Borel subalgebras of restricted Cartan-Type Lie algebras

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of [Formula: see text] under some conditions which is called [Formula: see text]-subalgebras (Definition 3.1). We classify the conjugacy classes of [Formula: see text]-subalgebras for [Formula: see text] and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of [Formula: see text]

LECTURES ON THE GEOMETRY AND MODULAR REPRESENTATION THEORY OF ALGEBRAIC GROUPS

These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the first author's notes from a lecture series delivered by the second author at the Simons Centre for Geometry and Physics in August 2019. We intend them to complement more detailed treatments.

On the Finiteness of Quantum K-Theory of a Homogeneous Space

We show that the product in the quantum K-ring of a generalized flag manifold $G/P$ involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the $J$-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.

Poisson-Commutative Subalgebras of 𝒮(𝔤) associated with Involutions

The symmetric algebra ${\mathcal{S}}({{\mathfrak{g}}})$ of a reductive Lie algebra ${{\mathfrak{g}}}$ is equipped with the standard Poisson structure, that is, the Lie–Poisson bracket. Poisson-commutative subalgebras of ${\mathcal{S}}({{\mathfrak{g}}})$ attract a great deal of attention because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset{\mathcal{S}}({{\mathfrak{g}}})$ is bounded by the “magic number” ${\boldsymbol{b}}({{\mathfrak{g}}})$ of ${{\mathfrak{g}}}$. There are two classical constructions of $\mathcal C$ with ${\textrm{tr.deg}}\,{\mathcal C}={\boldsymbol{b}}({{\mathfrak{g}}})$. The 1st one is applicable to $\mathfrak{gl}_n$ and $\mathfrak{so}_n$ and uses the Gelfand–Tsetlin chains of subalgebras. The 2nd one is known as the “argument shift method” of Mishchenko–Fomenko. We generalise the Gelfand–Tsetlin approach to chains of almost arbitrary symmetric subalgebras. Our method works for all types. Starting from a symmetric decompositions ${{\mathfrak{g}}}={{\mathfrak{g}}}_0\oplus{{\mathfrak{g}}}_1$, Poisson-commutative subalgebras ${{\mathcal{Z}}},\tilde{{\mathcal{Z}}}\subset{\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ of the maximal possible transcendence degree are constructed. If the ${{\mathbb{Z}}}_2$-contraction ${{\mathfrak{g}}}_0\ltimes{{\mathfrak{g}}}_1^{\textsf{ab}}$ has a polynomial ring of symmetric invariants, then $\tilde{{\mathcal{Z}}}$ is a polynomial maximal Poisson-commutative subalgebra of ${\mathcal{S}}({{\mathfrak{g}}})^{{{\mathfrak{g}}}_0}$ and its free generators are explicitly described.

The parabolic exotic t-structure

International audience Let G be a connected reductive algebraic group over an algebraically closed field k, with simply connected derived subgroup. The exotic t-structure on the cotangent bundle of its flag variety T^*(G/B), originally introduced by Bezrukavnikov, has been a key tool for a number of major results in geometric representation theory, including the proof of the graded Finkelberg-Mirkovic conjecture. In this paper, we study (under mild technical assumptions) an analogous t-structure on the cotangent bundle of a partial flag variety T^*(G/P). As an application, we prove a parabolic analogue of the Arkhipov-Bezrukavnikov-Ginzburg equivalence. When the characteristic of k is larger than the Coxeter number, we deduce an analogue of the graded Finkelberg-Mirkovic conjecture for some singular blocks. Soit G un groupe algébrique réductif connexe sur un corps k algébriquement clos. La t-structure exotique sur le fibré cotangent de sa variété de drapeaux T^*(G/B), introduite à l'origine par Bezrukavnikov, a été un outil clé pour de nombreux résultats majeurs en théorie géométrique des représentations, en particulier la démonstration de la conjecture de Finkelberg-Mirkovic graduée. Dans cet article, nous étudions (sous de légères hypothèses techniques) une t-structure analogue sur le fibré cotangent de la variété de drapeaux partiels T^*(G/P). Comme application, nous prouvons un analogue parabolique de l'équivalence de Arkhipov-Bezrukavnikov-Ginzburg. Lorsque la caractéristique de k est supérieure au nombre de Coxeter, nous déduisons un analogue de la conjecture de Finkelberg-Mirkovic graduée pour certains blocs singuliers.