In this paper, we prove some orthogonality relations for representations arising from deep level Deligne–Lusztig schemes of Coxeter type. This generalizes previous results of Lusztig [Lus04], and of Chan and the second author [CI21b]. Applications include the study of smooth representations of p-adic groups in the cohomology of p-adic Deligne–Lusztig spaces and their relation to the local Langlands correspondences. Also, the geometry of deep level Deligne–Lusztig schemes gets accessible, in the spirit of Lusztig’s work [Lus76].

The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2, 3)$-generators for the finite odd-dimensional orthogonal groups $\Omega _{2k+1}(q)$, $k\geq 4$. As a byproduct, we also obtain $(2,3)$-generators for $\Omega _{4k}^+(q)$ with $k\geq 3$ and q odd, and for $\Omega _{4k+2}^\pm (q)$ with $k\geq 4$ and $q\equiv \pm 1~ \mathrm {(mod~ 4)}$.

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.

We study reductive subgroups H of a reductive linear algebraic group G – possibly nonconnected – such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre. This generalises results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.

We fix an error on a $3$ -cocycle in the original version of the paper ‘Endoscopy for Hecke categories, character sheaves and representations’. We give the corrected statements of the main results.

Lusztig’s algorithm of computing generalized Green functions of reductive groups involves an ambiguity on certain scalars. In this paper, for reductive groups of classical type with arbitrary characteristic, we determine those scalars explicitly, and eliminate the ambiguity. Our results imply that all the generalized Green functions of classical type are computable.

For a connected reductive group G over a finite field, we study automorphic vector bundles on the stack of G-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack of G-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

We determine the smallest irreducible Brauer characters for finite quasi-simple orthogonal type groups in non-defining characteristic. Under some restrictions on the characteristic we also prove a gap result showing that the next larger irreducible Brauer characters have a degree roughly the square of those of the smallest non-trivial characters.

For a reductive group $G$ over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$, after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$.

We show that a nearly square independent and identically distributed random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz [6]. Our result extends to sparse matrices as well as to matrices of dependent entries.

We show that for any n and q, the number of real conjugacy classes in $ \rm{PGL}(\it{n},\mathbb{F}_q) $ is equal to the number of real conjugacy classes of $ \rm{GL}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SL}(\it{n},\mathbb{F}_q) $, refining a result of Lehrer [J. Algebra36(2) (1975), 278–286] and extending the result of Gill and Singh [J. Group Theory14(3) (2011), 461–489] that this holds when n is odd or q is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $ \rm{PGU}(\it{n},\mathbb{F}_q) $, and equal to the number of real conjugacy classes of $ \rm{U}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SU}(\it{n},\mathbb{F}_q) $, refining results of Gow [Linear Algebra Appl.41 (1981), 175–181] and Macdonald [Bull. Austral. Math. Soc.23(1) (1981), 23–48]. We also give a generating function for this common quantity.

We apply our theory of partial flag spaces developed with W. Goldring to study a group-theoretical generalization of the canonical filtration of a truncated Barsotti–Tate group of level 1. As an application, we determine explicitly the normalization of the Zariski closures of Ekedahl–Oort strata of Shimura varieties of Hodge-type as certain closed coarse strata in the associated partial flag spaces.

Fix an arbitrary finite group A of order a, and let X(n, q) denote the set of homomorphisms from A to the finite general linear group GLn(q). The size of X(n, q) is a polynomial in q. In this note, it is shown that generically this polynomial has degree n2(1 – a−1) − εr and leading coefficient mr, where εr and mr are constants depending only on r := n mod a. We also present an algorithm for explicitly determining these constants.

Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_{p}}$ of the finite field of prime order $p$ and let $F:\mathbf{G}\rightarrow \mathbf{G}$ be a Frobenius endomorphism with $G=\mathbf{G}^{F}$ the corresponding $\mathbb{F}_{q}$-rational structure. One of the strongest links we have between the representation theory of $G$ and the geometry of the unipotent conjugacy classes of $\mathbf{G}$ is a formula, due to Lusztig (Adv. Math. 94(2) (1992), 139–179), which decomposes Kawanaka’s Generalized Gelfand–Graev Representations (GGGRs) in terms of characteristic functions of intersection cohomology complexes defined on the closure of a unipotent class. Unfortunately, the formula given in Lusztig (Adv. Math. 94(2) (1992), 139–179) is only valid under the assumption that $p$ is large enough. In this article, we show that Lusztig’s formula for GGGRs holds under the much milder assumption that $p$ is an acceptable prime for $\mathbf{G}$ ($p$ very good is sufficient but not necessary). As an application we show that every irreducible character of $G$, respectively, character sheaf of $\mathbf{G}$, has a unique wave front set, respectively, unipotent support, whenever $p$ is good for $\mathbf{G}$.

Suppose that $\widetilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\widetilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\widetilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $\left( \tilde{G},\Gamma \right)$, and consider any group $G$ satisfying the axioms. If both $\widetilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\widetilde{{{G}^{*}}}$ and ${{G}^{*}}$. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in ${{G}^{*}}\,(k)$ to the analogous set for $\widetilde{{{G}^{*}}}\,(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\widetilde{G}\,(k)$, one obtains a mapping of such packets.

Let $G(q)$ be a finite Chevalley group, where $q$ is a power of a good prime $p$, and let $U(q)$ be a Sylow $p$-subgroup of $G(q)$. Then a generalized version of a conjecture of Higman asserts that the number $k(U(q))$ of conjugacy classes in $U(q)$ is given by a polynomial in $q$ with integer coefficients. In [S. M. Goodwin and G. Röhrle, J. Algebra 321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of $k(U(q))$. By implementing it into a computer program using $\mathsf{GAP}$, they were able to calculate $k(U(q))$ for $G$ of rank at most five, thereby proving that for these cases $k(U(q))$ is given by a polynomial in $q$. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of $k(U(q))$ for finite Chevalley groups of rank six and seven, except $E_7$. We observe that $k(U(q))$ is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write $k(U(q))$ as a polynomial in $q-1$, then the coefficients are non-negative.

Under the assumption that $k(U(q))$ is a polynomial in $q-1$, we also give an explicit formula for the coefficients of $k(U(q))$ of degrees zero, one and two.

We show that there are only finitely many primes $p$ such that $\mathrm{PSL} (2, p)$ has a minimal generating set of size four.

Supplementary materials are available with this article.

We calculate all decomposition matrices of the cyclotomic Hecke algebras of the rank two exceptional complex reflection groups in characteristic zero. We prove the existence of canonical basic sets in the sense of Geck–Rouquier and show that all modular irreducible representations can be lifted to the ordinary ones.