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 in 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^o 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 C of a nilpotent section X in the Lie algebra of a suitable semisimple group scheme over a Noetherian, normal, local ring A. When the centralizer of X is equidimensional on Spec(A), a crucial result is that locally in the etale topology there is a smooth A-subgroup scheme L of CC such that L_t is a Levi factor of C_t for each t in Spec(A).
To any pair of commuting n x n nilpotent matrices it is associated a pair of partitions of n. We describe a maximal nilpotent subalgebra of the centralizer of a given nilpotent n x n matrix and prove a conjecture of Polona Oblak which consists in an algorithm for the determination of the maximum partition which forms with a given partition a pair with the previous property.
We introduce a method to determine the maximum nilpotent orbit which intersects a variety of nilpotent matrices described by a strictly upper triangular matrix over a polynomial ring. We show that the result only depends on the ranks of its submatrices and we introduce conditions on a subvariety so that it intersects the same orbit. Then we describe a maximal nilpotent subalgebra of the centralizer of any nilpotent matrix; the previous method allows us to show that the maximum nilpotent orbit which intersects that centralizer only depends on which entries are identically 0 in that subalgebra. …
Let $G$ be a connected reductive group over an algebraically closed field $\Bbbk$. Under mild restrictions on the characteristic of $\Bbbk$, we show that any $G$-module with a good filtration also has a good filtration as a module for the reductive part of the centralizer of a nilpotent element $x$ in its Lie algebra.
In this paper, by establishing an explicit and combinatorial description of the centralizer of a distinguished nilpotent pair in a classical simple Lie algebra, we solve in the classical case Panyushev's Conjecture which says that distinguished nilpotent pairs are wonderful, and the classification problem on almost principal nilpotent pairs. More precisely, we show that disinguished nilpotent pairs are wonderful in types A, B and C, but they are not always wonderful in type D. Also, as the corollary of the classification of almost principal nilpotent pairs, we have that almost principal nilpot…
Let G be a semisimple algebraic group over a field K whose characteristic is very good for G, and let sigma be any G-equivariant isomorphism from the nilpotent variety to the unipotent variety; the map sigma is known as a Springer isomorphism. Let y in G(K), let Y in Lie(G)(K), and write C_y = C_G(y) and C_Y= C_G(Y) for the centralizers. We show that the center of C_y and the center of C_Y are smooth group schemes over K. The existence of a Springer isomorphism is used to treat the crucial cases where y is unipotent and where Y is nilpotent. Now suppose G to be quasisplit, and write C for the …
We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the Lie algebra gl_N(C). In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova (math.RT/0610049).
Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a finite-dimensional simple Lie superalgebra of type $D(2,1;α)$, $G(3)$ or $F(4)$ over $\mathbb{C}$. Let $G$ be the simply connected semisimple algebraic group over $\mathbb{C}$ such that $\mathrm{Lie}(G)=\mathfrak{g}_{\bar{0}}$. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$ especially. We also determine the labelled Dynkin diagram for $e$. We prove theorems relating the dimension of $\left(\mathfra…
Let \mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}} be a basic classical Lie superalgebra over an algebraically closed field \mathbb{K} whose characteristic p>0 is a good prime for \mathfrak{g}. Let G_{\bar{0}} be the reductive algebraic group over \mathbb{K} such that \mathrm{Lie}(G_{\bar{0}})=\mathfrak{g}_{\bar{0}}. Suppose e\in\mathfrak{g}_{\bar{0}} is nilpotent. Write \mathfrak{g}^{e} for the centralizer of e in \mathfrak{g} and \mathfrak{z}(\mathfrak{g}^{e}) for the centre of \mathfrak{g}^{e}. We calculate a basis for \mathfrak{g}^{e} and \mathfrak{z}(\mathfrak{g}^{e}) by …
Let $\bar{G}$ be the simple algebraic supergroup $\mathrm{SL}(m|n)$ or $\mathrm{OSp}(m|2n)$ over $\mathbb{C}$. Let $\mathfrak{g}=\mathrm{Lie}(\bar{G})=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ and let $G=\bar{G}(\mathbb{C})$ where $\mathbb{C}$ is considered as a superalgebra concentrated in even degree. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$. In particular, we give bases for $\mathfrak{g}^{e}$, $\mathfrak{z}(\mathfrak{g}^{e})$ and $\left(\mathfrak{z}(\ma…
We use affine W-algebras to quantize Mishchenko-Fomenko subalgebras for centralizers of nilpotent elements in simple Lie algebras under certain assumptions that are satisfied for all cases in type A and all minimal nilpotent cases outside type $E_8$.
