On the properties of irreducible representations of special linear and symplectic groups that are not large with respect to the field characteristic and regular unipotent elements from subsystem subgroups

T. S. Busel ; I. D. Suprunenko

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 88-97

Résumé

We study the properties of irreducible representations of special linear and symplectic groups that are not large with respect to the ground field characteristic and regular unipotent elements of nonprime order from subsystem subgroups of types $A_l$ and $C_l$, respectively, with certain conditions on $l$. Assume that $K$ is an algebraically closed field of characteristic $p>2$, $G=A_r(K)$ or $C_r(K)$, $l$ for $G=A_r(K)$ and $l$ for $G=C_r(K)$, $H\subset G$ is a subsystem subgroup with two simple components $H_1$ and $H_2$ of types $A_l$ and $A_{l-r-1}$ or $C_l$ and $C_{r-l}$, respectively, and $x$ is a regular unipotent element from $H_1$. Suppose that $l+1=ap^s+b$ for $G=A_r(K)$ and $2l=ap^s+b$ for $G=C_r(K)$ where $a$, $p\leq b\leq p^s$, and $s>1$. An irreducible representation $\varphi$ of $G$ is said to be $(p,x)$-special if all the weights of the restriction of $\varphi$ to a nice $A_1$-subgroup containing $x^{p^s}$ are less than $p$ (here the set of weights of a group of type $A_1$ is canonically identified with the set of integers). Denote by $d_{\rho}(z)$ the minimal polynomial of the image of an element $z$ in a representation $\rho$ and call the composition factor $\psi$ of the restriction of $\varphi$ to $H$ large for $z\in H$ if $d_{\psi}(z)=d_{\varphi}(z)$. The main results of the paper are Theorems 1 and 2. $\bf{Theorem~1.}$ Let $\varphi$ be a $(p,x)$-special representation of $G$. Then the restriction of $\varphi$ to $H$ has no composition factors that are large for $x$ and nontrivial for $H_2$. $\bf{Theorem~2.}$ Under the assumptions of Theorem 1, the number of maximum size Jordan blocks of the element $\varphi(x)$ does not exceed a certain integer which depends only upon $p$, $b$, and the coefficients at the highest weight and does not depend on the group rank. We explain why the case studied here should be considered separately. For instance, for $p$-restricted representations of the corresponding groups with large highest weights with respect to the characteristic, assertions opposite to Theorems 1 and 2 are valid. The results on the block structure of the images of unipotent elements in representations of algebraic groups can be used for solving recognition problems for representations and linear groups based of the presence of certain special matrices.

Détail

Mots-clés : unipotent elements
Keywords: Jordan block sizes, special linear group, symplectic group.

Bibliographie

[1] Burbaki N., Gruppy i algebry Li, gl. IV-VI, Mir, M., 1972, 334 pp. | MR

[2] Burbaki N., Gruppy i algebry Li, gl. VII-VIII, Mir, M., 1978, 342 pp. | MR

[3] Steinberg R., Lektsii o gruppakh Shevalle, Mir, M., 1975, 263 pp.

[4] Suprunenko I.D., “Minimalnye polinomy elementov poryadka $p$ v neprivodimykh predstavleniyakh grupp Shevalle nad polyami kharakteristiki $p$”, Tr. In-ta matematiki SO RAN. Voprosy algebry i logiki, 30 (1996), 126–163, Novosibirsk | Zbl

[5] Andersen H.H., Jorgensen J., Landrock P., “The projective indecomposable modules of $SL(2,p^n)$”, Proc. London Math. Soc., 46 (1983), 38–52 | DOI | MR | Zbl

[6] Jantzen J.C., “Darstellungen halbeinfacher algebraicher Gruppen und zugeordnete kontravariante Formen”, Bonner math. Schr., 67 (1973) | MR | Zbl

[7] Jantzen J.C., Representations of algebraic groups, Math. Surveys and Monographs, 107, Second ed., 2003, 576 pp. | MR | Zbl

[8] Liebeck M.W., Seitz G.M., Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, Math. Surveys and Monographs, 180, American Math. Soc., Providence, 2012, 380 pp. | DOI | MR | Zbl

[9] Korhonen M., “Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic”, Proc. Amer. Math. Soc., 147 (2019), 4205–4219 | DOI | MR | Zbl

[10] Seitz G.M., “Unipotent elements, tilting modules, and saturation”, Invent. Math., 141:3 (2000), 467–502 | DOI | MR | Zbl

[11] Smith S., “Irreducible modules and parabolic subgroups”, J. Algebra, 75 (1982), 286–289 | DOI | MR | Zbl

[12] Suprunenko I.D., The minimal polynomials of unipotent elements in irreducible representations of the classical groups in odd characteristic, Memoirs Amer. Math. Soc., 200, no. 939, 2009, 154 pp. | DOI | MR

[13] Suprunenko I.D., “Special composition factors in restrictions of representations of special linear and symplectic groups to subsystem subgroups with two simple components”, Trudy Instituta matematiki, 26:1 (2018), 115–133 | MR

[14] Suprunenko I.D., Zalesskii A.E., “On restricting representations of simple algebraic groups to semisimple subgroups with two simple components”, Trudy Instituta matematiki, 13:2 (2005), 109–115

[15] Testerman D., Zalesski A.E., “Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block”, J. Group Theory, 21 (2018), 1–20 | DOI | MR | Zbl