Remarks on computing irreducible characters. (English) Zbl 0773.20011
Let \(G\) be a connected reductive algebraic group defined over a finite field \(F_ q\) and let \(G(F_ q)\) be the finite group of all rational points of \(G\). Then the author presents a method for computing the character table of \(G(F_ q)\) under the assumption that \(\text{char }F_ q=p\) is sufficiently large.
Let \({\mathcal A}\) be the vector space of class functions \(G(F_ q)\to \overline {Q}_ \ell\) (\(\ell\) a prime number different from \(p\)). Let \(A_ 1\) be the orthonormal basis of \({\mathcal A}\) consisting of “almost characters” [see the author, Characters of reductive groups over a finite field (Ann. Math. Stud. 107, 1984; Zbl 0556.20033)]. Let \({\mathcal A}^ 0\) be the space of all functions in \({\mathcal A}\) which are orthogonal to the Deligne-Lusztig virtual character \(R_ L^ G(f)\) for any proper Levi subgroup \(L\) of \(G\) defined over \(F_ q\) and any virtual character \(f\) on \(L(F_ q)\). Let \({\mathcal Z}\) be the centre of \(G\). The group \({\mathcal Z}(F_ q)\) acts naturally on the vector space \({\mathcal A}\) by \((t_ z f)(g)=f(zf)\) for all \(z\in {\mathcal Z}(F_ q)\), \(f\in{\mathcal A}\), \(g\in G(F_ q)\). It is known that this action of \({\mathcal Z}(F_ q)\) stabilizes the sets \(A_ 1\) and \(A_ 2=A_ 1\cap{\mathcal A}^ 0\). Picking an element \(\beta\in A_ 2\) in each \({\mathcal Z}(F_ q)\)-orbit on \(A_ 2\). Denote by \(I_ \beta\) the isotropy group in \({\mathcal Z}(F_ q)\). For each character \(\zeta: {\mathcal Z}(F_ q)\to \overline {Q}_ \ell^*\), define \(\beta(\zeta)=| I_ \beta|^{-1} \sum_{z\in {\mathcal Z}(F_ q)} \zeta(z^{-1})t_ z\beta\). Let \(A_ 3\) be the set of all such functions \(\beta(\zeta)\). Thus the main result of this paper is to reduce the computation of the character table of \(G(F_ q)\) to the finding of several explicit relations. The last one of such relations is the expressions of elements of \(A_ 3\) as linear combinations of irreducible characters of \(G(F_ q)\). Note that the key theorem for the above reduction was obtained by Kawanaka in the case when \(G\) has rank \(\leq 8\). The author proves it in the general case. But the techniques applied in their proofs are different.
Let \({\mathcal A}\) be the vector space of class functions \(G(F_ q)\to \overline {Q}_ \ell\) (\(\ell\) a prime number different from \(p\)). Let \(A_ 1\) be the orthonormal basis of \({\mathcal A}\) consisting of “almost characters” [see the author, Characters of reductive groups over a finite field (Ann. Math. Stud. 107, 1984; Zbl 0556.20033)]. Let \({\mathcal A}^ 0\) be the space of all functions in \({\mathcal A}\) which are orthogonal to the Deligne-Lusztig virtual character \(R_ L^ G(f)\) for any proper Levi subgroup \(L\) of \(G\) defined over \(F_ q\) and any virtual character \(f\) on \(L(F_ q)\). Let \({\mathcal Z}\) be the centre of \(G\). The group \({\mathcal Z}(F_ q)\) acts naturally on the vector space \({\mathcal A}\) by \((t_ z f)(g)=f(zf)\) for all \(z\in {\mathcal Z}(F_ q)\), \(f\in{\mathcal A}\), \(g\in G(F_ q)\). It is known that this action of \({\mathcal Z}(F_ q)\) stabilizes the sets \(A_ 1\) and \(A_ 2=A_ 1\cap{\mathcal A}^ 0\). Picking an element \(\beta\in A_ 2\) in each \({\mathcal Z}(F_ q)\)-orbit on \(A_ 2\). Denote by \(I_ \beta\) the isotropy group in \({\mathcal Z}(F_ q)\). For each character \(\zeta: {\mathcal Z}(F_ q)\to \overline {Q}_ \ell^*\), define \(\beta(\zeta)=| I_ \beta|^{-1} \sum_{z\in {\mathcal Z}(F_ q)} \zeta(z^{-1})t_ z\beta\). Let \(A_ 3\) be the set of all such functions \(\beta(\zeta)\). Thus the main result of this paper is to reduce the computation of the character table of \(G(F_ q)\) to the finding of several explicit relations. The last one of such relations is the expressions of elements of \(A_ 3\) as linear combinations of irreducible characters of \(G(F_ q)\). Note that the key theorem for the above reduction was obtained by Kawanaka in the case when \(G\) has rank \(\leq 8\). The author proves it in the general case. But the techniques applied in their proofs are different.
