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Equivariant vector bundles on Drinfeld’s upper half space. (English) Zbl 1136.22009

Let \(K\) be a finite extension of \({\mathbb Q}_p\), let \({\mathcal X}\) be Drinfeld’s symmetric space of dimension \(d\) over \(K\), the complement of all \(K\)-rational hyperplanes in projective \(d\)-space \({\mathbb P}_K^d\). This is a \(K\)-rigid analytic space, and the action of the \(p\)-adic group \(G=\text{ GL}_{d+1}(K)\) on \({\mathbb P}_K^d\) induces an action of \(G\) on \({\mathcal X}\). Let \({\mathcal F}\) be a \(G\)-equivariant vector bundle on \({\mathbb P}_K^d\). Its space \(H^0({\mathcal X},{\mathcal F})\) of sections on \({\mathcal X}\) becomes a \(K\)-Fréchet space endowed with a \(G\)-action. The purpose of this paper is to construct a (finite) \(G\)-equivariant filtration of \(H^0({\mathcal X},{\mathcal F})\) by closed subspaces and to identify the resulting graded pieces as \(G\)-representations. Namely, the graded pieces of the strong dual of this filtration are certain subspaces, cut out by the vanishing of certain differential equations, in locally analytic representations which are induced from locally algebraic representations with respect to maximal parabolic subgroups.
The case where \({\mathcal F}\) is the canonical bundle had been treated earlier by P. Schneider and J. Teitelbaum [Astérisque 278, 51–125 (2002); correction Astérisque 295, 191–299 (2004; Zbl 1051.14024)] (and Pohlkamp used their methods to deal with the case where \({\mathcal F}\) is the structure sheaf). The case of dimension \(d=1\) had been analysed by Morita.
The approach taken here is different from the more explicit one followed by Schneider and Teitelbaum. It is based on the long exact exicision sequence comparing the cohomology of \({\mathcal F}\) on \({\mathbb P}_K^d\), on \({\mathcal X}\) and (the local cohomology) on \({\mathbb P}_K^d-{\mathcal X}\subset {\mathbb P}_K^d\). The filtration \({\mathcal F}({\mathcal X})^{\bullet}\) on \(H^0({\mathcal X},{\mathcal F})\) is then induced from a spectral sequence which results from a certain acyclic resolution of the constant sheaf \({\mathbb Z}\) on (the adic space associated to) \({\mathbb P}_K^d-{\mathcal X}\). The filtration has length \(d+1\). The strong dual \(({\mathcal F}({\mathcal X})^j/{\mathcal F}({\mathcal X})^{j+1})'\) of a typical graded piece is then shown to sit in an exact sequence \[ 0\to v_{P(j+1,1,\ldots,1)}^G (H^{d-j}( {\mathbb P}_K^d,{\mathcal F})')\to ({\mathcal F}({\mathcal X})^j/{\mathcal F}({\mathcal X})^{j+1})'\to C^{an}(G,P_{\underline{j+1}};U'_j)^{\partial_j=0}\to0. \] Here \(v_{P(j+1,1,\ldots,1)}^G(H^{d-j}( {\mathbb P}_K^d,{\mathcal F})')\) is what the author calls a generalized Steinberg representation with coefficients in the finite dimensional algebraic \(G\)-module \(H^{d-j}( {\mathbb P}_K^d,{\mathcal F})'\). The representation \(C^{an}(G,P_{\underline{j+1}};U'_j)^{\partial_j=0}\) is, as indicated above, the subspace cut out by an annhilation condition \(\partial_j=0\) of certain differential opertors, inside a locally analytic representation \(C^{an}(G,P_{\underline{j+1}};U'_j)\) induced from a locally algebraic representation \(U'_j\) of a maximal parabolic subgroup \(P_{\underline{j+1}}\) in \(G\).
In the case where \({\mathcal F}\) arises from an irreducible representation of the Levi subgroup of a maximal parabolic subgroup \(P_{(1,d)}\) for which \({\mathbb P}_K^d\cong G/P_{(1,d)}\), more details on \(C^{an}(G,P_{\underline{j+1}};U'_j)^{\partial_j=0}\) are worked out.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
14G22 Rigid analytic geometry
11F85 \(p\)-adic theory, local fields
20G25 Linear algebraic groups over local fields and their integers

Citations:

Zbl 1051.14024
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References:

