Principal Nilpotent Research Articles

Fomenko–Mischenko Theory, Hessenberg Varieties, and Polarizations

The symmetric algebra g (denoted S(\g)) over a Lie algebra \g (frak g) has the structure of a Poisson algebra. Assume \g is complex semi-simple. Then results of Fomenko- Mischenko (translation of invariants) and A.Tarasev construct a polynomial subalgebra \cal H = \bf C[q_1,...,q_b] of S(\g) which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of \g. Let G be the adjoint group of \g and let \ell = rank \g. Identify \g with its dual so that any G-orbit O in \g has the structure (KKS) of a symplectic manifold and S(\g) can be identified with the affine algebra of \g. An element x \in \g is strongly regular if \{(dq_i)_x\}, i=1,...,b, are linearly independent. Then the set \g^{sreg} of all strongly regular elements is Zariski open and dense in \g, and also \g^{sreg \subset \g^{reg} where \g^{reg} is the set of all regular elements in \g. A Hessenberg variety is the b-dimensional affine plane in \g, obtained by translating a Borel subalgebra by a suitable principal nilpotent element. This variety was introduced in [K2]. Defining Hess to be a particular Hessenberg variety, Tarasev has shown that Hess \subset \g^sreg. Let R be the set of all regular G-orbits in \g. Thus if O \in R, then O is a symplectic manifold of dim 2n where n= b-\ell. For any O\in R let O^{sreg} = \g^{sreg}\cap O. We show that O^{sreg} is Zariski open and dense in O so that O^{sreg} is again a symplectic manifold of dim 2n. For any O \in R let Hess (O) = Hess \cap O. We prove that Hess(O) is a Lagrangian submanifold of O^{sreg} and Hess =\sqcup_{O \in R} Hess(O). The main result here shows that there exists, simultaneously over all O \in R, an explicit polarization (i.e., a "fibration" by Lagrangian submanifolds) of O^{sreg} which makes O^{sreg} simulate, in some sense, the cotangent bundle of Hess(O).

Cohomology of flag varieties and the Brylinski–Kostant filtration

Let G be a reductive algebraic group over C and let N be a G-module. For any subspace M of N, the Brylinski–Kostant filtration on M is defined through the action of a principal nilpotent element in Lie G. This filtration is related to a q-analog of weight multiplicity due to Lusztig. We generalize this filtration to other nilpotent elements and show that this generalized filtration is related to “parabolic” versions of Lusztig's q-analog of weight multiplicity. Along the way we also generalize results of Broer on cohomology vanishing of bundles on cotangent bundles of partial flag varieties. We conclude by computing some explicit examples.

ON SOME $\ell$-ADIC REPRESENTATIONS OF $\mathop{\rm Gal}(\overline{/mathbb{Q}})$ ATTACHED TO NONCONGRUENCE SUBGROUPS

The l-adic parabolic cohomology groups attached to noncongruence subgroups of SL2(Z) are finite-dimensional l-adic representations of Gal ( Q ¯ / K ) for some number field K. We exhibit examples (with K = Q) for which the primitive parts give Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of the Galois group relies on Katz's classification theorem for semisimple subalgebras of sln containing a principal nilpotent element, for which we give a short conceptual proof, suggested by I. Grojnowski. 2000 Mathematics Subject Classification 11F80 (primary), 11G18 (secondary).

Regular points in affine Springer fibers

studied by Kazhdan and Lusztig in [KL]. For x = gG(O) ∈ X the G(O)-orbit (for the adjoint action) of Ad(g−1)(u) in g(O) depends only on x, and its image under g(O) g(C) is a well-defined G(C)-orbit in g(C). We say that x ∈Xu is regular if the associated orbit is regular in g(C). (Recall that an element of g(C) is regular if the nilpotent part of its Jordan decomposition is a principal nilpotent element in the centralizer of the semisimple part of its Jordan decomposition.) We write X reg for the (Zariski open) subset of regular elements in X. From now on we assume that u is regular semisimple with centralizer T, a maximal torus in G over F. Assume further that u is integral, by which we mean that X is nonempty. Kazhdan and Lusztig [KL] show that X is then a locally finite union of projective algebraic varieties, and in [KL, Sec. 4, Cor. 1] they show that the open subset X reg of X u is nonempty (and hence dense in at least one irreducible component of X). The action of T(F ) on X clearly preserves the subsets X andX reg. Bezrukavnikov [B] proved thatX u reg forms a single orbit under T(F ). (Actually Kazhdan–Lusztig and Bezrukavnikov consider only topologically nilpotent elements u, but the general case can be reduced to their special case by using the topological Jordan decomposition of u.) The goal of this paper is to characterize regular elements in X (for integral regular semisimple u as just described). When T is elliptic (in other words, F anisotropic modulo the center of G), the characterization gives no new information. At the other extreme, in the split case, the characterization gives a clear picture of what it means for a point in X to be regular. We will now state our characterization in the split case, leaving the more technical general statement to the next section (see Theorem 1). Fix a split maximal torus