Let X and Y be commuting nilpotent K-endomorphisms of a vector space V, where K is a field of characteristic p >= 0. If F=K(t) is the field of rational functions on the projective line, consider the K(t)-endomorphism A=X+tY of V. If p=0, or if the (p-1)-st power of A is 0, we show here that X and Y are tangent to the unipotent radical of the centralizer of A in GL(V). For all geometric points (a:b) of a suitable open subset of the projective line, it follows that X and Y are tangent to the unipotent radical of the centralizer of aX+bY. This answers a question of J. Pevtsova.
Let g be a finite-dimensional simple Lie algebra of rank r over an algebraically closed field of characteristic zero, and let e be a nilpotent element of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} the algebra of symmetric invariants of g^e. We say that e is good if the nullvariety of some r homogeneous elements of S(g^e)^{g^e} in the dual of g^{e} has codimension r. If e is good then S(g^e)^{g^e} is polynomial. The main result of this paper stipulates that if for some homogeneous generators of S(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^f ar…
We prove analogues of fundamental results of Kostant on the universal centralizer of a connected reductive algebraic group for algebraically closed fields of positive characteristic (with mild assumptions), and for integral coefficients. As an application, we use these results to obtain a "mixed modular" analogue of the derived Satake equivalence of Bezrukavnikov-Finkelberg.
Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.
We introduce a new family of affine $W$-algebras associated with the centralizers of arbitrary nilpotent elements in $\mathfrak{gl}_N$. We define them by using a version of the BRST complex of the quantum Drinfeld--Sokolov reduction. A family of free generators of the new algebras is produced in an explicit form. We also give an analogue of the Fateev--Lukyanov realization for these algebras by applying a Miura-type map.
We organize the nilpotent orbits in the exceptional complex Lie algebras into series using the triality model and show that within each series the dimension of the orbit is a linear function of the natural parameter a=1,2,4,8, respectively for f_4,e_6,e_7,e_8. We also obtain explicit representatives in a uniform manner. We observe similar regularities for the centralizers of nilpotent elements in a series and graded components in the associated grading of the ambient Lie algebra. More strikingly, for a greater than one, the degrees of the unipotent characters of the corresponding Chevalley gro…
Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in Lie(G)(F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals …
We study the Loewy structure of the centralizer algebra kP^Q for P a p-group with subgroup Q and k a field of characteristic p. Here kP^Q is a special type of Hecke algebra. The main tool we employ is the decomposition of kP^Q as a split extension of a nilpotent ideal I by the group algebra kC_P(Q). We compute the Loewy structure for several classes of groups, investigate the symmetry of the Loewy series, and give upper and lower bounds on the Loewy length of $P^Q. Several of these results were discovered through the use of MAGMA, especially the general pattern for most of our computations. As…
Given a complex projective algebraic variety, write H(X) for its cohomology with complex coefficients and IH(X) for its Intersection cohomology. We first show that, under some fairly general conditions, the canonical map H(X)\to IH(X) is injective. Now let Gr = G((z))/G[[z]] be the loop Grassmannian for a complex semisimple group G, and let X be the closure of a G[[z]]-orbit in Gr. We prove, using the general result above, a conjecture of D. Peterson describing the cohomology algebra H(X) in terms of the centralizer of the principal nilpotent in the Langlands dual of Lie(G). In the last sectio…
We consider ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra. The goal of this paper is two-fold. First, we study the ad-nilpotent ideals lying inside the Heisenberg ideal. The Heisenberg ideal is the nilpotent radical of the centralizer of the highest root vector. Second, we study the ideals having the property that the corresponding domain of the Shi arrangement consists of a single alcove. Such ideals (and the corresponding elements of the affine Weyl group) are called minimax.
