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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 prove a character formula for some closed fine Deligne–Lusztig varieties. We apply it to compute fixed points for fine Deligne–Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport–Zink spaces arising from the arithmetic Gan–Gross–Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.
We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups
$\text{SO}^{\pm }(n,q)$
and
$\unicode[STIX]{x1D6FA}^{\pm }(n,q)$
. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group
$G$
is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups
$G$
other than the groups
$\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$
with
$q$
even. We prove computationally that for small
$m$
this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups
$\text{SO}^{\pm }(4m+2,q)$
with
$q$
odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in
$\text{SO}^{\pm }(n,q)$
and
$\unicode[STIX]{x1D6FA}^{\pm }(n,q)$
for all
$q$
, and we use these to compute the asymptotic behavior for the number of involutions in these groups when
$q$
is fixed and
$n$
grows.
We show that integral monodromy groups of Kloosterman
$\ell$
-adic sheaves of rank
$n\geqslant 2$
on
$\mathbb{G}_{m}/\mathbb{F}_{q}$
are as large as possible when the characteristic
$\ell$
is large enough, depending only on the rank. This variant of Katz’s results over
$\mathbb{C}$
was known by works of Gabber, Larsen, Nori and Hall under restrictions such as
$\ell$
large enough depending on
$\operatorname{char}(\mathbb{F}_{q})$
with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz’s ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman–Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.
We prove that a finite coprime linear group
$G$
in characteristic
$p\geq \frac{1}{2}(|G|-1)$
has a regular orbit. This bound on
$p$
is best possible. We also give an application to blocks with abelian defect groups.
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.
We complete the classification of the finite special linear groups
$\text{SL}_{n}(q)$
which are
$(2,3)$
-generated, that is, which are generated by an involution and an element of order
$3$
. This also gives the classification of the finite simple groups
$\text{PSL}_{n}(q)$
which are
$(2,3)$
-generated.
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
$\mathbf{G}$
.
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.
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.
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 present a new algorithm for constructing a Chevalley basis for any Chevalley Lie algebra over a finite field. This is a necessary component for some constructive recognition algorithms of exceptional quasisimple groups of Lie type. When applied to a simple Chevalley Lie algebra in characteristic p⩾5, our algorithm has complexity involving the seventh power of the Lie rank, which is likely to be close to best possible.
Let G be a finite d-dimensional classical group and p a prime divisor of ∣G∣ distinct from the characteristic of the natural representation. We consider a subfamily of p-singular elements in G (elements with order divisible by p) that leave invariant a subspace X of the natural G-module of dimension greater than d/2 and either act irreducibly on X or preserve a particular decomposition of X into two equal-dimensional irreducible subspaces. We proved in a recent paper that the proportion in G of these so-called p-abundant elements is at least an absolute constant multiple of the best currently known lower bound for the proportion of all p-singular elements. From a computational point of view, the p-abundant elements generalise another class of p-singular elements which underpin recognition algorithms for finite classical groups, and it is our hope that p-abundant elements might lead to improved versions of these algorithms. As a step towards this, here we present efficient algorithms to test whether a given element is p-abundant, both for a known prime p and for the case where p is not known a priori.
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.
We fix a prime p and consider a connected reductive algebraic group G over a perfect field k which is defined over 𝔽p. Let M be a finite-dimensional rational G-module M, a comodule for k[G]. We seek to somewhat unravel the relationship between the restriction of M to the finite Chevalley subgroup G(𝔽p)⊂G and the family of restrictions of M to Frobenius kernels G(r) ⊂G. In particular, we confront the conundrum that if M is the Frobenius twist of a rational G-module N,M=N(1), then the restrictions of M and N to G(𝔽p) are equal whereas the restriction of M to G(1) is trivial. Our analysis enables us to compare support varieties (and the finer non-maximal support varieties) for G(𝔽p) and G(r) of a rational G-module M where the choice of r depends explicitly on M.
In this paper we construct the maximal subgroups of geometric type of the orthogonal groups in dimension d over GF(q) in O(d3+d2log q+log qlog log q) finite field operations.