A conjectural generalization of the n! result to arbitrary groups

The aim of this paper is to formulate a conjecture for an arbitrary simple Lie algebra g in terms of the geometry of principal nilpotent pairs. When g is specialized to sln, this conjecture readily implies the n! result and it is very likely that, in fact, it is equivalent to the n! result in this case. In addition, this conjecture can be thought of as generalizing an old result of Kostant. In another direction, we show that to prove the validity of the n! result for an arbitrary n and an arbitrary partition of n, it suffices to show its validity only for the staircase partitions.

Centralizers of distinguished nilpotent pairs and related problems

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 distinguished 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 nilpotent pairs do not exist in the simply-laced case and that the centralizer of an almost principal nilpotent pair in a classical simple Lie algebra is always abelian.

A Classification of the Principal Nilpotent Pairs in Simple Lie Algebras and Related Problems

Let g be a semisimple Lie algebra over an algebraically closed field K of characteristic zero and G be its adjoint group. The notion of a principal nilpotent pair is a double counterpart of the notion of a regular (= principal) nilpotent element in g. Roughly speaking, a principal nilpotent pair e = (e1, e2) consists of two commuting elements in g that can independently be contracted to the origin and such that their simultaneous centralizer has the minimal possible dimension, that is, rkg. The definition and the basic results are due to V. Ginzburg [3]. He showed that the theory of principal nilpotent pairs yields a refinement of well-known results by B. Kostant on regular nilpotent elements in g and has interesting applications to representation theory. In particular, he proved that the number of G-orbits of principal nilpotent pairs is finite and gave a classification for g = sl(V). Trying to achieve a greater generality, Ginzburg also introduced a wider class of distinguished nilpotent pairs and, again, classified them for sl(V). (The precise definitions for all notions related to nilpotent pairs are found in §1.)

Nilpotent pairs in semisimple Lie algebras and their characteristics

Recently, V. Ginzburg introduced the notion of a principal nilpotent pair in a semisimple Lie algebra g. Our aim is to contribute to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pair (e1, e2) consists of two commuting elements in g that can independently be contracted to the origin. We show that any nilpotent pair has a characteristic (h1, h2), which is unique within G-conjugacy. Generalizing Dynkin's approach in case of sl2-triples, we prove that the number of G-orbits of characteristics of nilpotent pairs is finite and provide some estimates for the numerical labels αj(hi), where αj is a suitably chosen set of simple roots of g. It was observed by Ginzburg that the number of G-orbits of nilpotent pairs is infinite. This means this class is too wide to have a reasonable theory. To resolve this difficulty, we introduce wonderful (nilpotent) pairs. It is proven that two wonderful pairs having the same characteristic are necessarily conjugate. This implies that there are finitely many G-orbits of wonderful pairs. A number of nice properties of wonderful pairs shows that these can be regarded as a right double analogue of nilpotent orbits. We also consider several natural classes of wonderful pairs and describe characteristics for principal and almost principal nilpotent pairs.

Principal nilpotent pairs in a semisimple Lie algebra 1

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue of the Kostant partition function, and propose the corresponding two-parameter analogue of the weight multiplicity formula. In a different direction, each principal nilpotent pair gives rise to a harmonic polynomial on the Cartesian square of the Cartan subalgebra, that transforms under an irreducible representation of the Weyl group. In the special case of sl_n, the conjugacy classes of principal nilpotent pairs and the irreducible representations of the Symmetric group, S_n, are both parametrised (in a compatible way) by Young diagrams. In general, our theory provides a natural generalization to arbitrary Weyl groups of the classical construction of simple S_n-modules in terms of Young's symmetrisers.

A result on exponents of finite-dimensional simple lie algebras and its application to Kac-Moody algebras

The centralizer of the principal nilpotent element of a finite-dimensional simple Lie algebra is a naturally graded Cartan subalgebra, the degrees of the non-trivial components being the so-called exponents. If every nonzero element of a homogeneous subspace is a linear combination of all root spaces of the corresponding height, the exponent is called complete. We will show that there are precisely three non-complete exponents and apply this to affine Kac-Moody algebras.

