Let $g_e$ be the centraliser of a nilpotent element $e$ in a finite dimensional simple Lie algebra $g$ of rank $l$ over an algebraically closed field of characteristic 0. We investigate the algebra $S(g_e)^{g_e}$ of symmetric invariants of $g_e$ and prove that if $g$ is of type $A$ or $C$, then $S(g_e)^{g_e}$ is always a graded polynomial algebra in $l$ variables. We show that this continues to hold for some nilpotent elements in the Lie algebras of other types. In type $A$ we prove that $S(g_e)^{g_e}$ is freely generated by a regular sequence in $S(g_e)$ and describe the tangent cone at $e$ to the nilpotent variety of $g$.
In this paper, we study the index for several natural classes of non-reductive subalgebras of semisimple Lie algebras. Namely, we look at parabolic subalgebras, centralisers of nilpotent elements, and the normalisers of the centralisers. We discuss a conjecture of Elashvili to the effect that the index of any centraliser is equal to the rank of the semisimple algebra in question. It is shown that Elashvili's conjecture is true for `small' and `large' orbits. Some properties of the index for the normaliser of the centraliser are proved. In particular, we prove that, for a regular nilpotent elem…
The index of a finite-dimensional Lie algebra $g$ is the minimum of dimensions of stabilisers $g_α$ of elements $α\in g^*$. Let $g$ be a reductive Lie algebra and $z(x)$ a centraliser of a nilpotent element $x\in g$. Elashvili has conjectured that the index of the centraliser $z(x)$ equals the index of $g$, i.e., the rank of $g$. Here Elashvili's conjecture is proved for reductive Lie algebras of classical type. It is shown that in cases $g=gl_n$ and $g=sp_{2n}$ the coadjoint action of $z(x)$ has a generic stabiliser. Also, we give an example of a nilpotent element $x\in so_8$ such that the co…
We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the reductive quotients as stated in [Liebeck, Seitz; Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras].
In this thesis we discuss some properties of centralisers in classical Lie algebas and related structures. Our results follow three distinct but related themes: the modular representation theory of centralisers, the sheets of simple Lie algebras and the representation theory of finite W-algebras and enveloping algebras.
Let $g$ be a classical Lie algebra, i.e., either $gl_n$, $sp_n$, or $so_n$ and let $e\in g$ be a nilpotent element. We study various properties of centralisers $g_e$. The first four sections deal with rather elementary questions, like the centre of $g_e$, commuting varieties associated with $g_e$, or centralisers of commuting pairs. The second half of the paper addresses problems related to different Poisson structures on $g_e^*$ and symmetric invariants of $g_e$.
We interpret a result of S. Oehms as a statement about the symplectic ideal. We use this result to prove a double centraliser theorem for the symplectic group acting on \bigoplus_{r=0}^s\otimes^rV, where V is the natural module for the symplectic group. This result was obtained in characteristic zero by H. Weyl. Furthermore we use this to extend to arbitrary connected reductive groups G with simply connected derived group the earlier result of the author that the algebra K[G]^g of infinitesimal invariants in the algebra of regular functions on G is a unique factorisation domain.
We extend the classical result asserting that the twisting operator preserves certain Deligne--Lusztig character values for truncated formal power series; along the way we discuss some properties of centralisers. This leads us to the construction of an action of $\mathrm{GL}_n(\mathbb{F}_q[[π]]/π^r)$ on a Springer fibre intersected by Deligne--Lusztig varieties; we determine the primitivities of the induced cohomological representations for single cycles. The case of $\mathrm{SL}_2$ over finite dual numbers is presented with a criterion on semisimple orbit representations.
We use the rational tableaux introduced by Stembridge to give a bideterminant basis for a normal reductive monoid and for its variety of noninvertible elements. We also obtain a bideterminant basis for the full coordinate ring of the general linear group and for all its truncations with respect to saturated sets. Finally, we deduce an alternative proof of the double centraliser theorem for the rational Schur algebra and the walled Brauer algebra over an arbitrary infinite base field which was first obtained by Dipper, Doty and Stoll.
Let $\g$ be a classical Lie algebra, $e$ a nilpotent of $\g$ element and $\gt g_e$ the centraliser of $e$ in $\g$. We prove that $\g_e=[\g_e,\g_e]$ if and only if $e$ is rigid. It is also shown that if $e$ is contained in $[\g_e,\g_e]$, then the nilpotent radical of $\g_e$ coincides with $[\g(1)_e,\g_e]$, where $\g(1)_e$ is an eigenspace of a characteristic of $e$ with the eigenvalue 1.
We present an application of Hodge theory towards the study of irreducible unitary representations of reductive Lie groups. We describe a conjecture about such representations and discuss some progress towards its proof.
