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Register a new userOn $mathfrak{sl}_2$-triples for classical algebraic groups in positive characteristic
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Let $k$ be an algebraically closed field of characteristic $p > 2$, let $n in mathbb Z_{>0} $, and take $G$ to be one of the classical algebraic groups $mathrm{GL}_n(k)$, $mathrm{SL}_n(k)$, $mathrm{Sp}_n(k)$, $mathrm O_n(k)$ or $mathrm{SO}_n(k)$, with $mathfrak g = operatorname{Lie} G$. We determine the maximal $G$-stable closed subvariety $mathcal V$ of the nilpotent cone $mathcal N$ of $mathfrak g$ such that the $G$-orbits in $mathcal V$ are in bijection with the $G$-orbits of $mathfrak{sl}_2$-triples $(e,h,f)$ with $e,f in mathcal V$. This result determines to what extent the theorems of Jacobson--Morozov and Kostant on $mathfrak{sl}_2$-triples hold for classical algebraic groups over an algebraically closed field of ``small odd characteristic.
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We use analogues of Enrights and Arkhipovs functors to determine the quiver and relations for a category of $mathfrak{sl}_2 ltimes L(4)$-modules which are locally finite (and with finite multiplicities) over $mathfrak{sl}_2$. We also outline serious obstacles to extend our result to $mathfrak{sl}_2 ltimes L(k)$, for $k>4$.
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.
Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $mathfrak{sl}_2$ at generic $q$. We show that for any $rge 1$, the endomorphism algebra of $Motimes V^{otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $rm{TLB}_r(q,Q)$ as the endomorphism algebras of the objects in a quotient category of the category of coloured ribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.
The higher rank Racah algebra $R(n)$ introduced recently is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra $sR(n)$, is then introduced. Using results from classical invariant theory, this $sR(n)$ algebra is shown to be isomorphic to the centralizer $Z_{n}(mathfrak{sl}_2)$ of the diagonal embedding of $U(mathfrak{sl}_2)$ in $U(mathfrak{sl}_2)^{otimes n}$. This leads to a first and novel presentation of the centralizer $Z_{n}(mathfrak{sl}_2)$ in terms of generators and defining relations. An explicit formula of its Hilbert-Poincare series is also obtained and studied. The extension of the results to the study of the special Askey-Wilson algebra and its higher rank generalizations is discussed.
We classify the simple bounded weight modules of ${mathfrak{sl}(infty})$, ${mathfrak{o}(infty)}$ and ${mathfrak{sp}(infty)}$, and compute their annihilators in $U({mathfrak{sl}(infty}))$, $U({mathfrak{o}(infty))}$, $U({mathfrak{sp}(infty))}$, respectively.
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