Let $G$ be a simple simply connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}=\text{Lie}(G)$. We discuss various properties of nilpotent orbits in $\mathfrak{g}$, which have previously only been considered over $\mathbb{C}$. Using computational methods, we extend to positive characteristic various calculations of de Graaf with nilpotent orbits in exceptional Lie algebras. In particular, we classify those orbits which are reachable as well as those which satisfy a certain related condition due to Panyushev, and determine the codimension of the derived subalgebra $[\mathfrak{g}_{e},\mathfrak{g}_{e}]$ in the centraliser $\mathfrak{g}_{e}$ of any nilpotent element $e\in \mathfrak{g}$. Some of these calculations are used to show that the list of rigid nilpotent orbits in $\mathfrak{g}$, the classification of sheets of $\mathfrak{g}$ and the distribution of the nilpotent orbits amongst them are independent of good characteristic, remaining the same as in the characteristic zero case. We also give a comprehensive account of the theory of sheets in reductive Lie algebras over algebraically closed fields of good characteristic.

Let $E$ be an elliptic curve without complex multiplication (CM) over a number field $K$ , and let $G_{E}(\ell )$ be the image of the Galois representation induced by the action of the absolute Galois group of  $K$ on the $\ell$ -torsion subgroup of  $E$ . We present two probabilistic algorithms to simultaneously determine $G_{E}(\ell )$ up to local conjugacy for all primes $\ell$ by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine $G_{E}(\ell )$ up to one of at most two isomorphic conjugacy classes of subgroups of $\mathbf{GL}_{2}(\mathbf{Z}/\ell \mathbf{Z})$ that have the same semisimplification, each of which occurs for an elliptic curve isogenous to $E$ . Under the GRH, their running times are polynomial in the bit-size $n$ of an integral Weierstrass equation for  $E$ , and for our Monte Carlo algorithm, quasilinear in $n$ . We have applied our algorithms to the non-CM elliptic curves in Cremona’s tables and the Stein–Watkins database, some 140 million curves of conductor up to  $10^{10}$ , thereby obtaining a conjecturally complete list of 63 exceptional Galois images $G_{E}(\ell )$ that arise for $E/\mathbf{Q}$ without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images $G_{E}(\ell )$ that arise for non-CM elliptic curves over quadratic fields with rational $j$ -invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the $j$ -invariant is irrational.

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in Hernandez and Leclerc (Duke Math. J.154 (2010), 265–341; Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statitics, Volume 40, pp. 175–193 (2013)), in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting) to $\mathfrak{sl}_{n+1}[t]$-stable prime Demazure modules in level-two integrable highest-weight representations of the classical affine Lie algebra. It was proved in Chari et al. (arXiv:1408.4090) that a stable Demazure module is isomorphic to the fusion product of stable prime Demazure modules. Our next result proves that such a fusion product is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.

Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ , which is closely related to the cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{n}(x)$ and to primitive prime divisors of $q^{n}-1$ . Our definition of $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$ , we provide an algorithm for determining all pairs $(n,q)$ with $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)\leq cn^{k}$ . This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

Let $R$ be a commutative ring, let $F$ be a locally compact non-archimedean field of finite residual field $k$ of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. We show that the pro-$p$-Iwahori Hecke $R$-algebra of $G=\mathbf{G}(F)$ admits a presentation similar to the Iwahori–Matsumoto presentation of the Iwahori Hecke algebra of a Chevalley group, and alcove walk bases satisfying Bernstein relations. This was previously known only for a $F$-split group $\mathbf{G}$.

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

Let $\mathbf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series.

Let $G$ be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal $G$-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of $G$. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple $G$, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev–Manin’s spaces of weighted pointed curves and with Kausz’s compactification of $GL_{n}$.

In this paper, we prove that the finite simple groups $\text{PSp}_{6}(q)$, ${\rm\Omega}_{7}(q)$ and $\text{PSU}_{7}(q^{2})$ are $(2,3)$-generated for all $q$. In particular, this result completes the classification of the $(2,3)$-generated finite classical simple groups up to dimension 7.

In this paper, we investigate the abstract homomorphisms of the special linear group SLn($\mathfrak{O}$) over complete discrete valuation rings with finite residue field into the general linear group GLm($\mathbb{R}$) over the field of real numbers. We show that for m < 2n, every such homomorphism factors through a finite index subgroup of SLn($\mathfrak{O}$). For $\mathfrak{O}$ with positive characteristic, this result holds for all m ∈ ${\mathbb N}$.