Let G be a reductive algebraic group over the complex field and O_h be a closed adjoint orbit through a semisimple element h. By a result of Borho and Kraft (1979), it is known that the asymptotic cone of the orbit O_h is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer Z_G(h) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair (G, K).
Let g be a complex semisimple Lie algebra and let G' be the Langlands dual group. We give a description of the cohomology algebra of an arbitrary spherical Schubert variety in the loop Grassmannian for G' as a quotient of the form Sym(g^e)/J. Here, J is an appropriate ideal in the symmetric algebra of g^e, the centralizer of a principal nilpotent in g. We also discuss a `topological' proof of Kostant's famous result on the structure of the polynomial algebra on g.
For local non-archimedean fields $k$ of sufficiently large residual characteristic, we explicitly parametrize and count the rational nilpotent adjoint orbits in each algebraic orbit of orthogonal and special orthogonal groups. We separately give an explicit algorithmic construction for representatives of each orbit. We then, in the general setting of groups $\mathrm{GL}_n(D)$, $\mathrm{SL}_n(D)$ (where $D$ is a central division algebra over $k$) or classical groups, give a new characterisation of the "building set" (defined by DeBacker) of an $\mathfrak{sl}_2(k)$-triple in terms of the buildin…
Let $g$ be a finite dimensional complex reductive Lie algebra and <.,.> an invariant non degenerated bilinear form on $g\times g$ which extends the Killing form of $[g,g]$. We define a subcomplex $E\_{\bullet}(g)$ of the canonical complex $C\_{\bullet}(g)$ of $g$. There exists a well defined sub-module $B\_{g}$ of the module of polynomial maps from $g\times g$ to $g$ which is free of rank equal to the dimension b of the borel subalgebras of $g$. Moreover, $B\_{g}$ is contained in the space of cycles of the canonical complex of $g$. The complex $E\_{\bullet}(g)$ is the ideal of $C\_{\bullet}(g)…
We classify good Z-gradings of basic Lie superalgebras over an algebraically closed field of characteristic zero. Good Z-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the construction of super W-algebras. We also describe the centralizer of a nilpotent even element and of an sl(2)-triple in gl(m|n) and osp(m|2n).
