Tensor ideals, Deligne categories and invariant theory

We derive several tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne's universal categories RepO, RepGL and RepP. These results are then used to obtain new insight into the second fundamental theorem of invariant theory for the algebraic supergroups of types A,B,C,D,P. We also find short proofs for the classification of tensor ideals in RepS and in the category of tilting modules for SL2(k) with char(k)>0 and for Uq(sl2) with q a root of unity. In general, for a simple Lie algebra g of type ADE, we show that the lattice of such tensor ideals for Uq(g) corresponds to the lattice of submodules in a parabolic Verma module for the corresponding affine Kac-Moody algebra.Comment: v5: proof main results is now independent of previous classifications of tensor ideals in object

The orthosymplectic superalgebra in harmonic analysis

We introduce the orthosymplectic superalgebra osp(m|2n) as the algebra of Killing vector fields on Riemannian superspace R^{m|2n} which stabilize the origin. The Laplace operator and norm squared on R^{m|2n}, which generate sl(2), are orthosymplectically invariant, therefore we obtain the Howe dual pair (osp(m|2n),sl(2)). We study the osp(m|2n)-representation structure of the kernel of the Laplace operator. This also yields the decomposition of the supersymmetric tensor powers of the fundamental osp(m|2n)-representation under the action of sl(2) x osp(m|2n). As a side result we obtain information about the irreducible osp(m|2n)-representations L_(k,0,...,0). In particular we find branching rules with respect to osp(m-1|2n) and an interesting formula for the Cartan product inside the tensor powers of the natural representation of osp(m|2n). We also prove that integration over the supersphere is uniquely defined by its orthosymplectic invariance.Comment: partial overlap with arXiv:1202.066

Bott-Borel-Weil theory and Bernstein-Gel'Fand-Gel'Fand reciprocity for the Lie superalgebras

The main focus of this paper is Bott-Borel-Weil (BBW) theory for basic classical Lie superalgebras. We take a purely algebraic self-contained approach to the problem. A new element in this study is twisting functors, which we use in particular to prove that the top of the cohomology groups of BBW theory for generic weights is described by the recently introduced star action. We also study the algebra of regular functions, related to BBW theory. Then we introduce a weaker form of genericness, relative to the Borel subalgebra and show that the virtual BGG reciprocity of Gruson and Serganova becomes an actual reciprocity in the relatively generic region. We also obtain a complete solution of BBW theory for (m|2), D(2, 1; alpha), F(4) and G(3) with distinguished Borel subalgebra. Furthermore, we derive information about the category of finite-dimensional (m|2)-modules, such as BGG-type resolutions and Kostant homology of Kac modules and the structure of projective modules

The Fourier Transform on Quantum Euclidean Space

We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's relations and a new type of q-Hankel transforms using the first and second q-Bessel functions. The behavior of the Fourier transforms with respect to partial derivatives and multiplication with variables is studied. The Fourier transform acts between the two representation spaces for the harmonic oscillator on quantum Euclidean space. By using this property it is possible to define a Fourier transform on the entire Hilbert space of the harmonic oscillator, which is its own inverse and satisfies the Parseval theorem

The primitive spectrum of basic classical Lie superalgebras

We prove Conjecture 5.7 in [arXiv:1409.2532], describing all inclusions between primitive ideals for the general linear superalgebra in terms of the Ext1-quiver of simple highest weight modules. For arbitrary basic classical Lie superalgebras, we formulate two types of Kazhdan-Lusztig quasi-orders on the dual of the Cartan subalgebra, where one corresponds to the above conjecture. Both orders can be seen as generalisations of the left Kazhdan-Lusztig order on Hecke algebras and are related to categorical braid group actions. We prove that the primitive spectrum is always described by one of the orders, obtaining for the first time a description of the inclusions. We also prove that the two orders are identical if category O admits `enough' abstract Kazhdan-Lusztig theories. In particular, they are identical for the general linear superalgebra, concluding the proof of the conjecture