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Cohomology of tilting modules over quantum groups and \(t\)-structures on derived categories of coherent sheaves. (English) Zbl 1123.17002
Let \(G\) be a complex semisimple group of adjoint type with Lie algebra \(\mathfrak{g}\) and let \(q\) be a primitive root of unity of odd order \(l\) greater than the Coxeter number of \(\mathfrak{g}\) and prime to \(3\) if \(\mathfrak{g}\) has a factor of type \(G_2\). Then V. Ginzburg and S. Kumar [Duke Math. J. 69, No. 1, 179–198 (1993; Zbl 0774.17013)] proved that the cohomology algebra of the small quantum group \(u_q\) corresponding to \(\mathfrak{g}\) is isomorphic to the coordinate algebra \(\mathcal{O}(\mathcal{N})\) of the nilpotent cone \(\mathcal{N}\) of \(\mathfrak{g}\). Note that every irreducible object of the category \(\mathcal{P}Coh\) of \(G\)-equivariant coherent sheaves on \(\mathcal{N}\) with respect to the middle perversity is determined by a certain pair \((O,\mathcal{L})\) where \(O\) is some \(G\)-orbit in \(\mathcal{N}\) and \(\mathcal{L}\) is some irreducible \(G\)-equivariant vector bundle on \(O\). The main result of the paper under review is a description of the \(\mathcal{O}(\mathcal{N})\)-module \(H^\bullet(u_q,T)\) for a tilting object \(T\) in the principal block \({\mathfrak U}_q \mathrm{-mod}^0\) of the category of finite-dimensional graded modules over Lusztig’s quantum group \({\mathfrak U}_q\) which goes as follows. If \(T\) is an indecomposable tilting module of \({\mathfrak U}_q\mathrm{-mod}^0\), then either \(H^\bullet(u_q,T)=0\) or \(H^\bullet (u_q,T)\) is isomorphic to the total cohomology of the irreducible object of \(\mathcal{P} Coh\) corresponding to the pair \((O_T,\mathcal{L}_T)\) uniquely determined by \(T\). As a consequence the author proves a sheaf-theoretic version of a conjecture of J. E. Humphreys [in AMS/IP Stud. Adv. Math. 4, 69–80 (1997; Zbl 0919.17013)] (verified by V. V. Ostrik [Funct. Anal. Appl. 32, No. 4, 237–246 (1998; Zbl 0981.17010)] for type \(A\)) which states that the support of the cohomology of an indecomposable tilting module of \({\mathfrak U}_q\mathrm{-mod}^0\) as a coherent sheaf on \(\mathcal{N}\) is the closure of the nilpotent orbit corresponding to the two-sided cell in the affine Weyl group given via Lusztig’s bijection.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
14A22 Noncommutative algebraic geometry
16S38 Rings arising from noncommutative algebraic geometry
18E30 Derived categories, triangulated categories (MSC2010)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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