We illustrate the Lie theoretic capabilities of the computational algebra system GAP4 by reporting on results on nilpotent orbits of simple Lie algebras that have been obtained using computations in that system. Concerning reachable elements in simple Lie algebras we show by computational means that the simple Lie algebras of exceptional type have the Panyushev property. We computationally prove two propositions on the dimension of the abelianization of the centralizer of a nilpotent element in simple Lie algebras of exceptional type. Finally we obtain the closure ordering of the orbits in the…
A uniform parametrization for the irreducible spin representations of Weyl groups in terms of nilpotent orbits is recently achieved by Ciubotaru (2011). This paper is a generalization of this result to other real reflection groups. Let $(V_0, R, V_0^{\vee}, R^{\vee})$ be a root system with the real reflection group $W$. We define a special subset of points in $V_0^{\vee}$ which will be called solvable points. Those solvable points, in the case $R$ crystallographic, correspond to the nilpotent orbits whose elements have a solvable centralizer in the corresponding Lie algebra. Then a connection …
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
Let G be a semisimple group over an algebraically closed field of very good characteristic for G. In the context of geometric invariant theory, G. Kempf has associated optimal cocharacters of G to an unstable vector in a linear G-representation. If the nilpotent element X in Lie(G) lies in the image of the differential of a homomorphism SL(2) --> G, we say that homomorphism is optimal for X, or simply optimal, provided that its restriction to a suitable torus of SL(2) is optimal for X in Kempf's sense. We show here that any two SL(2)-homomorphisms which are optimal for X are conjugate under th…
Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e \in \mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(\mathfrak g,e)$ associated to $e$ and the problem of determining the variety $\mathcal E(\mathfrak g,e)$ of 1-dimensional representations of $U(\mathfrak g,e)$. For $\mathfrak g$ of low rank, we report on computer calculations that have been used to determine the structure of $\mathcal E(\mathfrak g,e)$, and the action of the component group $Γ_e$ of the centralizer of $e$ on $\mathcal E(\mathfrak g,e)$. As a consequence, we provide two examples where the n…
Let G be a simple, simply-connected algebraic group over the complex numbers with Lie algebra $\mathfrak g$. The main result of this article is a proof that each irreducible representation of the fundamental group of the orbit O through a nilpotent element $e \in \mathfrak g$ lifts to a representation of a Jacobson-Morozov parabolic subgroup of G associated to e. This result was shown in some cases by Barbasch and Vogan in their study of unipotent representations for complex groups and, in general, in an unpublished part of the author's doctoral thesis. In the last section of the article, we s…
Let K be a connected compact Lie group, and G be its complexification. The homology of the based loop group ΩK with integer coefficients is naturally a \ZZ-Hopf algebra. After possibly inverting 2 or 3, we identify H_*(ΩK,\ZZ) with the Hopf algebra of algebraic functions on B^\vee_e, where B^\vee is a Borel subgroup of the Langlands dual group scheme G^\vee of G and B^\vee_e is the centralizer in B^\vee of a regular nilpotent element e\in\Lie B^\vee. We also give a similar interpretation for the equivariant homology of ΩK under the maximal torus action.
This paper provides a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type $D$. Precisely, let $ G_ 0 =Spin(2n,\mathbb C)$ be the Spin complex group viewed as a real group, and $K\cong G_0$ be the complexification of the maximal compact subgroup of $G_0$. We compute $K$-spectra of the regular functions on some small nilpotent orbits $\mathcal O$ transforming according to characters $ψ$ of $C_{ K}(\mathcal O)$ trivial on the connected component of the identity $C_{ K}(\mathcal O)^0$. We then match them w…
Let $N$ be a step two connected and simply connected non commutative nilpotent Lie group which is square-integrable modulo the center. Let $Z$ be the center of $N$. Assume that $N=P\rtimes M$ such that $P$, and $M$ are simply connected, connected abelian Lie groups, $M$ acts non-trivially on $P$ by automorphisms and $\dim P/Z=\dim M$. We study band-limited subspaces of $L^2(N)$ which admit Parseval frames generated by discrete translates of a single function. We also find characteristics of band-limited subspaces of $L^2(N)$ which do not admit a single Parseval frame. We also provide some cond…
Finite W-algebras are certain associative algebras arising in Lie theory. Each W-algebra is constructed from a pair of a semisimple Lie algebra g (our base field is algebraically closed and of characteristic 0) and its nilpotent element e. In this paper we classify finite dimensional irreducible modules with integral central character over W-algebras. In more detail, in a previous paper the first author proved that the component group A(e) of the centralizer of the nilpotent element under consideration acts on the set of finite dimensional irreducible modules over the W-algebra and the quotien…
The results in this paper provide a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits. Precisely, let $\widetilde{G_0} =\widetilde{Spin}(a,b)$ with $a+b=2n$, the nonlinear double cover of $Spin(a,b)$, and let $\widetilde{K}=Spin(a, \mathbb C)\times Spin(b, \mathbb C)$ be the complexification of the maximal compact subgroup of $\widetilde{G_0}$. We consider the nilpotent orbit $\mathcal O_c$ parametrized by $[3 \ 2^{2k} \ 1^{2n-4k-3}]$ with $k>0$. We provide a list of unipotent representations that are genuine, and prove that t…
Let $N$ be a connected and simply connected nilpotent Lie group, and let $K$ be a subgroup of the automorphism group of $N$. We say that the pair $(K,N)$ is a nilpotent Gelfand pair if $L^1_K(N)$ is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs $(K,N)$ where the $K$-orbits in the center of $N$ have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specifically, we show that the one-to-one correspondence between the set $Δ(K,N)$ of bounded $K$-spherical functions on $N$ an…
We define algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g. The new slices are transversal to the conjugacy classes in an algebraic group G with Lie algebra g. These slices are associated to (the conjugacy classes of) elements s of the Weyl group W of g. For such slices we prove an analogue of the Kostant cross-section theorem for the action of a unipotent group.
