BGG Resolution Research Articles

UNITARY REPRESENTATIONS OF CYCLOTOMIC HECKE ALGEBRAS AT ROOTS OF UNITY: COMBINATORIAL CLASSIFICATION AND BGG RESOLUTIONS

AbstractWe relate the classes of unitary and calibrated representations of cyclotomic Hecke algebras, and, in particular, we show that for the most important deformation parameters these two classes coincide. We classify these representations in terms of both multipartition combinatorics and as the points in the fundamental alcove under the action of an affine Weyl group. Finally, we cohomologically construct these modules via BGG resolutions.

The modular Weyl–Kac character formula

We classify and explicitly construct the irreducible graded representations of anti-spherical Hecke categories which are concentrated one degree. Each of these homogeneous representations is one-dimensional and can be cohomologically constructed via a BGG resolution involving every (infinite dimensional) standard representation of the category. We hence determine the complete first row of the inverse parabolic p-Kazhdan–Lusztig matrix for an arbitrary Coxeter group and an arbitrary parabolic subgroup. This generalises the Weyl–Kac character formula to all Coxeter systems (and their parabolics) and proves that this generalised formula is rigid with respect to base change to fields of arbitrary characteristic.

On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras

We provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type A via BGG resolutions, solving a conjecture of Berkesch–Griffeth–Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo–Mumford regularity of certain symmetric linear subspace arrangements.

A computer algorithm for the BGG resolution

We present a computer algorithm to explicitly compute the BGG resolution and its cohomology. We give several applications, in particular computation of various sheaf cohomology groups on flag varieties. An implementation of the algorithm is available at https://github.com/RikVoorhaar/bgg-cohomology.

BGG complexes in singular blocks of category [formula omitted

Using translation from the regular block, we construct and analyze properties of BGG complexes in singular blocks of BGG category O. We provide criteria, in terms of the Kazhdan-Lusztig-Vogan polynomials, for such complexes to be exact. In the Koszul dual picture, exactness of BGG complexes is expressed as a certain condition on a generalized Verma flag of an indecomposable projective object in the corresponding block of parabolic category O.In the second part of the paper, we construct BGG complexes in a more general setting of balanced quasi-hereditary algebras and show how our results for singular blocks can be used to construct BGG resolutions of simple modules in S-subcategories in O.

BGG complexes in singular infinitesimal character for type A

We give a geometric construction of the Bernstein-Gel’fand-Gel’fand (BGG) resolutions in singular infinitesimal character in the case of 1-graded complex Lie algebras of type A.

Character formulas and Bernstein–Gelfand–Gelfand resolutions for Cherednik algebra modules

We study blocks of category O for the Cherednik algebra having the property that every irreducible module in the block admits a BGG resolution, and as a consequence prove a character formula conjectured by Oblomkov-Yun.

Completely exceptional 2nd order PDEs via conformal geometry and BGG resolution

By studying the development of shock waves out of discontinuity waves, in 1954 P. Lax discovered a class of PDEs, which he called “completely exceptional”, where such a transition does not occur after a finite time. A straightforward integration of the completely exceptional conditions allowed Boillat to show that such PDEs are actually of Monge–Ampère type. In this paper, we first recast these conditions in terms of characteristics, and then we show that the completely exceptional PDEs, with 2 or 3 independent variables, can be described in terms of the conformal geometry of the Lagrangian Grassmannian, where they are naturally embedded. Moreover, for an arbitrary number of independent variables, we show that the space of r th degree sections of the Lagrangian Grassmannian can be resolved via a BGG operator. In the particular case of 1st degree sections, i.e., hyperplane sections or, equivalently, Monge–Ampère equations, such operator is a close analogue of the trace-free second fundamental form.

Representations of rational Cherednik algebras with minimal support and torus knots

In this paper we obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen–Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type An−1 for c=mn, and an expression of its quasispherical part (i.e., the isotypic part of “hooks”) in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SLm. Our third result is the construction of the Koszul–BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul–BGG resolution from [3] and [21], and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul–BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras Hmn(Sn) and Hnm(Sm). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry was essentially established in [8] in the spherical case, and in [24] in the case GCD(m,n)=1, and it has a natural interpretation in terms of invariants of torus knots.

Two-sided BGG resolutions of admissible representations

We prove the conjecture of Frenkel, Kac and Wakimoto on the existence of two-sided BGG resolutions of G G -integrable admissible representations of affine Kac-Moody algebras at fractional levels. As an application we establish the semi-infinite analogue of the generalized Borel-Weil theorem for minimal parabolic subalgebras which enables an inductive study of admissible representations.

Bernstein–Gelfand–Gelfand resolutions for basic classical Lie superalgebras

We study Kostant cohomology and Bernstein–Gelfand–Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras. For each choice of parabolic subalgebra and representation of such a Lie superalgebra, there is a natural definition of the boundary and coboundary operators, which define (co)homology of the nilradical of the parabolic subalgebra. We prove that complete reducibility of the homology groups is a necessary condition to have a resolution of an irreducible module in terms of (generalised) Verma modules. Every such a resolution is then given by modules induced by these homology groups. We also prove that if these homology groups are completely reducible, a sufficient condition for the existence of this resolution is the property that these groups are isomorphic to the kernel of the Kostant quabla operator, which is equivalent with disjointness of the boundary and coboundary operators. Then we use these results to derive very explicit conditions under which BGG resolutions exist, which are particularly useful for the superalgebras of type I. For the unitarisable representations of gl(m|n) and osp(2|2n) we derive conditions on the parabolic subalgebra under which the BGG resolutions exist. We also apply the obtained theory to construct specific examples of BGG resolutions for osp(m|2n). Finally we state some results for typical modules of gl(m|n) and osp(2|2n).

