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Springer Correspondence for Disconnected Reductive Groups 数理解析研究所講究録 1310 巻 2003 年 42-53 42 SPRINGER CORRESPONDENCE FOR DISCONNECTED REDUCTIVE GROUPS KARINE SORLIN ABSTRACT. The Springer correspondence is amap from the set of unipotent conjugacy classes of areductive algebraic group to the set of irreducible complex characters of the Weyl group. This paper gives ar\'esum\'e of the construction of the Springer correspon- dence for disconnected reductive groups of the form $G.\langle\sigma\rangle$ when $\sigma$ is aquasi-semi-simple unipotent automorphism of $G$ and more precisely for the connected component $G\sigma$ . This correspondence is then described for $GL(n)\sigma$ in characteristic 2when $\sigma$ is an autouxor- phism of order 2. INTRODUCTION Disconnected algebraic reductive groups naturally appear as extensions of connected reductive groups by an automorphism. Such groups were used by G. Malle to solve the Inverse Galois Problem in [10]. In this paper, we consider adisconnected reductive alge- braic group of the form $G.\langle\sigma\rangle$ , where $G$ is aconnected reductive group over an algebraic closed field of characteristic $p>0$ and $\sigma$ is aunipotent quasi-semi-simple automorphism of $G$ . An automorphism $\sigma$ of $G$ is said to be quasi-semi-simple if there is a $\sigma$ -stable pair $(T, B)$ , where $T$ is amaximal torus included in aBorel subgroup $B$ of $G$ . If $G=GL(n)$ , the subgroup $B$ of upper triangular matrices is aBorel subgroup of $G$ and the subgroup of diagonal matrices is amaximal torus of $G$ included in $B$ . We then consider the aut0- $A-rJ^{t}A^{-1}J^{-1}$ $J$ morphism $\sigma$ of $G$ defined by , where is the matrix $J=(\begin{array}{ll} 11 \end{array})$ $B$ Then $\sigma$ stabilises both $T$ and , so it is quasi-semi-simple. The subject of this paper is the Springer correspondence for the disconnected group $G.\langle\sigma\rangle$ and more precisely for the connected component Go. In the connected case, the Springer correspondence has been studied from many dif- ferent points of view. It is easely described in the case of $GL(n)$ . The Weyl group of $GL(n)$ is isomorphic to the symmetric group $S_{n}$ and irreducible representations of $S_{n}$ are parametrized by partitions of $\mathrm{n}$ . But unipotent conjugacy classes of $GL(n)$ are also para- metrized by partitions of $n$ thanks to Jordan’s normal form of matrices. The Springer 43 KARINE SORLIN correspondence is then the induced bijection from the set of unipotent conjugacy classes of $GL(n)$ to the set of irreducible representations of $S_{n}$ . The Springer correspondence for aconnected reductive group $G$ is not abijection in general. It is an injection from the set of irreducible representations of the Weyl group of $G$ $G$ to the set of pairs $(u, \rho_{u})$ , where $u$ is arepresentative of aunipotent conjugacy class $G$ $(u)=\mathrm{Z}\mathrm{c}(u)/\mathrm{Z}c(u)^{0}$ $\mathrm{Z}c(u)$ in and $\rho$ is an irreducible representation of Ac , where is the centralizer of $u$ in $G$ . $G\sigma$ Let us now consider the case of when $\sigma$ is aunipotent quasi-semi-simple auto morphism of $G$ . The Springer correspondence is then amap from the set of irreducible representations of the subgroup $W(G)^{\sigma}$ of $\sigma$ -fixed elements of the Weyl group $W(G)$ of $G$ to the set of pairs $(u\sigma, \rho_{u\sigma})$ , where ua is arepresentative of aunipotent G-conjugacy $G\sigma$ $=\mathrm{Z}c(u\sigma)/\mathrm{Z}c(u\sigma)^{0}$ class in and $\rho_{u\sigma}$ is an irreducible representation of Ag (u) , where Zc $(u\sigma)$ is the centralizer of $u\sigma$ in $G$ . This correspondence induces an injective map from the set of unipotent conjugacy classes in Go to the set of irreducible representations of $W(G)^{\sigma}$ . We will call this injection the restricted Springer correspondence. In the first section of this paper, we give some main results from [17] and [3] about quasi-semi-simple automorphisms of aconnected reductive group. The second section is asummary of the construction of the Springer correspondence for Go with $\sigma$ unipotent quasi-semi-simple, which was done in [14] and [15]. In the third section, we state two properties of the correspondence which were proved in [15] and which were used in [12] to calculate the restricted Springer correspondence for any reductive group $G.\langle\sigma\rangle$ when $\sigma$ is unipotent quasi-semi-simple and when the connected components of $G$ are of classical type. Following [12], this restrictive Springer correspondence is described in section 4for $GL(n)\sigma$ in characteristic 2, when ais aquasi-semi-simple automorphism of order 2of $GL(n)$ . 