Let N be a finite group of odd order and A a finite group that acts on N such that the orders of N and A are coprime. Isaacs constructed a natural correspondence between the set Irr_A(N) of irreducible complex characters invariant under the action of A, and the irreducible characters of the centralizer of A in N, Irr(C_N(A)). We show that this correspondence preserves Schur indices over the rational numbers. Moreover, suppose that the semidirect product AN is a normal subgroup of the finite group G and set U= N_G(A). Let χ\in Irr_A(N) and χ* \in Irr(C_N(A)) correspond. Then there is a canonica…
We study ad-nilpotent ideals of a parabolic subalgebra of a simple Lie algebra. Any such ideal determines an antichain in a set of positive roots of the simple Lie algebra. We give a necessary and sufficient condition for an antichain to determine an ad-nilpotent ideal of the parabolic. We write down all such antichains for the classical simple Lie algebras and in particular recover the results of D. Peterson. In section 2 of the paper we study the unique ideal in a parabolic which is irreducible as a module for the reductive part and give several equivalent statements that are satisfied by th…
Let $\mathfrak g$ be a simple Lie algebra over an algebraically closed field $\bf k$ of characteristic zero and $\bf G$ its adjoint group. Let $\mathfrak q$ be a biparabolic subalgebra of $\mathfrak g$. The algebra $Sy(\mathfrak q)$ of semi-invariants on $\mathfrak q^*$ is polynomial in most cases, in particular when $\mathfrak g$ is simple of type $A$ or $C$. On the other hand $\mathfrak q$ admits a canonical truncation $\mathfrak q_Λ$ such that $Sy(\mathfrak q)=Sy(\mathfrak q_Λ)=Y(\mathfrak q_Λ)$ where $Y(\mathfrak q_Λ)$ denotes the algebra of invariant functions on $\mathfrak q_Λ^*$. An ada…
Let G denote a connected, quasi-split reductive group over a field F that is complete with respect to a discrete valuation and that has a perfect residue field. Under mild hypotheses, we produce a subset of the Lie algebra g(F) that picks out a G(F)-conjugacy class in every stable, regular, topologically nilpotent conjugacy class in g(F). This generalizes an earlier result obtained by DeBacker and one of the authors under stronger hypotheses. We then show that if F is p-adic, then the characteristic function of this set behaves well with respect to endoscopic transfer.
In this paper we determine, for all $r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting $r$-tuples of nilpotent elements of $\mathfrak{gl}_n$. Our main result is that in characteristic $\neq 2,3$, this nilpotent commuting variety has dimension $(r+1)\lfloor \frac{n^2}{4}\rfloor$ for $n\geq 4$, $r\geq 7$. We use this to find the dimension of the (ordinary) $r$-th commuting varieties of $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ for the same range of values of $r$ and $n$. Our principal motivation is the connection between nilpotent commuting variet…
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 $g_e$.
