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Gaussian maps and tensor products of irreducible representations. (English) Zbl 0764.20022

Let \(X\) be a smooth projective variety in characteristic 0; \(L\) and \(M\) line bundles over \(X\). Then a natural filtration of \(\Gamma(L)\otimes \Gamma(M)\) is provided by the \(R_ i(L,M)=\Gamma(X\times X,I^ i\otimes (L\otimes M))\), where \(I\) is the ideal sheaf of the diagonal. There are natural maps \(\Phi_ i: R_ i(L,M)\to\Gamma(S^ i \Omega_ x^ 1\otimes L\otimes M)\) (with kernel \(R_{i+1}(L,M)\)) generalising \(\Phi_ 1\), which is a sort of Gauss map. If \(M=L\), the \(R_{2i}\) induce a filtration for \(S^ 2\Gamma(L)\) and the \(R_{2i+1}\) one for \(\Lambda^ 2\Gamma(L)\). Also \(\Phi_ i\) is surjective if \(L\) and \(M\) are sufficiently ample. In the case when \(X=G/P\) with \(G\) a complex simply-connected simple Lie group and \(P\) a parabolic subgroup, \(\Gamma(L)\) is an irreducible \(G\)-module if \(L\) is effective. The author conjectures that if \(L\) and \(M\) are ample, then \(\Phi_ 1\) is surjective, and the main results are proofs of this conjecture when \(G=SL_ n\) and \(P\) a Borel subgroup or when \(P\) is a maximal parabolic corresponding to a miniscule root.
Corollaries of the conjecture (both assuming \(G\neq G_ 2\)) are: 1. If \(\lambda\), \(\mu\) are regular dominant weights and \(\alpha\) a positive root, then \(V(\lambda+\mu-\alpha)\) is a summand of \(V(\lambda)\otimes V(\mu)\); 2. If each of \(L\), \(M\) is at least \(r\) times an ample line bundle then \(\Phi_ i\) is surjective for \(i\leq r\); 3. If \(\lambda,\mu\geq r\delta\) are regular dominant weights and \(\beta\) a sum of \(i\leq r\) positive roots, then \(V(\lambda+\mu-\beta)\) is a summand of \(V(\lambda)\otimes V(\mu)\). It seems that the conjecture was proved in general in a recent preprint of Kumar. The author approaches this by defining the filtration \(F_ i=\Gamma(X,m^ i L)\) of \(\Gamma(L)\) (where \(m\) is the ideal sheaf defining the identity coset), and proving that \(R_ i(L,M)=\text{Ind}_ P^ G(F_ i\otimes\Gamma(M))\). A number of standard facts about representation theory are recalled, which suffice to permit a rather complete analysis of the cases where the conjecture is proved. Such complete information is difficult to obtain in general.

MSC:

20G05 Representation theory for linear algebraic groups
14B20 Formal neighborhoods in algebraic geometry
14L35 Classical groups (algebro-geometric aspects)
22E10 General properties and structure of complex Lie groups
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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References:

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