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**Derived categories and Deligne-Lusztig varieties. II.**
*(English)*
Zbl 1471.20028

The paper under review is a continuation of the work of the first and last author [Publ. Math., Inst. Hautes Étud. Sci. 97, 1–59 (2003; Zbl 1054.20024)]. Using the Deligne-Lusztig variety and its \(\ell\)-adic cohomology complex, the authors prove the existence of a Morita equivalence between certain blocks of algebraic groups and normalizers of the Levi subgroups which come actually from a splendid derived equivalence. This then has the advantage of inducing a compatible family of equivalences between the derived categories of centralizers of \(\ell\)-subgroups, for \(\ell\) being a prime different from the defining one, via the Brauer construction.

More precisely: Let \(G\) be a connected reductive algebraic group over an algebraic closure of a finite field, and let \(F\) be an endomorphism of \(G\) such that a power of \(F\) is a Frobenius endomorphism. Let \(\ell\) be a prime different from the characteristic of the defining field, and let \(K\) be a finite extension of the field of \(\ell\)-adic numbers. Let \(\mathcal O\) be the ring of integers in \(K\) over the \(\ell\)-adic integers and let \(k\) be the residue field of \(\mathcal O\). Let \(L\) be an \(F\)-stable Levi subgroup of \(G\), such that \(L\) is contained in a parabolic subgroup \(P\) with unipotent radical \(V\), and \(P\) is the semidirect product of \(V\) with \(L\). The Deligne-Lusztig variety \(Y_P\) is defined as the set of classes \(gV\in G/V\) such that \(g^{-1}F(g)\in V\cdot F(V)\). It has a left action of \(G^F\) and a right action of \(L^V\) by multiplication. Let \(\Lambda\) be any of the rings \(k\), \(\mathcal O\), \(K\). The complex \(R\Gamma_c(Y_P,\Lambda)\) computing the \(\ell\)-adic cohomology of \(Y_P\) then defines a functor from the bounded derived category of \(\Lambda L^F\)-modules to the bounded derived category of \(\Lambda G^F\)-modules by the left derived tensor product over \(\Lambda L^F\). On the level of Grothendieck groups this functor induces a map \(R_{L\subset P}^G\). Let \(G^*\) be a group Langlands dual to \(G\) with Frobenius \(F^*\) and let \(\mathrm{Irr}(G^F)\) be the set of irreducible characters of \(G^F\) over \(K\). P. Deligne and G. Lusztig [Ann. Math. (2) 103, 103–161 (1976; Zbl 0336.20029)] gave a decomposition of \(\mathrm{Irr}(G^F)\) into a disjoint union of subsets \(\mathrm{Irr}(G^F,(s))\) where \(s\) runs over the set of \({G^*}^{F^*}\)-conjugacy classes of semisimple elements of \({G^*}^{F^*}\). The unipotent characters are those in \(\mathrm{Irr}(G^F,1)\). Let \(L\) be an \(F\)-stable Levi subgroup of \(G\) with dual \(L^*\) in \(G^*\) containing the cantralizer of \(s\) in \(G^*\). Lusztig constructed a bijection between \(\mathrm{Irr}(L^F,(s))\) and \(\mathrm{Irr}(G^F,(s))\) given by \(R_{L\subset P}^G\) up to sign. If \(s\) is in the centre of \(L^*\) then there is a bijection between \(\mathrm{Irr}(L^F,(1))\) and \(\mathrm{Irr}(L^F,(s))\) given by multiplication by a linear character of \(L^F\). If \(s\) is a semisimple element of \({G^*}^{F^*}\) of order prime to \(\ell\). Then consider the union of \(\mathrm{Irr}(G^F,(t))\), where \((t)\) runs over conjugacy classes of semisimple elements of \({G^*}^{F^*}\) whose \(\ell'\)-part is \((s)\). M. Broué and J. Michel [J. Reine Angew. Math. 395, 56–67 (1989; Zbl 0654.20048)] showed that this union is a union of blocks. Let \(e_s^{G^F}\) be the corresponding central idempotent of \({\mathcal O}G^F\).

Here is the main theorem: Suppose that the connected component of the identity of the centraliser of \(s\) in \(G^*\) belongs to \(L^*\), and moreover \(L^*\) is minimal with respect to this property. Then, the action of \(L^F\) on the truncated complex \(G\Gamma_c(Y_P,{\mathcal O})e_s^{L^F}\) computing the cohomology extends to an action of its normaliser \(N_{G^F}(L,e_s^{L^F})=:N\) and the resulting complex induces a splendid Rickard equivalence between \({\mathcal O}G^Fe_s^{G^F}\) and \({\mathcal O}Ne_s^{L^F}\). Its degree \(\dim(Y_P)\) cohomology induces a Morita equivalence. Splendid equivalences induce moreover equivalences between the derived categories of the centralizers of \(\ell\)-subgroups via the Brauer construction.