Reviewer: Shi Jianyi (Shanghai)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20G40 | Linear algebraic groups over finite fields |
| 20C40 | Computational methods (representations of groups) (MSC2010) |
| 20C33 | Representations of finite groups of Lie type |
Keywords:
connected reductive algebraic groups; finite fields; rational points; character tables; class functions; Deligne-Lusztig virtual characters; Levi subgroups; irreducible charactersCitations:
Zbl 0556.20033References:
| [1] | Teruaki Asai, The unipotent class functions on the symplectic groups and the odd orthogonal groups over finite fields, Comm. Algebra 12 (1984), no. 5-6, 617 – 645. · Zbl 0559.20024 · doi:10.1080/00927878408823018 |
| [2] | W. M. Beynon and N. Spaltenstein, Green functions of finite Chevalley groups of type \?_{\?} (\?=6,7,8), J. Algebra 88 (1984), no. 2, 584 – 614. · Zbl 0539.20025 · doi:10.1016/0021-8693(84)90084-X |
| [3] | P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103 – 161. · Zbl 0336.20029 · doi:10.2307/1971021 |
| [4] | N. Kawanaka, Generalized Gel\(^{\prime}\)fand-Graev representations of exceptional simple algebraic groups over a finite field. I, Invent. Math. 84 (1986), no. 3, 575 – 616. · Zbl 0596.20028 · doi:10.1007/BF01388748 |
| [5] | Noriaki Kawanaka, Shintani lifting and Gel\(^{\prime}\)fand-Graev representations, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 147 – 163. |
| [6] | G. Lusztig, On the finiteness of the number of unipotent classes, Invent. Math. 34 (1976), no. 3, 201 – 213. · Zbl 0371.20039 · doi:10.1007/BF01403067 |
| [7] | G. Lusztig, A class of irreducible representations of a Weyl group, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 3, 323 – 335. · Zbl 0435.20021 |
| [8] | George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. · Zbl 0556.20033 |
| [9] | G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205 – 272. · Zbl 0547.20032 · doi:10.1007/BF01388564 |
| [10] | George Lusztig, Character sheaves. II, III, Adv. in Math. 57 (1985), no. 3, 226 – 265, 266 – 315. , https://doi.org/10.1016/0001-8708(85)90064-7 George Lusztig, Character sheaves. IV, Adv. in Math. 59 (1986), no. 1, 1 – 63. , https://doi.org/10.1016/0001-8708(86)90036-8 George Lusztig, Character sheaves. V, Adv. in Math. 61 (1986), no. 2, 103 – 155. , https://doi.org/10.1016/0001-8708(86)90071-X George Lusztig, Erratum: ”Character sheaves. V”, Adv. in Math. 62 (1986), no. 3, 313 – 314. · Zbl 0606.20036 · doi:10.1016/0001-8708(86)90105-2 |
| [11] | George Lusztig, On the character values of finite Chevalley groups at unipotent elements, J. Algebra 104 (1986), no. 1, 146 – 194. · Zbl 0603.20037 · doi:10.1016/0021-8693(86)90245-0 |
| [12] | G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 10, 157 – 166. Orbites unipotentes et représentations, I. |
| [13] | George Lusztig, Green functions and character sheaves, Ann. of Math. (2) 131 (1990), no. 2, 355 – 408. · Zbl 0695.20024 · doi:10.2307/1971496 |
| [14] | -, A unipotent support for irreducible representations, Adv. in Math (to appear). · Zbl 0789.20042 |
| [15] | T. Shoji, On the Green polynomials of classical groups, Invent. Math. 74 (1983), no. 2, 239 – 267. · Zbl 0525.20027 · doi:10.1007/BF01394315 |
| [16] | -, Some generalization of Asai’s result for classical groups, Algebraic Groups and Related Topics, Adv. Stud. Pure Math., vol. 6, Kinokuniya and North Holland, Tokyo and Amsterdam, 1985. · Zbl 0572.20027 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
