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Cross characteristic representations of symplectic and unitary groups. (English) Zbl 1025.20002
J. Algebra 257, No. 2, 291-347 (2002); Addendum 299, No. 1, 443-446 (2006).
In this very remarkable and elaborate paper the authors investigate low-dimensional representations in positive, non-defining characteristic for the symplectic and unitary groups. Furthermore, they deal with problems concerning minimal polynomials and quadratic modules.
Let $$p$$ and $$r$$ be two different primes and $$q=p^f$$ for some positive integer $$f$$; furthermore, let $$k$$ be an algebraically closed field of characteristic $$r$$. Then the main results read as follows.
Theorem 2.1. Let $$G=\text{Sp}_{2n}(q)$$ with $$n\geq 2$$ and $$q$$ odd. Let $$V$$ be an irreducible $$kG$$-module of dimension less than $$(q^n-1)(q^n-q)/(2(q+1))$$. Then $$V$$ is either the trivial module, or a Weil module of dimension $$(q^n\pm 1)/2$$.
Theorem 2.2. Let $$G=\text{Sp}_{2n}(q)$$ with $$n\geq 2$$ and $$q$$ odd. Let $$V$$ be an irreducible $$kG$$-module such that any long root subgroup of $$G$$ has at most $$(q-1)/2$$ nontrivial linear characters on $$V$$. Then $$V$$ is either trivial or a Weil module.
Theorem 2.3. Let $$G=\text{Sp}_{2n}(q)$$ with $$n\geq 3$$ and $$q$$ odd. For a long root subgroup $$Z:=\{x_\gamma(t)\mid t\in\mathbb{F}_q^*\}$$ let $$\Omega_1(Z)$$ denote the set of all linear characters of $$Z$$ of the form $$\lambda_a\colon x_\gamma(t)\to\varepsilon^{\text{tr}_{\mathbb{F}_q/\mathbb{F}_p}(at)}$$ where $$a\in\mathbb{F}_q^*$$ is a square and $$\varepsilon=\exp(2\pi i/p)$$; similarly $$\Omega_2(Z)$$ is the set of all $$\lambda_a$$ where $$a\in\mathbb{F}_q^*$$ is a nonsquare.
Let $$V$$ be an irreducible $$kG$$-module satisfying at least one of the following conditions:
(1) If $$Y=Y_1\times Y_2$$ with (distinct) long root subgroups $$Y_1$$ and $$Y_2$$ of $$G$$, then all nontrivial linear characters of $$Y$$ on $$V$$ are of the form $$\alpha_1\otimes\alpha_2$$ where either $$\alpha_j\in\Omega_1(Y_j)$$ for $$j\in \{1,2\}$$ or $$\alpha_j\in\Omega_2(Y_j)$$ for $$j\in\{1,2\}$$.
(2) For some $$j\in\{2,\dots,n-1\}$$ the restriction of $$V$$ to a standard subgroup $$\text{Sp}_{2j}(q)$$, i.e. the pointwise stabilizer of a nondegenerate $$(2n-2j)$$-dimensional subspace of the natural module, involves only irreducible Weil modules and maybe trivial modules.
(3) For the stabilizer $$P_n$$ of an $$n$$-dimensional totally isotropic subspace in the natural module any $$P_n$$-orbit of nontrivial linear $$O_p(P_n)$$-characters on $$V$$ is of length less than $$(q^n-1)(q^n-q)/(2(q+1))$$.
Then $$V$$ is either trivial or a Weil module.
Corollary 2.4. Let $$G=\text{Sp}_{2n}(q)$$ with $$n\geq 2$$ and $$q$$ odd. Let $$V$$ be an irreducible $$kG$$-module such that the restriction of $$V$$ to a standard subgroup $$\text{SL}_{2}(q)$$ involves only Weil modules of a given type and maybe the trivial module. Then $$V$$ is either trivial or a Weil module.
Theorem 2.5. Let $$G\in\{\text{SU}_n(q),\text{GU}_n(q)\}$$ with $$n\geq 4$$. Let $$V$$ be an irreducible $$kG$$-module with the following property:
For some $$j\in\{3,\dots,n-1\}$$ the restriction of $$V$$ to a standard subgroup $$\text{SU}_j(q)$$, i.e. the pointwise stabilizer in $$\text{SU}_n(q)$$ of a nondegenerate $$(n-j)$$-dimensional subspace of the natural module, involves only irreducible Weil modules and maybe trivial modules.
Then $$V$$ is either of dimension $$1$$ or a Weil module.