The commuting variety of a reductive Lie algebra ${\goth g}$ is the underlying variety of a well defined subscheme of $\gg g{}$. In this note, it is proved that this scheme is normal. In particular, its ideal of definition is a prime ideal.
We study the coadjoint representation of contractions of reductive Lie algebras associated with symmetric decompositions. Let $\frak g=\frak g_0\oplus \frak g_1$ be a symmetric decomposition of a reductive Lie algebra $\frak g$. Then the semi-direct product of $\frak g_0$ and the $\frak g_0$-module $\frak g_1$ is a contraction of $\frak g$. We conjecture that these contractions have many properties in common with reductive Lie algebras. In particular, it is proved that in many cases the algebra of invariants is polynomial. We also discuss the so-called "codim--2 property" for coadjoint represe…
For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in the study of the commuting variety. In thisnote, we prove that $X$ is Gorenstein with rational singularities.
This paper is an introduction, in a simplified setting, to Lusztig's theory of character sheaves. It develops a notion of character sheaves on reductive Lie algebras which is more general then such notion of Lusztig, and closer to Lusztig's theory of character sheaves on groups. The development is self contained and independent of the characteristic $p$ of the ground field. The results for Lie algebras are then used to give simple and uniform proofs for some of Lusztig's results on groups.
The generalized commuting and isospectral commuting varieties of a reductive Lie algebra have been introduced in a preceding article. In this note, it is proved that their normalizations are Gorenstein with rational singularities. Moreover, their canonical modules are free of rank 1. In particular, the usual commuting variety is Gorenstein with rational singularities and its canonical module is free of rank 1.
In this paper we prove two formulas for the characters of representations of reductive groups. Both express the character of a representation in terms of the same geometric data attached to it. When specialized to the case of a compact Lie group, one of them reduces to Kirillov's character formula in the compact case, and the other, to an application of the Atiyah-Bott fixed point formula to the Borel-Weil realization of the representation.
The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of Popov. In [Po], the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. [Po, Remark 1.13] suggests that the arguments and methods of [Po] are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain som…
Let $G$ be a simple simply-connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}={\rm Lie}(G)$. We discuss various properties of nilpotent orbits in $\mathfrak{g}$, which have previously only been considered over $\mathbb{C}$. Using a combination of theoretical and computational methods, we extend to positive characteristic various calculations of de Graaf with nilpotent orbits in exceptional Lie algebras. In particular, we classify those orbits which are reachable, those which satisfy a certain related condition due to Panyushev, and deter…
Let \q be a simple algebraic group of type A or C over a field of good positive characteristic. We show for any x \in \q =\Lie(Q) that the invariant algebra S(\q_x)^{\q_x} is generated by the p^{th} power subalgebra and the mod p reduction of the characteristic zero invariant algebra. The latter algebra is known to be polynomial \cite{PPY} and we show that it remains so after reduction. Using a theory of symmetrisation in positive characteristic we prove the analogue of this result in the enveloping algebra, where the p-centre plays the role of the p^{th} power subalgebra. In Zassenhaus' found…
We carry an intrinsic approach to the study of the connectedness of the moduli space $\mathcal{M}_G$ of $G$-Higgs bundles, over a compact Riemann surface, when $G$ is a complex reductive (not necessarily connected) Lie group. We prove that the number of connected components of $\mathcal{M}_G$ is indexed by the corresponding topological invariants. In particular, this gives an alternative proof of the counting by J. Li of the number of connected components of the moduli space of flat $G$-connections in the case in which $G$ is connected and semisimple.
We say that a finite dimensional Lie algebra is quasi-reductive if it has a linear form whose stabilizer for the coadjoint representation, modulo the center, is a reductive Lie algebra with a center consisting of semisimple elements. Parabolic subalgebras of a semisimple Lie algebra are not always quasi-reductive (except in types A or C by work of Panyushev). The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso. In this paper, we investigate the quasi-reductivity of biparabolic subalgebras…
Let ${\goth g}$ be a finite dimensional complex reductive Lie algebra and $\dv ..$ an invariant non degenerated bilinear form on ${\goth g}\times {\goth g}$ which extends the Killing form of $[{\goth g},{\goth g}]$. We define the homology complex $C_{\bullet}({\goth g})$. Its space is the algebra $\tk {\Bbb C}{\e Sg}\tk {\Bbb C}{\e Sg}\ex {}{\goth g}$ where $\e Sg$ and $\ex {}{\goth g}$ are the symmetric and exterior algebras of ${\goth g}$. The differential of $C_{\bullet}({\goth g})$ is the $\tk {\Bbb C}{\e Sg}\e Sg$-derivation which associates to the element $v$ of ${\goth g}$ the function …
We give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that we take draws heavily for the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows us to find a bridge between multiloop algebras and the work of J. Tits on reductive groups over complete local fields…
We study the possibility of factoring a covariant distribution on reductive Lie algebras as finite sum of products of an invariant distribution by a covariant polynomial.
