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Register a new userInvariant differential operators in positive characteristic
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We consider an analog of the problem Veblen formulated in 1928 at the IMC: classify invariant differential operators between natural objects (spaces of either tensor fields, or jets, in modern terms) over a real manifold of any dimension. For unary operators, the problem was solved by Rudakov (no nonscalar operators except the exterior differential); for binary ones, by Grozman (there are no operators of orders higher than 3, operators of order 2 and 3 are, bar an exception in dimension 1, compositions of order 1 operators which, up to dualization and permutation of arguments, form 8 families). In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblens problem in the 1-dimensional case over the ground field of positive characteristic. In addition to analogs of the Berezin integral (strangely overlooked so far) and binary operators constructed from them, we discovered two more (up to dualization) types of indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators.
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The notion of $n$-th indicator for a finite-dimensional Hopf algebra was introduced by Kashina, Montgomery and Ng as gauge invariance of the monoidal category of its representations. The properties of these indicators were further investigated by Shimizu. In this short note, we show that the indicators appearing in positive characteristic all share the same sequence pattern if we assume the coradical of the Hopf algebra is a local Hopf subalgebra.
We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $bf k$. If ${rm char}({bf k})=p>0$, we use this method to construct generalizations ${rm Ver}_{p^n}$, ${rm Ver}_{p^n}^+$ of the incompressible abelian symmetric tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan and Georgiev-Mathieu for $n=1$. Namely, ${rm Ver}_{p^n}$ is the abelian envelope of the quotient of the category of tilting modules for $SL_2(bf k)$ by the $n$-th Steinberg module, and ${rm Ver}_{p^n}^+$ is its subcategory generated by $PGL_2(bf k)$-modules. We show that ${rm Ver}_{p^n}$ are reductions to characteristic $p$ of Verlinde braided tensor categories in characteristic zero, which explains the notation. We study the structure of these categories in detail, and in particular show that they categorify the real cyclotomic rings $mathbb{Z}[2cos(2pi/p^n)]$, and that ${rm Ver}_{p^n}$ embeds into ${rm Ver}_{p^{n+1}}$. We conjecture that every symmetric tensor category of moderate growth over $bf k$ admits a fiber functor to the union ${rm Ver}_{p^infty}$ of the nested sequence ${rm Ver}_{p}subset {rm Ver}_{p^2}subsetcdots$. This would provide an analog of Delignes theorem in characteristic zero and a generalization of the result of arXiv:1503.01492, which shows that this conjecture holds for fusion categories, and then moreover the fiber functor lands in ${rm Ver}_p$.
In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic p>0. In particular, we show that every admissible pair for such a group G gives rise to an L-packet of character sheaves on G, and that, conversely, every L-packet of character sheaves on G arises from a (non-unique) admissible pair. In the appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck-Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third appendix proves that the naive definition of the equivariant constructible derived category with respect to a unipotent algebraic group is equivalent to the correct one.
It is conjectured by Adams-Vogan and Prasad that under the local Langlands correspondence, the L-parameter of the contragredient representation equals that of the original representation composed with the Chevalley involution of the L-group. We verify a variant of their prediction for all connected reductive groups over local fields of positive characteristic, in terms of the local Langlands parameterization of Genestier-Lafforgue. We deduce this from a global result for cuspidal automorphic representations over function fields, which is in turn based on a description of the transposes of V. Lafforgues excursion operators.
The Capelli problem for the symmetric pairs $(mathfrak{gl}times mathfrak{gl},mathfrak{gl})$ $(mathfrak{gl},mathfrak{o})$, and $(mathfrak{gl},mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(mathfrak{g},mathfrak{g}):=(mathfrak{gl}(m|2n),mathfrak{osp}(m|2n))$ and $(mathfrak{gl}(m|n)timesmathfrak{gl}(m|n),mathfrak{gl}(m|n))$, acting on $W:=S^2(mathbb C^{m|2n})$ and $mathbb C^{m|n}otimes(mathbb C^{m|n})^*$. We also give an affirmative answer to the abstract Capelli problem for these cases.
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