Mathematics

We describe the cohomology of the sheaf of twisted differential operators on the quantized flag manifold at a root of unity whose order is a prime power. It follows from this and our previous results that for the De Concini-Kac type quantized enveloping algebra, where the parameter q is specialized to a root of unity whose order is a prime power, the number of irreducible modules with a certain specified central character coincides with the dimension of the total cohomology group of the corresponding Springer fiber. This gives a weak version of a conjecture of Lusztig concerning non-restricted representations of the quantized enveloping algebra.

In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished p -characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category O . For this, we prove an equivalence between two categorifications of a parabolically induced module over the affine Hecke algebra conjectured by the first named author. For the general nilpotent p -character, we get character formulas by explicitly computing the duality operator on a suitable equivariant K-group.

Let G be Sp(2n,R) or S O ??(2n) . We compute the Dirac index of a large class of unitary representations considered by Vogan in Section 8 of [Vog84], which include all weakly fair A q (λ) modules and (weakly) unipotent representations of G as two extreme cases. We conjecture that these representations exhaust all unitary representations of G with nonzero Dirac cohomology. In general, for certain irreducible unitary module of an equal rank group, we clarify the link between the possible cancellations in its Dirac index, and the parities of its spin-lowest K -types.

The unitary dual of GL(n,R) was classified by Vogan in the 1980s. Focusing on the irreducible unitary representations of GL(n,R) with half-integral infinitesimal characters, we find that Speh representations and the special unipotent representations are building blocks. By looking at the K -types of them, and by using a Blattner-type formula, we obtain all the irreducible unitary (g,K) -modules with non-zero Dirac cohomology of GL(n,R) , as well as a formula for (one of) their spin-lowest K -types. Moreover, analogous to the GL(n,C) case given in [DW1], we count the number of the FS-scattered representations of GL(n,R) .

Let L denote a finite lattice with at least two points and let A denote the incidence algebra of L . We prove that L is distributive if and only if A is an Auslander regular ring, which gives a homological characterisation of distributive lattices. In this case, A has an explicit minimal injective coresolution, whose i -th term is given by the elements of L covered by precisely i elements. We give a combinatorial formula of the Bass numbers of A . We apply our results to show that the order dimension of a distributive lattice L coincides with the global dimension of the incidence algebra of L . Also we categorify the rowmotion bijection for distributive lattices using higher Auslander-Reiten translates of the simple modules.

The paper contains results that characterize the Donkin-Koppinen filtration of the coordinate superalgebra K[G] of the general linear supergroup G=GL(m|n) by its subsupermodules C Γ = O Γ (K[G]) . Here, the supermodule C Γ is the largest subsupermodule of K[G] whose composition factors are irreducible supermodules of highest weight λ , where λ belongs to a finitely-generated ideal Γ of the poset X(T ) + of dominant weights of G . A decomposition of G as a product of subsuperschemes U − × G ev × U + induces a superalgebra isomorphism ϕ ∗ :K[ U − ]⊗K[ G ev ]⊗K[ U + ]≃K[G] . We show that C Γ = ϕ ∗ (K[ U − ]⊗ M Γ ⊗K[ U + ]) , where M Γ = O Γ (K[ G ev ]) . Using the basis of the module M Γ , given by generalized bideterminants, we describe a basis of C Γ . Since each C Γ is a subsupercoalgebra of K[G] , its dual C ∗ Γ = S Γ is a (pseudocompact) superalgebra, called the generalized Schur superalgebra. There is a natural superalgebra morphism π Γ :Dist(G)→ S Γ such that the image of the distribution algebra Dist(G) is dense in S Γ . For the ideal X(T ) + l , of all weights of fixed length l , the generators of the kernel of π X(T ) + l are described.

We consider Donovan's conjecture in the context of blocks of groups G with defect group D and normal subgroups N⊲G such that G= C D (D∩N)N , extending similar results for blocks with abelian defect groups. As an application we show that Donovan's conjecture holds for blocks with defect groups of the form Q 8 × C 2 n or Q 8 × Q 8 defined over a discrete valuation ring.

Let A be the ring of adeles of a number field F . Given a self-dual irreducible, automorphic, cuspidal representation τ of $\GL_n(\BA)$, with trivial central characters, we construct its full inverse image under the weak Langlands functorial lift from the appropriate split classical group G . We do this by a new automorphic descent method, namely the double descent. This method is derived from the recent generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan \cite{CFGK17}, which represent the standard L -functions for $G\times \GL_n$. Our results are valid also for double covers of symplectic groups.

We study the double centraliser property and the annihilators ideals of certain permutation modules for symmetric groups and their quantum analogues. In version 2 some remarks have been added on cell ideals and annihilators.

For the exceptional finite-dimensional modular Lie superalgebras g(A) with indecomposable Cartan matrix A , and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd elements with zero square. These homologies are~key ingredients in the Duflo--Serganova approach to the representation theory. There were two definitions of defect of Lie superalgebras in the literature with different ranges of application. We suggest a third definition and an easy-to-use way to find its value. In positive characteristic, we found out one more reason to consider the space of roots over reals, unlike the space of weights, which should be considered over the ground field. We proved that the rank of the homological element (decisive in calculating the defect of a given Lie superalgebra) should be considered in the adjoint module, not the irreducible module of least dimension (although the latter is sometimes possible to consider, e.g., for p=0 ). We also computed the above homology for the only case of simple Lie superalgebras with symmetric root system not considered so far over the field of complex numbers, and its modular versions: psl(a|a+pk) for a and k small, and p=2,3,5 .