Related papers: Divided Difference Operator for the Highest root H…
In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $G_{2(2)}$. We use both the minimal and the maximal Heisenberg parabolic subalgebras. We…
It is proved that given a divergence operator on the structural sheaf of graded commutative algebras of a supermanifold, it is possible to construct a generating operator for the Krashil'shchik-Schouten bracket. This is a particular case of…
The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order…
We study the complex symmetric structure of weighted composition--differentiation operators of order $n $ on the weighted Bergman spaces $A_{\alpha}^2$ with respect to some conjugations. Then we provide some examples of these operators.
The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we…
We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of…
We compute the Hochschild-Kostant-Rosenberg decomposition of the Hochschild cohomology of generalised Grassmannians, i.e. partial flag varieties associated to maximal parabolic subgroups in a simple algebraic group. We explain how the…
By reading a standard formula for the ring of Grothendieck differential operators in a derived way, we construct a derived (sheaf of) ring of Grothendieck differential operators for Noetherian schemes $X$ separated and finite-type over a…
In this paper, we introduce a new differential-difference operator $T_\xi$ $(\xi \in \mathbb{R}^N)$ by using projections associated to orthogonal subsystems in root systems. Similarly to Dunkl theory, we show that these operators commute…
This article explores the relationship between Hessenberg varieties associated with semisimple operators with two eigenvalues and orbit closures of a spherical subgroup of the general linear group. We establish the specific conditions under…
For a Hopf algebra A, we define the structures of differential complexes on two dual exterior Hopf algebras: 1) an exterior extension of A and 2) an exterior extension of the dual algebra A^*. The Heisenberg double of these two exterior…
In this paper we study mixed Hodge structures on the cohomology of locally symmetric varieties and give an application to modular forms. After proving vanishing of some Hodge numbers, we focus on the weight filtration on the last Hodge…
This memoir is a summary of recent work, including collaborations with Erik van Erp, Christian Voigt and Marco Matassa, compiled for the "Habilitation \`a diriger des recherches". We present various different approaches to constructing…
We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group…
We consider several differential operators on compact almost-complex, almost-Hermitian and almost-K\"ahler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces…
We construct explicit differential operators on hermitian modular forms, extending methods developed for Siegel modular forms. These differential operators are closely related to the two-variable spherical pluriharmonic polynomials. We…
We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal…
We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed with the help…
A study of diff($S^1$) covariant properties of pseudodifferential operator of integer degree is presented. First, it is shown that the action of diff($S^1$) defines a hamiltonian flow defined by the second Gelfand-Dickey bracket if and only…
For $k$ a field of positive characteristic and $X$ a smooth variety over $k$, we compute the Hochschild cohomology of Grothendieck's differential operators on $X$. The answer involves the derived inverse limit of the Frobenius acting on the…