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.

In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

We prove that among all compact homogeneous spaces for an effective transitive action of a Lie group whose Levi subgroup has no compact simple factors, the seven-dimensional flat torus is the only one that admits an invariant torsion-free $\text{G}_{2(2)}$-structure.

We show a precise formula, in the form of a monomial, for certain families of parabolic Kazhdan–Lusztig polynomials of the symmetric group. The proof stems from results of Lapid–Mínguez on irreducibility of products in the Bernstein–Zelevinski ring. By quantizing those results into a statement on quantum groups and their canonical bases, we obtain identities of coefficients of certain transition matrices that relate Kazhdan–Lusztig polynomials to their parabolic analogues. This affirms some basic cases of conjectures raised recently by Lapid.

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:(\mathbb{C}^{n},0)\rightarrow (\mathbb{C},0)$. The Yau algebra $L(V)$ is defined to be the Lie algebra of derivations of the moduli algebra $A(V):={\mathcal{O}}_{n}/(f,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{1},\ldots ,\unicode[STIX]{x2202}f/\unicode[STIX]{x2202}x_{n})$, that is, $L(V)=\text{Der}(A(V),A(V))$. It is known that $L(V)$ is finite dimensional and its dimension $\unicode[STIX]{x1D706}(V)$ is called the Yau number. We introduce a new series of Lie algebras, that is, $k$th Yau algebras $L^{k}(V)$, $k\geq 0$, which are a generalization of the Yau algebra. The algebra $L^{k}(V)$ is defined to be the Lie algebra of derivations of the $k$th moduli algebra $A^{k}(V):={\mathcal{O}}_{n}/(f,m^{k}J(f)),k\geq 0$, that is, $L^{k}(V)=\text{Der}(A^{k}(V),A^{k}(V))$, where $m$ is the maximal ideal of ${\mathcal{O}}_{n}$. The $k$th Yau number is the dimension of $L^{k}(V)$, which we denote by $\unicode[STIX]{x1D706}^{k}(V)$. In particular, $L^{0}(V)$ is exactly the Yau algebra, that is, $L^{0}(V)=L(V),\unicode[STIX]{x1D706}^{0}(V)=\unicode[STIX]{x1D706}(V)$. These numbers $\unicode[STIX]{x1D706}^{k}(V)$ are new numerical analytic invariants of singularities. In this paper we formulate a conjecture that $\unicode[STIX]{x1D706}^{(k+1)}(V)>\unicode[STIX]{x1D706}^{k}(V),k\geq 0.$ We prove this conjecture for a large class of singularities.

For every n, we evaluate the smallest k such that the congruence inclusion $\alpha (\beta \circ _n \gamma ) \subseteq \alpha \beta \circ _{k} \alpha \gamma $ holds in a variety of reducts of lattices introduced by K. Baker. We also study varieties with a near-unanimity term and discuss identities dealing with reflexive and admissible relations.