In our paper, we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq {\mathbb {Z}}/q{\mathbb {Z}}$ in the case of composite q. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

We study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. Applying this procedure, we find the quantum symmetries of the halved cube graph, the folded cube graph, and the Hamming graphs.

Let $\textbf {G}$ be a simply connected semisimple algebraic group over a field of characteristic greater than the Coxeter number. We construct a monoidal action of the diagrammatic Hecke category on the principal block $\operatorname {Rep}_0(\textbf {G})$ of $\operatorname {Rep}(\textbf {G})$ by wall-crossing functors. This action was conjectured to exist by Riche and Williamson. Our method uses constructible sheaves and relies on Smith–Treumann theory.

For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.

Let BG be the classifying space of an algebraic group G over the field ${\mathbb C}$ of complex numbers. There are smooth projective approximations X of $BG\times {\mathbb P}^{\infty}$, by Ekedahl. We compute a new stable birational invariant of X defined by the difference of two coniveau filtrations of X, by Benoist and Ottem. Hence we give many examples such that two coniveau filtrations are different.

We construct an action of the affine Hecke category on the principal block $\mathrm {Rep}_0(G_1T)$ of $G_1T$-modules where G is a connected reductive group over an algebraically closed field of characteristic $p> 0$, T a maximal torus of G and $G_1$ the Frobenius kernel of G. To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.

Let $G$ be a reductive group over an algebraically closed field $k$ of separably good characteristic $p>0$ for $G$. Under these assumptions, a Springer isomorphism $\phi : \mathcal {N}_{\mathrm {red}}(\mathfrak {g}) \rightarrow \mathcal {V}_{\mathrm {red}}(G)$ from the nilpotent scheme of $\mathfrak {g}$ to the unipotent scheme of $G$ always exists and allows one to integrate any $p$-nilpotent element of $\mathfrak {g}$ into a unipotent element of $G$. One should wonder whether such a punctual integration can lead to an integration of restricted $p$-nil $p$-subalgebras of $\mathfrak {g}= \operatorname {Lie}(G)$. We provide a counter-example of the existence of such an integration in general, as well as criteria to integrate some restricted $p$-nil $p$-subalgebras of $\mathfrak {g}$ (that are maximal in a certain sense). This requires the generalisation of the notion of infinitesimal saturation first introduced by Deligne and the extension of one of his theorems on infinitesimally saturated subgroups of $G$ to the previously mentioned framework.

We consider rational representations of a connected linear algebraic group $\mathbb {G}$ over a field $k$ of positive characteristic $p > 0$. We introduce a natural extension $M \mapsto \Pi (\mathbb {G})_M$ to $\mathbb {G}$-modules of the $\pi$-point support theory for modules $M$ for a finite group scheme $G$ and show that this theory is essentially equivalent to the more ‘intrinsic’ and ‘explicit’ theory $M \mapsto \mathbb {P}\mathfrak{C}(\mathbb {G})_M$ of supports for an algebraic group of exponential type, a theory which uses $1$-parameter subgroups $\mathbb {G}_a \to \mathbb {G}$. We extend our support theory to bounded complexes of $\mathbb {G}$-modules, $C^\bullet \mapsto \Pi (\mathbb {G})_{C^\bullet }$. We introduce the tensor triangulated category $\mathit {StMod}(\mathbb {G})$, the Verdier quotient of the bounded derived category $D^b(\mathit {Mod}(\mathbb {G}))$ by the thick subcategory of mock injective modules. Our support theory satisfies all the ‘standard properties’ for a theory of supports for $\mathit {StMod}(\mathbb {G})$. As an application, we employ $C^\bullet \mapsto \Pi (\mathbb {G})_{C^\bullet }$ to establish the classification of $(r)$-complete, thick tensor ideals of $\mathit {stmod}(\mathbb {G})$ in terms of locally $\mathit {stmod}(\mathbb {G})$-realizable subsets of $\Pi (\mathbb {G})$ and the classification of $(r)$-complete, localizing subcategories of $\mathit {StMod}(\mathbb {G})$ in terms of locally $\mathit {StMod}(\mathbb {G})$-realizable subsets of $\Pi (\mathbb {G})$.

We introduce the combinatorial notion of a q-factorization graph intended as a tool to study and express results related to the classification of prime simple modules for quantum affine algebras. These are directed graphs equipped with three decorations: a coloring and a weight map on vertices, and an exponent map on arrows (the exponent map can be seen as a weight map on arrows). Such graphs do not contain oriented cycles and, hence, the set of arrows induces a partial order on the set of vertices. In this first paper on the topic, beside setting the theoretical base of the concept, we establish several criteria for deciding whether or not a tensor product of two simple modules is a highest-$\ell $-weight module and use such criteria to prove, for type A, that a simple module whose q-factorization graph has a totally ordered vertex set is prime.

Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type $\mathsf {F}_4$ , $\mathsf {E}_6$ , or $\mathsf {E}_7$ . We study these objects over an arbitrary base ring R, with particular attention to the case $R = \mathbb {Z}$ . We prove in this generality results previously in the literature in the special case where R is a field of characteristic different from 2 and 3.

We categorify the commutation of Nakajima’s Heisenberg operators $P_{\pm 1}$ andtheir infinitely many counterparts in the quantum toroidal algebra $U_{q_1,q_2}(\ddot {gl_1})$ acting on the Grothendieck groups of Hilbert schemes from [10, 24, 26, 32]. By combining our result with [26], one obtains a geometric categorical $U_{q_1,q_2}(\ddot {gl_1})$ action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semidivisorial log terminal singularities.

We generalize the works of Pappas–Rapoport–Zhu on twisted affine Grassmannians to the wildly ramified case under mild assumptions. This rests on a construction of certain smooth affine $\mathbb {Z}[t]$ -groups with connected fibers of parahoric type, motivated by previous work of Tits. The resulting $\mathbb {F}_p(t)$ -groups are pseudo-reductive and sometimes non-standard in the sense of Conrad–Gabber–Prasad, and their $\mathbb {F}_p [\hspace {-0,5mm}[ {t} ]\hspace {-0,5mm}] $ -models are parahoric in a generalized sense. We study their affine Grassmannians, proving normality of Schubert varieties and Zhu’s coherence theorem.

Let G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs $(M,{\mathcal O})$ where M is the identity component of the centralizer of a semisimple element in G and ${\mathcal O}$ is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.

An affine variety with an action of a semisimple group G is called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group ${\mathbb {K}^{*}}$ commuting with the G-action. We show that X is determined by the ${\mathbb {K}^{*}}$-variety $X^U$ of fixed points under a maximal unipotent subgroup $U \subset G$. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient $X /\!\!/ G$.

If G is of type ${\mathsf {A}_n}$ ($n\geq 2$), ${\mathsf {C}_{n}}$, ${\mathsf {E}_{6}}$, ${\mathsf {E}_{7}}$, or ${\mathsf {E}_{8}}$, we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If $n \geq 5$, every smooth affine $\operatorname {\mathrm {SL}}_n$-variety of dimension $< 2n-2$ is an $\operatorname {\mathrm {SL}}_n$-vector bundle over the smooth quotient $X /\!\!/ \operatorname {\mathrm {SL}}_n$, with fiber isomorphic to the natural representation or its dual.

Motivated by the desire to understand the geometry of the basic loci in the reduction of Shimura varieties, we study their “group-theoretic models”—generalized affine Deligne–Lusztig varieties—in cases where they have a particularly nice description. Continuing the work of Görtz and He (2015, Cambridge Journal of Mathematics 3, 323–353) and Görtz, He, and Nie (2019, Peking Mathematical Journal 2, 99–154), we single out the class of cases of Coxeter type, give a characterization in terms of the dimension, and obtain a complete classification. We also discuss known, new, and open cases from the point of view of Shimura varieties/Rapoport–Zink spaces.

Let K be a number field, let A be a finite-dimensional K-algebra, let $\operatorname {\mathrm {J}}(A)$ denote the Jacobson radical of A and let $\Lambda $ be an $\mathcal {O}_{K}$ -order in A. Suppose that each simple component of the semisimple K-algebra $A/{\operatorname {\mathrm {J}}(A)}$ is isomorphic to a matrix ring over a field. Under this hypothesis on A, we give an algorithm that, given two $\Lambda $ -lattices X and Y, determines whether X and Y are isomorphic and, if so, computes an explicit isomorphism $X \rightarrow Y$ . This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: Given a number field K, a positive integer n and two matrices $A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$ , determine whether A and B are similar over $\mathcal {O}_{K}$ , and if so, return a matrix $C \in \mathrm {GL}_{n}(\mathcal {O}_{K})$ such that $B= CAC^{-1}$ . We give explicit examples that show that the implementation of the latter algorithm for $\mathcal {O}_{K}=\mathbb {Z}$ vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.

We compute the number of points over finite fields of the character stack associated to a compact surface group and a reductive group with connected centre. We find that the answer is a polynomial on residue classes (PORC). The key ingredients in the proof are Lusztig’s Jordan decomposition of complex characters of finite reductive groups and Deriziotis’s results on their genus numbers. As a consequence of our main theorem, we obtain an expression for the E-polynomial of the character stack.

We prove that any element in a sufficiently large transitive finite simple permutation group is a product of two derangements.

For any $n>1$ we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group $\operatorname {\mathrm {Sp}}(n,1)$ . We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the icosian ring in the uniform case for all $n>1$ .