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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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