1. CONNECTED REDUCTIVE GROUPS WITH AQUASI-SEMI-SIMPLE AUTOMORPHISM Theorem 1.1 ([17] 9.4, [3] Theorem 1.8, (i)). If $G$ is a connected reductive group and $G^{\sigma}$ $\sigma$ $G$ $\sigma$ a quasi-semi-simple automorphism of $G$ , then the group of -fixed points of is reductive. We consider aconnected reductive group $G$ endowed with aquasi-semi-simple aut0- $G$ morphism $\sigma$ . We fix apair $T\subset B$ , where $T$ is a $\mathrm{c}\mathrm{r}$ -stable maximal torus of included $U$ $B$ $\Phi$ in a $\mathrm{c}\mathrm{r}$ -stable Borel subgroup $B$ . We denote by the unipotent radical of , by the set of roots of $G$ relative to $T$ and by $\Pi$ the basis of simple roots relative to $(T, B)$ . For $\in\Phi$ $K^{+}arrow U_{\alpha}$ $U_{\alpha}$ any cr we fix an automorphism $x_{\alpha}$ : where is the unipotent subgroup corresponding to $\alpha$ . $\Pi$ The automorphism $\sigma$ induces an automorphism of 4which stabilizes . 44 SPRINGER CORRESPONDENCE FOR DISCONNECTED REDUCTIVE GROUPS $\sigma^{i}$ $\Phi$ $i$ $\alpha$ Definition 1.2. Let $\alpha$ be in . If is the order of the a-0rbit of , then, induces an automorphism of Ua. We define $C_{\sigma,\alpha}\in k^{\cross}$ by: $\sigma^{i}(x_{\alpha}(\lambda))=x_{\alpha}(C_{\sigma,\alpha}\lambda)$ Example 1.3. Let us consider the case of G $=$ $GL(4)$ , when $\sigma$ is defined by $=(\begin{array}{llll}0 0 0 10 0 1 00 \mathrm{l} 0 01 0 0 0\end{array})$ $\sigma$ : A $\in GL(4)\vdasharrow J^{t}A^{-1}J^{-1}$ with J . We denote by $T$ the set of diagonal matrices, and by $B$ the set of upper triangular matrices. Then the simple roots of $G$ relative to $(T, B)$ are the $\alpha_{i}$ : $t\vdash*t_{i+1}/t_{i}$ for $i\in$ $\{1,2,3\}$ , where $t=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(t_{1}, t_{2}, t_{3}, t4)$ . An easy matrices computation gives: $\sigma(x_{\alpha_{1}}(\lambda))=$ $\sigma(x_{\alpha_{3}}(\lambda))=x_{\alpha_{1}}(-\lambda)$ $x_{\alpha_{3}}(-\lambda)$ , $\sigma(x_{\alpha 2}(\lambda))=x_{\alpha_{2}}(-\lambda)$ and . Hence, we have two a-0rbits of simple roots: $\{\alpha_{1}, \alpha_{3}\}$ and $\{\alpha_{2}\}$ . Let us calculate the coefficients $C_{\sigma,\alpha}$ for both orbits. $\sigma^{2}(x_{\alpha_{1}}(\lambda))=x_{\alpha_{1}}(C_{\sigma,\alpha_{1}}\lambda)$ $C_{\sigma,\alpha_{1}}=1$ The coefficient $C_{\sigma,\alpha_{1}}$ is defined by , so we get . $\sigma(x_{\alpha_{2}}(\lambda))=x_{\alpha_{2}}(C_{\sigma,\alpha_{2}}\lambda)$ $C_{\sigma,\alpha_{2}}=-1$ The coefficient $C_{\sigma,\alpha_{2}}$ is defined by , so we get . Theorem 1.4 ([17] 8.2, [3] Theorem 1.8, (v)). There is a natural surjection between the set of $\sigma$ -orbits such that: $\beta$ $\beta'\in\Phi$ $\sigma$ $\alpha$ $C_{\sigma,\alpha}=\pm 1$ where –1 is allowed only if there are , in the -orbit of $(*)\{$ such that $\beta+\beta'\in\Phi$ , onto the set of roots of $(G^{\sigma})^{0}$ relative to $(T^{\sigma})^{0}$ . This is a bijection and any $C_{\sigma,\alpha}$ is equal to 1except in the case when the Dynkin diagram of $G$ has $k$ components of type $A_{2n}$ which are permuted by $\sigma$ . Theorem 1.5 ([3] Theorem 1.15, [4] Proposition 2.1). A quasi-semi-simple automorphism $\sigma$ of $G$ is said to be quasi-central if one of the follow $ing$ equivalent properties is verified: $G$ $\sigma'=\sigma\circ \mathrm{a}\mathrm{d}(g)$ $w$ 1) If $\sigma$ ’is a quasi-semi-simple automorphism of such that , here $g\in G$ , then $\dim(G^{\sigma})^{0}\geq\dim(G^{\sigma'})^{0}$ . $B^{(G^{\sigma})^{0}}$ 2) The rnap $B\mapsto B\cap(G^{\sigma})^{0}$ defines an isomorphism from the variety of Borel $B_{\sigma}^{G}$ $G$ subgroups of $(G^{\sigma})^{0}$ to the variety of $\sigma$ -stable Borel subgroups of . 3) Let $W(G)$ be the Weyl group of G. Then, the group $W(G)^{\sigma}$ of $\sigma$ -fixed points of $W(G)$ is isomorphic to the Weyl group of $(G^{\sigma})^{0}$ . $T$ 4) There is a $\sigma$ -stable pair $T\subset B$ uthere is a maximal torus and $B.a$ Borel subgroup of $G$ such that the condition $(*)$ of Theorem 1.4 is true for any simple root $\alpha$ of $G$ relative to $(T, B)$ . 4: Property 4) is true for any $\sigma$ -stable pair (T, B). Example 1.6. With the same hypotheses as in Example 1.3 and in characteristic different from 2, $\sigma$ is not quasi-central 45 KARINE SORLIN $A\in GL(4)\vdash\not\simeq J^{t}A^{-1}J^{-1}$ We still consider $G=GL(4)$ but now ais defined by $\sigma$ : with $J=($ $000-1$ $\frac{00}{0}1$ $0001$ $0001$ ). $\{\alpha_{1}, \alpha_{3}\}$ $\{\alpha_{2}\}$ In this case, we still have two $\sigma$ -orbits of simple roots and , but now we have $C_{\sigma,\alpha_{1}}=C_{\sigma,\alpha_{2}}=1$ .
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