Deficient homomorphisms between generalized Verma modules

A class of homomorphisms between generalized Verma modules that have an unusual degeneracy is identified. Homomorphisms in this class are called deficient homomorphisms. A family of maximally deficient homomorphisms is constructed. A necessary condition on a parabolic subalgebra is identified for the associated category of generalized Verma modules to admit deficient homomorphisms.

DEMAZURE MODULES OF LEVEL TWO AND PRIME REPRESENTATIONS OF QUANTUM AFFINE

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154 (2010), 265–341; Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to $\mathfrak{sl}_{n+1}[t]$-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chari et al. (arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.

Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology

For a fixed parabolic subalgebra 𝔭 of $\mathfrak {gl}(n,\mathbb {C})$ we prove that the centre of the principal block 𝒪0𝔭 of the parabolic category 𝒪 is naturally isomorphic to the cohomology ring H*(ℬ𝔭) of the corresponding Springer fibre. We give a diagrammatic description of 𝒪0𝔭 for maximal parabolic 𝔭 and give an explicit isomorphism to Braden’s description of the category PervB(G(k,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov’s algebra ℋn is realised as the endomorphism ring of some object from PervB(G(n,n)) which corresponds under localisation and the Riemann–Hilbert correspondence to a full projective–injective module in the corresponding category 𝒪0𝔭. From there one can deduce that Khovanov’s tangle invariants are obtained from the more general functorial invariants in [C. Stroppel, Categorification of the Temperley Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126(3) (2005), 547–596] by restriction.

A LIMIT FORMULA FOR EVEN NILPOTENT ORBITS

Let Gℝ be a real form of a complex, semisimple Lie group G. Assume [Formula: see text] is an even nilpotent coadjoint Gℝ-orbit. We prove a limit formula, expressing the canonical measure on [Formula: see text] as a limit of canonical measures on semisimple coadjoint orbits, where the parameter of orbits varies over the negative chamber defined by the parabolic subalgebra associated with [Formula: see text].

UNFOLDED FORM OF CONFORMAL EQUATIONS IN M DIMENSIONS AND 𝔬(M + 2)-MODULES

A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra 𝔣. Under certain conditions, 𝔣-invariant systems of differential equations are shown to be associated with 𝔣-modules that are integrable with respect to some parabolic subalgebra of 𝔣. The suggested construction is motivated by the unfolded formulation of dynamical equations developed in the higher spin gauge theory and provides a starting point for generalization to the nonlinear case. It is applied to the conformal algebra 𝔬(M, 2) to classify all linear conformally invariant differential equations in the Minkowski space. Numerous examples of conformal equations are discussed from this perspective.