A Bott–Borel–Weil Theorem for Diagonal Ind-groups

Dimitrov, Ivan ; Penkov, Ivan

Canadian journal of mathematics, Tome 63 (2011) no. 6, pp. 1307-1327

Résumé

A diagonal ind-group is a direct limit of classical affine algebraic groups of growing rank under a class of inclusions that contains the inclusion $$SL(n)\,\to \,SL(2n),\,\,M\mapsto \,\left( \begin{matrix} M & 0\\ 0 & M\\\end{matrix} \right)$$ as a typical special case. If $G$ is a diagonal ind-group and $B\,\subset \,G$ is a Borel ind-subgroup, we consider the ind-variety $G/B$ and compute the cohomology ${{H}^{\ell }}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ of any $G$ -equivariant line bundle ${{\mathcal{O}}_{-\lambda }}$ on $G/B$ . It has been known that, for a generic $\lambda $ , all cohomology groups of ${{\mathcal{O}}_{-\lambda }}$ vanish, and that a non-generic equivariant line bundle ${{\mathcal{O}}_{-\lambda }}$ has at most one nonzero cohomology group. The new result of this paper is a precise description of when ${{H}^{j}}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ is nonzero and the proof of the fact that, whenever nonzero, ${{H}^{j}}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ is a $G$ -module dual to a highest weight module. The main difficulty is in defining an appropriate analog ${{W}_{B}}$ of the Weyl group, so that the action of ${{W}_{B}}$ on weights of $G$ is compatible with the analog of the Demazure “action” of the Weyl group on the cohomology of line bundles. The highest weight corresponding to ${{H}^{j}}(G/B,\,{{\mathcal{O}}_{-\lambda }})$ is then computed by a procedure similar to that in the classical Bott–Borel–Weil theorem.

Détail

DOI : 10.4153/CJM-2011-032-6

Mots-clés : 22E65, 20G05

Bibliographie

[BZh] [BZh] Baranov, A. A. and A. G. Zhilinskii, Diagonal direct limits of simple Lie algebras. Comm. Algebra 27(1999), no. 6, 2749–2766. doi:10.1080/00927879908826590 Google Scholar

[B] [B] Bott, R., Homogeneous vector bundles. Ann. of Math. 66(1957), 203–248. doi:10.2307/1969996 Google Scholar

[D1] [D1] Demazure, M., Une démonstration algébrique d’un théorème de Bott. Invent. Math. 5(1968), 349–356. doi:10.1007/BF01389781 Google Scholar

[D2] [D2] Demazure, M., A very simple proof of Bott's theorem. Invent. Math. 33(1976), no. 3, 271–272. doi:10.1007/BF01404206 Google Scholar

[DP] [DP] Dimitrov, I. and I. Penkov, Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups. Int. Math. Res. Not. , no. 55, 2935–2953. Google Scholar

[DPW] [DPW] Dimitrov, I., I. Penkov, and J. A.Wolf, A Bott–Borel–Weil theory for direct limits of algebraic groups. Amer. J. Math. 124(2002), no. 5, 955–998. doi:10.1353/ajm.2002.0025 Google Scholar

[DR] [DR] Dimitrov, I. and M. Roth, Cup products of line bundles on homogeneous varieties and generalized PRV components of multiplicity one. arXiv:0909.2280v1. Google Scholar

[NRW] [NRW] Natarajan, L., E. Rodr´ıguez-Carrington, and J. A.Wolf, The Bott–Borel–Weil theorem for direct limit groups. Trans. Amer. Math. Soc. 353(2001), no. 11, 4583–4622. doi:10.1090/S0002-9947-01-02452-7 Google Scholar

[S] [S] Serre, J.-P., Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d’aprés Armand Borel et André Weil). In: Séminaire Bourbaki, 2, Exp. No. 100, Soc. Math. France, Paris, 1995, pp. 447–454. Google Scholar

[Sh] [Sh] Shafarevich, I., On some infinite dimensional groups. II. (Russian) Izv. Akad. Nauk USSR Ser. Mat. 45(1981), no. 1, 214–226, 240. Google Scholar

[T] [T] Tsanov, V., Embeddings of semisimple complex Lie groups and cohomological components of modules. arXiv:1005.4225v2. Google Scholar

[W] [W] Wolf, J. A., Principal series representations of direct limit groups. Compos. Math. 141(2005), no. 6, 1504–1530. doi:10.1112/S0010437X05001430 Google Scholar