A deterministic construction for normal bases of abelian extensions

In this paper we give a deterministic construction for a Normal basis of Kf over K, where K' is an abelian extension of finite degree n over some commutative, infinite field K. No particular property for K is needed.We use polynomials already defined in [2], [5, chapter 5], In these papers the field K was finite. These polynomials were used in order to construct SDMC codes [5,chapter 6], to determine bivariate codes [2],[3], or to obtain the best possible principal nilpotent codes [6], Our construction is deduced from new properties of these polynomials

Limits of weight spaces, Lusztig’s 𝑞-analogs, and fiberings of adjoint orbits

Let G be a connected complex semisimple algebraic group, and T a maximal torus inside a Borel subgroup B, with g, t, and b their Lie algebras. Let V be a representation in the category a for g. The t-decomposition V = met* VA of V into a direct sum of finite-dimensional weight spaces is central in the representation theory of g. In this paper, we introduce on weight spaces a new structure, the principal filtration Je (V) , where e is a principal nilpotent in g chosen to be compatible with t; for example, e can be the sum of the simple root vectors relative to (t, b) . This filtration is constructed in a very simple way by taking J' (V') to be the space of vectors annihilated by the (p + 1)th power of e, for p > 0. Our approach is motivated by Kostant's fundamental work [Ki, K2] on actions of the principal TDS (three-dimensional subalgebra) and coordinate rings of regular adjoint orbits. In Theorem 3.4, we give a new description, in terms of the dimension jumps of the principal filtration of the weight space, of Lusztig's [L] q-analog mm (q) of dominant ,-weight multiplicity in a finite-dimensional irreducible g-representation V>. The polynomial m (q) was defined algebraically as an alternating sum over the Weyl group, through a q-analog of Kostant's weight multiplicity formula. We prove that m (q) is equal to the jump polynomial

Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups

The purpose of the present note is to give a general procedure for calculating the singular cohomology H*(K, IFp) with coefficients in an arbitrary finite field IFv of a connected compact Lie group K. Simultaneously, we calculate the Chow ring A(G, IFp)..=A(G)| p of the corresponding complex algebraic group G, and the degrees of the basic "generalized invariants" of the Weyl group W in arbitrary characteristic p. A mysterious connection between the degrees of generators of A(G, IFp) and the kernel of the principal nilpotent element in characteristic p is pointed out. The computation of real cohomology of compact Lie groups started by E. Cartan in 1929 was completed for classical groups by R. Brauer and by L. Pontryagin in 1935. H. Hopf and H. Samelson showed in 1941 that H*(K, F,.) is an exterior algebra on generators of degrees 2 d 1 1 . . . . ,2dr-l , where l = r a n k K . But only in 1949 C.T. Yen managed to compute the d; for all exceptional groups, case by case. On the other hand, C. Chevalley and J. Leray showed that the d i are nothing else but the degrees of the basic invariants over of the Weyl group W. A simple procedure for calculating the dl was given in the late fifties by A.J. Coleman and by B. Kostant. Nowadays this is common knowledge. The problem of computing the torsion in H*(K,Z) proved to be more recalcitrant. This was done by several authors: C.E. Miller settled SO,, in 1953 [,23], A. Borel settled Spin,, Ad Spin,, and the quotients of SU, and Sp, in 1954 [,6], P.F. Baum and W. Browder settled the remaining classical group, the semispin group, in 1965 [-4]. G 2 and F 4 were settled in 1954 [6], and the p > 3 torsion for E 6 and p>5tors ion for Ev,E s in 1961 [,7], by A. Borel. S. Araki computed the 2-torsion for E 6 , E 7 [-1] and the 3-torsion for E 7 , E s [2], and announced the 2-torsion for E 8 in a joint paper with Y. Shikata in 1961 [-3]. The details of the proof for the last result have appeared only recently [,18]. There was an error in [,1] in the case of Ad E 7, which has been fixed in [,13].

Ideaux principaux nilpoitnis de dimension maximale dans l'algebre Fq[G] d'un groupe abelien fini g

We characterize principal nilpotent ideals in finite abelian group algebra Fq [G] (Fq a finite field of ps elements, p prime), with greatest possible dimension. We prove also that the-se ideals are all isomorphic.