae for the dimensions of H and K.