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Bernstein-Gelfand-Gelfand reciprocity on perverse sheaves. (English) Zbl 0651.14010
In this paper, the authors generalize the results of [BGG]: I. N. Bernshteĭn, I. M. Gel’fand and S. I. Gel’fand [Funct. Anal. Appl. 10, 87–92 (1976); translation from Funkts. Anal. Prilozh. 10, No. 2, 1–8 (1976; Zbl 0353.18013)] to the cateory of perverse sheaves \(P(X)\) on an analytic space \(X\) with a Whitney stratification \(\mathcal S\) satisfying \(\pi_1(S)=0\) for all \(S\in\mathcal S\) and \(\pi_2(S)=0\) if \(\dim(S)<\dim(X)\). The results in [BGG] can be recovered from this by considering the flag manifold \(X\) associated to a complex semisimple group, stratified by the Schubert cells, and applying the Riemann-Hilbert correspondence. The main results are the following:
(Theorem 1.1) The category \(P(X)\) is equivalent to the category of finitely generated \(A\)-modules, for some associative \(\mathbb C\)-algebra with identity \(A\) which is of finite dimension over \(\mathbb C\). Furthermore, if \(L_1,\ldots, L_r\) are the irreducible objects of \(P(X)\), \(P_i\to L_i\) projective covers and \(C_{ij}=[P_i:L_j]\) is the multiplicity of \(L_j\) in the Jordan-Hölder series of \(P_i\), the matrix \(C=(C_{ij})\) is symmetric.
In fact, the authors give a more precise result in the case that \(\bar S-S\) is a Cartier divisor in \(\bar S\) for all \(S\in\mathcal S\): (Theorem 1.3) There exists \(M_1,\ldots, M_{\ell}\in P(X)\) such that the modules \(P_i\) have a decomposition series with factors isomorphic to the \(M^i_k\) and \([P_i:M_k]=[M_k:L_i]\) (BGG reciprocity). In particular \(C= {}^tDD\), \(D_{kj}=[M_k:L_j]\).
Furthermore every projective object in \(P(X)\) has a \(p\)-filtration (cf. [BGG]) and \(P(X)\) has projective dimension \(\leq 2\cdot \max \{\dim(X)-\dim(S)\mid S\in\mathcal S\}\). The \(M_k\) are explicitly obtained from \(\mathcal S\).
All the results in the paper are independent of the base field \(\mathbb C\), and are obtained using the inductive construction of perverse sheaves by R. MacPherson and K. Vilonen [Invent. Math. 84, 403–435 (1986; Zbl 0597.18005)].

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32C38 Sheaves of differential operators and their modules, \(D\)-modules
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
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