Cross characteristic representations of symplectic and unitary groups
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 \(3\).
scientific article; zbMATH DE number 1868066
scientific article