The goal of this paper is to extend the standard invariant-theoretic design, well-developed in the reductive case, to the setting of representation of certain non-reductive groups. This concerns the following notions and results: the existence of generic stabilisers and generic isotropy groups for finite-dimensional representations; structure of the fields and algebras of invariants; quotient morphisms and structure of their fibres. One of the main tools for obtaining non-reductive Lie algebras is the semi-direct product construction. We observe that there are surprisingly many non-reductive L…
In this paper, we give a simple formula for sectional curvatures on the general linear group, which is also valid for many other matrix groups. Similar formula is given for a reductive Lie group. We also discuss the relation between commuting matrices and zero sectional curvature.
We prove an explicit formula for the invariant $μ(\Lg)$ for finite-dimensional semisimple, and reductive Lie algebras $\Lg$ over $\C$. Here $μ(\Lg)$ is the minimal dimension of a faithful linear representation of $\Lg$. The result can be used to study Dynkin's classification of maximal reductive subalgebras of semisimple Lie algebras.
The simplicity of the induced modules for reductive Lie algebras over an algebraically closed field of positive characteristic is studied, and a necessary and sufficient condition for the simplicity is given.
Let g = Lie(G) be the Lie algebra of a simple algebraic group G over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g_e = Lie(G_e) where G_e stands for the stabiliser of e in G. For g classical, we give an explicit combinatorial formula for the codimension of [g_e, g_e] in g_e and use it to determine those e in g for which the largest commutative quotient U(g,e)^{ab} of the finite W-algebra U(g,e) is isomorphic to a polynomial algebra. It turns out that this happens if and only if e lies in a unique sheet of g. The nilpotent elements with this prop…
The nilpotent cone of a reductive Lie algebra has a desingularization given by thecotangent bundle of the flag variety. Analogously, the nullcone of a cartesianpower of the algebra has a desingularization given by a vector bundle over theflag variety. As for the nullcone, the subvariety of elements whose componentsare in a same Borel subalgebra, has a desingularization given by a vector bundle overthe flag variety. In this note, some properties of these varieties are given. Forthe study of the commuting variety, the analogous variety to the flag variety isthe closure in the Grassmannian of the…
This paper is a continuation of earlier work on generators of simple Lie algebras in arbitrary characteristic (see arXiv:0708.1711). We show that, in contrast to classical Lie algebras, simple graded Lie algebras of Cartan type S,H or K never enjoy the 'one-and-a-half generation' property. The methods rely on a study of centralisers in Cartan type Lie algebras.
The nilpotent bicone of a finite dimensional complex reductive Lie algebra g is the subset of elements in g x g whose subspace generated by the components is contained in the nilpotent cone of g. The main result of this note is that the nilpotent bicone is a complete intersection. This affirmatively answers a conjecture of Kraft-Wallach concerning the nullcone. In addition, we introduce and study the characteristic submodule of g. The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal o…
I connect an old result of mine on a Lie algebra generalization of the Amitsur-Levitski theorem with equations for sheets in a reductive Lie algebra and with recent results of Kostant-Wallach on the variety of singular elements in a reductive Lie algebra.
In this paper, we established the relationship among left-invariant flat connections on Lie groups, left-symmetric algebras, Frobenius Lie algebras and étale prehomogeneous vector spaces, gave a one-to-one correspondence between the left-symmetric Lie algebras with a right identity and the étale prehomogeneous vector spaces for a Lie group, and proved that, in essence, any left-symmetric structure on a reductive Lie algebra has a right identity, which implies that the classification of flat connections on a reductive Lie group $G$ amounts to that of étale prehomogeneous vector spaces for $G$. …
We prove for finite reductive groups $G$ of classical type, that every irreducible character of $L$ extends to its inertia group in $N$, where $L$ is an abelian centraliser of a Sylow $d$-torus $\mathbf S$ of $G$ and $N:=N_G(\mathbf S)$. This gives a precise description of the irreducible characters of $N$. Furthermore it enables us to verify the McKay conjecture in this situation for $G$ and some primes.
