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


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
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