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Register a new userInvariants and semi-invariants in the cohomology of the complement of a reflection arrangement
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Suppose V is a finite dimensional, complex vector space, A is a finite set of codimension one subspaces of V, and G is a finite subgroup of the general linear group GL(V) that permutes the hyperplanes in A. In this paper we study invariants and semi-invariants in the graded QG-module H^*(M(A)), where M(A) denotes the complement in V of the hyperplanes in A and H^* denotes rational singular cohomology, in the case when G is generated by reflections in V and A is the set of reflecting hyperplanes determined by G, or a closely related arrangement. Our main result is the construction of an explicit, natural (from the point of view of Coxeter groups) basis of the space of invariants, H^*(M(A))^G. In addition to leading to proof of the description of the space of invariants conjectured by Felder and Veselov for Coxeter groups that does not rely on computer calculations, this construction provides an extension of the description of the space of invariants proposed by Felder and Veselov to arbitrary finite unitary reflection groups.
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Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.
A canonical system of basic invariants is a system of invariants satisfying a set of differential equations. The properties of a canonical system are related to the mean value property for polytopes. In this article, we naturally identify the vector space spanned by a canonical system of basic invariants with an invariant space determined by a fundamental antiinvariant. From this identification, we obtain explicit formulas of canonical systems of basic invariants. The construction of the formulas does not depend on the classification of finite irreducible reflection groups.
We introduce the notion of a super-representation of a quiver. For super-representations of quivers over a field of characteristic zero, we describe the corresponding (super)algebras of polynomial semi-invariants and polynomial invariants.
The spaces of quasi-invariant polynomials were introduced by Chalykh and Veselov [Comm. Math. Phys. 126 (1990), 597-611]. Their Hilbert series over fields of characteristic 0 were computed by Feigin and Veselov [Int. Math. Res. Not. 2002 (2002), 521-545]. In this paper, we show some partial results and make two conjectures on the Hilbert series of these spaces over fields of positive characteristic. On the other hand, Braverman, Etingof and Finkelberg [arXiv:1611.10216] introduced the spaces of quasi-invariant polynomials twisted by a monomial. We extend some of their results to the spaces twisted by a smooth function.
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 are algebraically independent, with (e,h,f) an sl2-triple of g, then e is good. As applications, we obtain new examples of nilpotent elements that verify the above polynomiality condition, in in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type D_7.
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