Theorem 2.6. Let $$G=\text{SU}_n(q)$$ with $$n\geq 5$$ and let $$m=[n/2]$$. Furthermore, let $$P_m$$ be the stabilizer in $$G$$ of an $$m$$-dimensional totally isotropic subspace in the natural module and let $$Q_m=O_p(P_m)$$. Suppose that $$V$$ is an irreducible $$kG$$-module such that any $$P_m$$-orbit of nontrivial linear characters of $${\mathbf Z}(Q_m)$$ on $$V$$ is of length less than $$(q^n-1)(q^{n-1}-q)/(q^2-1)(q+1)$$ if $$n$$ is even, and $$(q^{n-1}-1)(q^{n-2}-q)/(q^2-1)(q+1)$$ if $$n$$ is odd.
Then $$V$$ is either trivial or a Weil module.
Theorem 2.7. Let $$G=\text{SU}_n(q)$$ with $$n\geq 5$$ and let $$m=[n/2]$$. Moreover, let $$\kappa_n(q,r)=1$$ if $$r$$ divides $$(q^{2m}-1)/(q^2-1)$$ and $$\kappa_n(q,r)=0$$ otherwise.
Suppose that $$V$$ is an irreducible $$kG$$-module of dimension less than
$$(q^n-1)(q^{n-1}-q)/(q^2-1)(q+1)$$ if $$n$$ is even and $$q=2$$,
$$(q^n-1)(q^{n-1}-q)/(q^2-1)(q+1)-1-\kappa_n(q,r)$$ if $$n$$ is even and $$q>2$$ ,
$$(q^n+1)(q^{n-1}-q^2)/(q^2-1)(q+1)-\kappa_n(q,r)$$ if $$n\geq 7$$ is odd,
$$(q^n+1)(q^{n-1}-q^2)/(q^2-1)(q+1)-1$$ if $$n=5$$.
Then $$V$$ is either trivial or a Weil module.
Theorem 3.1. Let $$G$$ be a finite quasi-simple group of Lie type of characteristic $$p>0$$ of simply connected type, and suppose that $$g\in G$$ is of order $$p$$. Let $$\Theta$$ be a nontrivial absolutely irreducible representation of $$G$$ in characteristic $$r\neq p$$ such that the degree $$d_\Theta(g)$$ of the minimal polynomial of $$\Theta(g)$$ is less than $$p$$.
Then $$p>2$$ and one of the following holds:
(1) $$G=\text{SU}_3(p)$$, $$g$$ is a transvection, and $$\Theta$$ is the reduction modulo $$r$$ of the (unique) complex representation of degree $$p(p-1)$$.
(2) $$G=\text{SL}_2(p)$$, and $$\Theta$$ is either a Weil representation or a representation of degree $$p-1$$.
(3) $$G=\text{SL}_2(p^2)$$, and $$\Theta$$ is a Weil representation.
(4) $$G=\text{Sp}_4(p)$$, and $$\Theta$$ is either a Weil representation, or the unique representation of degree $$p(p-1)^2/2$$.
(5) $$G=\text{Sp}_{2n}(p)$$, $$n\geq 3$$, $$g$$ is a transvection, and $$\Theta$$ is a Weil representation.
Moreover, in each of these cases there exists a representation $$\Theta$$ and an element $$g$$ satisfying the above conditions.
Theorem 3.2. Let $$G=\text{Sp}_{2n}(q)$$ with $$n\geq 2$$ and $$(n,q)\neq(2,3)$$, or $$G=\text{GU}_n(q)$$ with $$n\geq 3$$. Let $$s$$ be a prime not dividing $$q$$ and let $$g\in G$$ be a noncentral element such that $$g$$ belongs to a proper parabolic subgroup of $$G$$ and $$o(g{\mathbf Z}(G))$$ is a power of $$s$$. Let $$V$$ be a nontrivial absolutely irreducible $$G$$-module in characteristic coprime to $$q$$ such that the degree $$d_V(g)$$ of the minimal polynomial of the action of $$g$$ on $$V$$ is less than $$o(g{\mathbf Z}(G))$$. Then $$V$$ is a Weil module.
Theorem 3.3. Each of the groups $$2\text{Sp}_6(2)$$, $$2\Omega_8^+(2)$$, $$2J_2$$, $$2G_2(4)$$, $$2Sz$$, and $$2Co_1$$ has a unique irreducible quadratic $$\mathbb{F}_3$$-module $$V$$. In the first two cases $$V$$ can be obtained by reducing the root lattice of type $$E_8$$ modulo $$3$$, and in the last four cases $$V$$ can be obtained by reducing the Leech lattice modulo $$3$$.

##### MSC:
 20C33 Representations of finite groups of Lie type 20C20 Modular representations and characters 20C34 Representations of sporadic groups 20G05 Representation theory for linear algebraic groups 20G40 Linear algebraic groups over finite fields
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