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

We study the problem of determining, for a polynomial function $f$ on a vector space $V$, the linear transformations $g$ of $V$ such that $f\circ g=f$. When $f$ is invariant under a simple algebraic group $G$ acting irreducibly on $V$, we note that the subgroup of $\text{GL}(V)$ stabilizing $f$ often has identity component $G$, and we give applications realizing various groups, including the largest exceptional group $E_{8}$, as automorphism groups of polynomials and algebras. We show that, starting with a simple group $G$ and an irreducible representation $V$, one can almost always find an $f$ whose stabilizer has identity component $G$, and that no such $f$ exists in the short list of excluded cases. This relies on our core technical result, the enumeration of inclusions $G<H\leqslant \text{SL}(V)$ such that $V/H$ has the same dimension as $V/G$. The main results of this paper are new even in the special case where $k$ is the complex numbers.

From the mid-1990s onwards, the main focus of L. G. Kovács’ research was on Lie powers. This brief survey presents some of the key results on Lie powers obtained by Kovács and his collaborators, and discusses some subsequent developments and applications of this work.

Let $F$ be a non-Archimedean local field, and let $G^{\sharp }$ be the group of $F$-rational points of an inner form of $\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of $G^{\sharp }$, via restriction from an inner form $G$ of $\text{GL}_{n}(F)$.

For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth $G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of $G^{\sharp }$, and the idempotent is derived from a type for $G$. We show that the Hecke algebras for Bernstein components of $G^{\sharp }$ are similar to affine Hecke algebras of type $A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

We determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne–Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport–Zink spaces.

Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan–Gross–Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is said to be spherical if it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.

We construct cocompact lattices Γ'0 < Γ0 in the group G = PGLd$({\mathbb{F}_q(\!(t)\!)\!})$ which are type-preserving and act transitively on the set of vertices of each type in the building Δ associated to G. These lattices are commensurable with the lattices of Cartwright–Steger Isr. J. Math.103 (1998), 125–140. The stabiliser of each vertex in Γ'0 is a Singer cycle and the stabiliser of each vertex in Γ0 is isomorphic to the normaliser of a Singer cycle in PGLd(q). We show that the intersections of Γ'0 and Γ0 with PSLd$({\mathbb{F}_q(\!(t)\!)\!})$ are lattices in PSLd$({\mathbb{F}_q(\!(t)\!)\!})$, and identify the pairs (d, q) such that the entire lattice Γ'0 or Γ0 is contained in PSLd$({\mathbb{F}_q(\!(t)\!)\!})$. Finally we discuss minimality of covolumes of cocompact lattices in SL3$({\mathbb{F}_q(\!(t)\!)\!})$. Our proofs combine the construction of Cartwright–Steger Isr. J. Math.103 (1998), 125–140 with results about Singer cycles and their normalisers, and geometric arguments.

The celebrated Smith–Minkowski–Siegel mass formula expresses the mass of a quadratic lattice $(L,Q)$ as a product of local factors, called the local densities of $(L,Q)$. This mass formula is an essential tool for the classification of integral quadratic lattices. In this paper, we will describe the local density formula explicitly by observing the existence of a smooth affine group scheme $\underline{G}$ over $\mathbb{Z}_{2}$ with generic fiber $\text{Aut}_{\mathbb{Q}_{2}}(L,Q)$, which satisfies $\underline{G}(\mathbb{Z}_{2})=\text{Aut}_{\mathbb{Z}_{2}}(L,Q)$. Our method works for any unramified finite extension of $\mathbb{Q}_{2}$. Therefore, we give a long awaited proof for the local density formula of Conway and Sloane and discover its generalization to unramified finite extensions of $\mathbb{Q}_{2}$. As an example, we give the mass formula for the integral quadratic form $Q_{n}(x_{1},\dots ,x_{n})=x_{1}^{2}+\cdots +x_{n}^{2}$ associated to a number field $k$ which is totally real and such that the ideal $(2)$ is unramified over $k$.

We introduce support varieties for rational representations of a linear algebraic group $G$ of exponential type over an algebraically closed field $k$ of characteristic $p>0$. These varieties are closed subspaces of the space $V(G)$ of all 1-parameter subgroups of $G$. The functor $M\mapsto V(G)_{M}$ satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between $V(G)_{M}$ and the family of support varieties $V_{r}(G)_{M}$ obtained by restricting the $G$ action to Frobenius kernels $G_{(r)}\subset G$. These support varieties seem particularly appropriate for the investigation of infinite-dimensional rational $G$-modules.