A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of its dual of reductive type, that is such that the quotient of its stabiliser by the centre of Q is a reductive subgroup of GL(q), where q=Lie(Q). Due to results of M. Duflo, coadjoint representation of a quasi-reductive Q possesses a so called maximal reductive stabiliser and knowing this subgroup, defined up to a conjugation in Q, one can describe all coadjoint orbits of reductive type. In this paper, we consider quasi-reductive parabolic subalgebras of simple complex Lie alg…
Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras under injections which preserve the root spaces. It is known that a root-reductive Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of finite-dimensional simple Lie algebras as well as copies of the three simple infinite-dimensional root-reductive Lie algebras sl_infty, so_infty, and sp_infty. As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra gl_infty were introduced and studied in a paper of Neeb and Penkov. I…
A Lie algebra is said to be generalised reductive if it is a direct sum of a semisimple Lie algebra and a commutative radical. In this paper we extend the BGG category $\mathcal{O}$ over complex semisimple Lie algebras to the category $\mathcal {O'}$ over complex generalised reductive Lie algebras. Then we make a preliminary research on the highest weight modules and the projective modules in this new category $\mathcal {O'}$, and generalize some conclusions in the classical case. As a critical difference from the complex semisimple Lie algebra case, we prove that there is no projective module…
In math.RT/0205144 we observed that, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters. The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character as sheaves on the partial flag variety corresponding to the singularity of the ch…
We prove that the nilpotent commuting variety of a reductive Lie algebra over an algebraically closed field of good characteristic is equidimensional. In characteristic zero, this confirms a conjecture of Vladimir Baranovsky. As a by-product, we obtain tat the punctual (local) Hilbert scheme parametrising the ideals of colength $n$ in $k[[X,Y]]$ is irreducible over any algebraically closed field $k$.
This paper generalizes the classification in a paper of Dimitrov and Penkov of Borel subalgebras of gl_infty. Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras along inclusions preserving the root spaces with respect to nested Cartan subalgebras. A Borel subalgebra of a root-reductive Lie algebra is by definition a maximal locally solvable subalgebra. The main general result of this paper is that a Borel subalgebra of an infinite-dimensional indecomposable root-reductive Lie algebra is the simultaneous stabilizer of a certain type of generalized flag in…
We show that Arthur's Paley-Wiener theorem for K-finite compactly supported smooth functions on a real reductive Lie group G of the Harish-Chandra class can be deduced from the Paley-Wiener theorem we established in the more general setting of a reductive symmetric space. In addition, we formulate an extension of Arthur's theorem to K-finite compactly supported generalized functions (distributions) on G and show that this result follows from the analogous result for reductive symmetric spaces as well.
In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups.
Let $\mathfrak{g}$ be a root-reductive Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$ with a splitting Borel subalgebra $\mathfrak{b}$ containing a splitting maximal toral subalgebra $\mathfrak{h}$. We study the category $\bar{\mathcal{O}}$ consisting of all $\mathfrak{h}$-weight $\mathfrak{g}$-modules which are locally $\mathfrak{b}$-finite and have finite-dimensional $\mathfrak{h}$-weight spaces. The focus is on very special Borel subalgebras called the Dynkin Borel subalgebras. This paper serves as an initial passage to the understanding of categories $\ma…
For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the alternating or symmetric square representation is irreducible or decomposes into an irreducible representation and a trivial representation.
Let $G$ be a complex reductive algebraic group, $g$ its Lie algebra and $h$ a reductive subalgebra of $g$, $n$ a positive integer. Consider the diagonal actions $G:g^n, N_G(h):h^n$. We study a relation between the algebra $C[h^n]^{N_G(h)}$ and its subalgebra consisting of restrictions to $h^n$ of elements of $C[g^n]^G$.
For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. In this paper we give an almost general proof of that conjecture.
Let $G_\cpl$ be a connected complex reductive Lie group, and $G$ be a real form. Let $(π,V)$ be a finite-dimensional irreducible representation of $G$. Assume $(π,V)$ admits a $G$ invariant hermitian form. {In \cite{Signature-of-a-rep-of-reductive}, an analog of the Weyl dimension formula that instead computes the signature of the invariant form for finite dimensional representations of complex Lie groups is given. The goal of this paper is to give a short alternate proof as an application of a more general twisted character theory of \cite{Dirac-Index-and-Twisted-Characters}.
We describe Serre functors for (generalisations of) the category O associated with a semi-simple complex Lie algebra. In our approach, projective-injective modules play an important role. They control the Serre functor in the case of a quasi-hereditary algebra having a double centraliser property with respect to a symmetric algebra. As an application of the double centraliser property and our description of Serre functors, we prove three conjectures of Khovanov about the projective-injective modules in the parabolic category O for sl_n.
I prefer taking off this paper for the moment because of a mistake in the lemma 2.1 of the secund version. Precisely, in the proof of this lemma, it is not clear that the morphism $r\_j$ is flat, that I claim it.
Vogan and Barbasch-Vogan attach two similar invariants to representations of a reductive Lie group, one by an algebraic process, the other analytic. They conjectured that the two invariants determine each other in a definite manner. Here we prove the conjecture. Our arguments involve two finer invariants -- the characteristic cycles of representations -- which are interesting in their own right.