On a new normalization for tractor covariant derivatives

A regular normal parabolic geometry of type G/P on a manifold M gives rise to sequences D_i of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative \nabla^\omega on the corresponding tractor bundle V, where \omega is the normal Cartan connection. The first operator D_0 in the sequence is overdetermined and it is well known that \nabla^\omega yields the prolongation of this operator in the homogeneous case M = G/P . Our first main result is the curved version of such a prolongation. This requires a new normalization of the tractor covariant derivative on V . Moreover, we obtain an analogue for higher operators D_i . In that case one needs to modify the exterior covariant derivative d^{\nabla^\omega} by differential terms. Finally we demonstrate these results on simple examples in projective, conformal and Grassmannian geometry. Our approach is based on standard techniques of the BGG machinery.

Abelianization of the BGG resolution of representations of the Virasoro algebra

We construct a resolution that permits computing the t-character of representations of the Virasoro algebra from the (2, 2p + 1)-models, i.e., the characters of the associated graded spaces with respect to the Poincare-Birkhoff-Witt filtration.

Recursive properties of branching and BGG resolution

Recurrent relations for branching coefficients are based on a special type of singular element decomposition. We show that this decomposition can be used to construct the parabolic Verma modules and finally to obtain the generalized Weyl-Verma formulas for characters. We demonstrate how branching coefficients can determine the generalized BGG resolution sequence.

An analogue of the BGG resolution for locally analytic principal series

Let G be a connected reductive quasi-split algebraic group over a field L which is a finite extension of the p-adic numbers. We construct an exact sequence modelled on (the dual of) the BGG resolution involving locally analytic principal series representations for G(L). This leads to an exact sequence involving spaces of overconvergent p-adic automorphic forms for certain groups compact modulo centre at infinity.

Category 𝒪 for the Virasoro algebra: cohomology and Koszulity

We investigate blocks of the Category O for the Virasoro algebra over C. We demonstrate that the blocks have Kazhdan‐Lusztig theories and that the truncated blocks give rise to interesting Koszul algebras. The simple modules have BGG resolutions, and from this we compute the extensions between Verma modules and simple modules, and between pairs of simple modules.

Resolutions and Hilbert series of determinantal varieties and unitary highest weight modules

Let ( G, K) be a Hermitian symmetric pair and let g and k denote the corresponding complexified Lie algebras. Let g= k⊕ p +⊕ p − be the usual decomposition of g as a k -module. There is a natural correspondence between K C -orbits in p + and a distinguished family of unitarizable highest weight modules for g called the Wallach representations. We denote by Y k the closure of the K C -orbit in p + that is associated to the kth Wallach representation. In this article we give explicit formulas for the numerator polynomials of the Hilbert series of the varieties Y k by using BGG resolutions of unitarizable highest weight modules. A preliminary result gives a new branching formula for a certain two-parameter family of finite dimensional representations of the even orthogonal groups. Our work is an extension of previous work by Enright and Willenbring.

Semi-infinite induction and Wakimoto modules

The purpose of this paper is to suggest the construction and study properties of semi-infinite induction, which relates to semi-infinite cohomology the same way induction relates to homology and coinduction to cohomology. We prove a version of the Shapiro Lemma, showing that the semi-infinite cohomology of a module is isomorphic to that of the semi-infinitely induced module. A practical outcome of our construction is a simple construction of the Wakimoto modules, highest-weight modules used in double-sided BGG resolutions of irreducible modules.

Resolutions and characters of irreducible representations of the N = 2 superconformal algebra

We evaluate characters of irreducible representations of the N = 2 supersymmetric extension of the Virasoro algebra. We do so by deriving the BGG resolution of the admissible N = 2 representations and also a new “3, 5, 7, …” resolution in terms of twisted massive Verma modules. We analyse how the characters behave under the automorphisms of the algebra, whose most significant part is the spectral flow transformations. The possibility to express the characters in terms of theta functions is determined by their behaviour under the spectral flow. We also derive the identity expressing every sl ̂ (2) character as a linear combination of spectral-flow transformed N = 2 characters; this identity involves a finite number of N = 2 characters in the case of unitary representations. Conversely, we find an integral representation for the admissible N = 2 characters as contour integrals of admissible sl ̂ (2) characters.

Well-filtered algebras

We define a class of (lean) quasi-hereditary K- algebras A for which the standard filtration of the right regular representation may be described by a suitable directed quotient algebra A +. For this class, projective resolutions of simple left modules over A − will correspond to the so-called BGG resolutions over A, defined earlier by Bernstein, Gelfand and Gelfand. In the case when K is algebraically closed and A + is a subalgebra of A, A + coincides with the concept of a Borel subalgebra of König. We show that many algebras obtained by previously defined canonical constructions belong to this class and have additional structural properties.