One of the main tools is a result showing, under certain conditions, independence of the constructions under various choices of parabolic subgroups

More precisely: Let \(G\) be a connected reductive algebraic group over an algebraic closure of a finite field, and let \(F\) be an endomorphism of \(G\) such that a power of \(F\) is a Frobenius endomorphism. Let \(\ell\) be a prime different from the characteristic of the defining field, and let \(K\) be a finite extension of the field of \(\ell\)-adic numbers. Let \(\mathcal O\) be the ring of integers in \(K\) over the \(\ell\)-adic integers and let \(k\) be the residue field of \(\mathcal O\). Let \(L\) be an \(F\)-stable Levi subgroup of \(G\), such that \(L\) is contained in a parabolic subgroup \(P\) with unipotent radical \(V\), and \(P\) is the semidirect product of \(V\) with \(L\). The Deligne-Lusztig variety \(Y_P\) is defined as the set of classes \(gV\in G/V\) such that \(g^{-1}F(g)\in V\cdot F(V)\). It has a left action of \(G^F\) and a right action of \(L^V\) by multiplication. Let \(\Lambda\) be any of the rings \(k\), \(\mathcal O\), \(K\). The complex \(R\Gamma_c(Y_P,\Lambda)\) computing the \(\ell\)-adic cohomology of \(Y_P\) then defines a functor from the bounded derived category of \(\Lambda L^F\)-modules to the bounded derived category of \(\Lambda G^F\)-modules by the left derived tensor product over \(\Lambda L^F\). On the level of Grothendieck groups this functor induces a map \(R_{L\subset P}^G\). Let \(G^*\) be a group Langlands dual to \(G\) with Frobenius \(F^*\) and let \(\mathrm{Irr}(G^F)\) be the set of irreducible characters of \(G^F\) over \(K\). P. Deligne and G. Lusztig [Ann. Math. (2) 103, 103–161 (1976; Zbl 0336.20029)] gave a decomposition of \(\mathrm{Irr}(G^F)\) into a disjoint union of subsets \(\mathrm{Irr}(G^F,(s))\) where \(s\) runs over the set of \({G^*}^{F^*}\)-conjugacy classes of semisimple elements of \({G^*}^{F^*}\). The unipotent characters are those in \(\mathrm{Irr}(G^F,1)\). Let \(L\) be an \(F\)-stable Levi subgroup of \(G\) with dual \(L^*\) in \(G^*\) containing the cantralizer of \(s\) in \(G^*\). Lusztig constructed a bijection between \(\mathrm{Irr}(L^F,(s))\) and \(\mathrm{Irr}(G^F,(s))\) given by \(R_{L\subset P}^G\) up to sign. If \(s\) is in the centre of \(L^*\) then there is a bijection between \(\mathrm{Irr}(L^F,(1))\) and \(\mathrm{Irr}(L^F,(s))\) given by multiplication by a linear character of \(L^F\). If \(s\) is a semisimple element of \({G^*}^{F^*}\) of order prime to \(\ell\). Then consider the union of \(\mathrm{Irr}(G^F,(t))\), where \((t)\) runs over conjugacy classes of semisimple elements of \({G^*}^{F^*}\) whose \(\ell'\)-part is \((s)\). M. Broué and J. Michel [J. Reine Angew. Math. 395, 56–67 (1989; Zbl 0654.20048)] showed that this union is a union of blocks. Let \(e_s^{G^F}\) be the corresponding central idempotent of \({\mathcal O}G^F\).

Here is the main theorem: Suppose that the connected component of the identity of the centraliser of \(s\) in \(G^*\) belongs to \(L^*\), and moreover \(L^*\) is minimal with respect to this property. Then, the action of \(L^F\) on the truncated complex \(G\Gamma_c(Y_P,{\mathcal O})e_s^{L^F}\) computing the cohomology extends to an action of its normaliser \(N_{G^F}(L,e_s^{L^F})=:N\) and the resulting complex induces a splendid Rickard equivalence between \({\mathcal O}G^Fe_s^{G^F}\) and \({\mathcal O}Ne_s^{L^F}\). Its degree \(\dim(Y_P)\) cohomology induces a Morita equivalence. Splendid equivalences induce moreover equivalences between the derived categories of the centralizers of \(\ell\)-subgroups via the Brauer construction.

One of the main tools is a result showing, under certain conditions, independence of the constructions under various choices of parabolic subgroups

Reviewer: Alexander Zimmermann (Amiens)

### MSC:

20G05 | Representation theory for linear algebraic groups |

14L30 | Group actions on varieties or schemes (quotients) |

20G40 | Linear algebraic groups over finite fields |

18G80 | Derived categories, triangulated categories |

20G10 | Cohomology theory for linear algebraic groups |

20C33 | Representations of finite groups of Lie type |