Cohomology of Subregular Tilting Modules for Small Quantum Groups

Let R be an irreducible root system with the Coxeter number h. Let l > h be an odd integer (we assume that l is not divisible by 3 if R is of type G2). Let U be the quantum group of type 1 with divided powers associated to these data, see [10] (of type 1 means that the elements K l i are equal to 1). Let u ⊂ U be the Frobenius kernel, see loc. cit. Let 1 be the trivial U−module. The cohomology H(u,1) was computed by V.Ginzburg and S.Kumar in [5], see also [8]. They proved that the odd cohomology H(u,1) vanishes and the algebra of even cohomology H(u,1) is isomorphic to the algebra C[N ] of functions on the nilpotent cone N ⊂ g, where g is the semisimple Lie algebra associated to R. Moreover, this is an isomorphism of graded algebras with the grading on C[N ] corresponding to the natural C−action on N by dilatations. This isomorphism is compatible with natural G−structures of both algebras where G is simply connected group associated to R. Now let sa be the simple affine reflection lying in the affine Weyl group associated to R, l, see e.g. [2]. Let Θsa be the corresponding wall-crossing functor, see e.g. [12]. Let T = Θsa1. It is easy to see that cohomology H (u, T ) has a natural algebra structure; namely for any simple U−module L with highest weight lying on the affine wall of the fundamental alcove we have H(u, T ) = Ext•u(L,L). Since T is a U−module the cohomology H(u, T ) has a natural structure of G−module. Let O ⊂ N be the subregular nilpotent orbit. The main result of this note is the following Main Theorem. The odd cohomology H(u, T ) vanishes. The algebra H(u, T ) is isomorphic to the algebra C[O] of functions on the closure of O. This is an isomorphism of graded algebras with the grading on C[O] corresponding to the action of C by dilatations. This isomorphism is compatible with natural G−structures of both algebras. Remark. One can prove the analogous theorem for the Frobenius kernel G1 of an almost simple algebraic groupG over an algebraically closed field of characteristic p > h. We remark that C[O] = C[O] because of normality of O, see [4, 9]. In [6] W.H.Hesselink computed the structure of C[N ] as graded G-module. It is easy to deduce the Hesselink Theorem from the Ginzburg-Kumar Theorem (or rather from Andersen-Jantzen vanishing Theorem, see [1]). In the same way we are able to compute the structure of C[O] as graded G−module, see Corollary 3 below.