[1] Bănică, C., Stănăşilă, O.: Algebraic methods in the global theory of complex spaces. Translated from the Romanian. Editura Academiei, Bucharest. John Wiley &Sons, London New York Sydney (1976)
[2] Bott, R.: Homogeneous vector bundles. Ann. Math. (2) 66, 203–248 (1957) · Zbl 0094.35701 · doi:10.2307/1969996
[3] Breuil, C., Schneider, P.: First steps towards p-adic Langlands functoriality. J. Reine Angew. Math. 610, 149–180 (2007) · Zbl 1180.11036 · doi:10.1515/CRELLE.2007.070
[4] Borel, A., Wallach, N.: Continuous cohomology, discrete subgroups, and representations of reductive groups, Second edition. Math. Surv. Monogr., vol. 67. Am. Math. Soc., Providence, RI (2000) · Zbl 0980.22015
[5] Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean Analysis. Grundlehren Math. Wiss., vol. 261. Springer, Berlin Heidelberg Tokyo New York (1984) · Zbl 0539.14017
[6] Cartier, P.: Representations of p-adic groups: a survey. In: Automorphic Forms, Representations and L-Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1. Proc. Sympos. Pure Math., vol. XXXIII, pp. 111–155. Am. Math. Soc., Providence, RI (1979)
[7] Drinfeld, V.G.: Coverings of p-adic symmetric regions. Funct. Anal. Appl. 10, 29–40 (1976)
[8] Grothendieck, A.: Élements de géométrie algébrique. III, Étude cohomologique des faisceaux cohérents, I. Publ. Math., Inst. Hautes Étud. Sci. 11, 349–511 (1961)
[9] Féaux de Lacroix, C.T.: Einige Resultate über die topologischen Darstellungen p-adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem p-adischen Körper. Schriftenr. Math. Inst. Univ. Münster, 3. Serie, vol. 23. Univ. Münster, Mathematisches Institut, Münster (1999) · Zbl 0963.22009
[10] Fulton, W., Harris, J.: Representation theory. A first course. Grad. Texts Math., vol. 129. Springer, New York (1991) · Zbl 0744.22001
[11] Gel’fand, I.M., Varchenko, A.N.: Heaviside functions of a configuration of hyperplanes. Funct. Anal. Appl. 21(4), 255–270 (1987); translation from Funkts. Anal. Prilozh. 21(4), 1–18 (1987) · Zbl 0664.14030
[12] Hartshorne, R.: Algebraic Geometry. Grad. Texts Math., vol. 52. Springer, New York Heidelberg (1977) · Zbl 0367.14001
[13] Huber, R.: Étale Cohomology of Rigid Analytic Varieties and Adic spaces. Aspects Math., vol. E 30. Vieweg, Wiesbaden (1996) · Zbl 0868.14010
[14] Jantzen, J.C.: Representations of algebraic groups. Pure Appl. Math., vol. 131. Academic Press, Boston (1987) · Zbl 0654.20039
[15] Jensen, C.U.: Les foncteurs dérivés de \(\varprojlim\) et leurs applications en théorie des modules. Lect. Notes Math., vol. 254. Springer, Berlin New York (1972)
[16] de Jong, J., van der Put, M.: Étale cohomology of rigid analytic spaces. Doc. Math. 1, 1–56 (1996) · Zbl 0922.14012
[17] Kempf, G.: The Grothendieck–Cousin complex of an induced representation. Adv. Math. 29, 310–396 (1978) · Zbl 0393.20027 · doi:10.1016/0001-8708(78)90021-X
[18] Kiehl, R.: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math. 2, 191–214 (1967) · Zbl 0202.20101 · doi:10.1007/BF01425513
[19] Kiehl, R.: Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie. Invent. Math. 2, 256–273 (1967) · Zbl 0202.20201 · doi:10.1007/BF01425404
[20] Lepowsky, J.: Generalized Verma modules, the Cartan–Helgason theorem, and the Harish–Chandra homomorphism. J. Algebra 49, 470–495 (1977) · Zbl 0381.17005 · doi:10.1016/0021-8693(77)90253-8
[21] Morita, Y.: A p-adic theory of hyperfunctions I. Publ. Res. Inst. Math. Sci. 17, 1–24 (1981) · Zbl 0457.12010 · doi:10.2977/prims/1195186702
[22] Morita, Y.: Analytic representations of SL2 over a p-adic number field. II. In: Automorphic Forms of Several Variables (Katata, 1983). Prog. Math., vol. 46, pp. 282–297. Birkhäuser Boston, Boston (1984)
[23] Morita, Y.: p-adic theory of hyperfunctions. II. In: Algebraic Analysis, vol. I, pp. 457–472. Academic Press, Boston (1988)
[24] Mumford, D., Fogarty, J., Kirwan, F.: Geometric invariant theory. Ergeb. Math. Grenzgeb., vol. 34. Springer, Berlin (1994) · Zbl 0797.14004
[25] Orlik, S.: Kohomologie von Periodenbereichen über endlichen Körpern. J. Reine Angew. Math. 528, 201–233 (2000) · Zbl 1056.11039 · doi:10.1515/crll.2000.091
[26] Orlik, S.: On extensions of generalized Steinberg representations. J. Algebra 293(2), 611–630 (2005) · Zbl 1080.22008 · doi:10.1016/j.jalgebra.2005.03.028
[27] Orlik, S.: The cohomology of period domains for reductive groups over local fields. Invent. Math. 162, 523–549 (2005) · Zbl 1093.14065 · doi:10.1007/s00222-005-0452-1
[28] Orlik, S., Rapoport, M.: Period domains over finite and over local fields. Preprint November 2006 · Zbl 1222.14102
[29] Pohlkamp, K.: Randwerte holomorpher Funktionen auf p-adischen symmetrischen Räumen. Diplomarbeit Universität Münster (2004)
[30] Van der Put, M.: Serre duality for rigid analytic spaces. Indag. Math., New Ser. 3, 219–235 (1992) · Zbl 0762.32016 · doi:10.1016/0019-3577(92)90011-9
[31] Quillen, D.: Homotopy Properties of the Poset of Nontrivial p-Subgroups of a Group. Adv. Math. 28, 101–128 (1978) · Zbl 0388.55007 · doi:10.1016/0001-8708(78)90058-0
[32] Rapoport, M., Zink, T.: Period spaces for p-divisible groups. Ann. Math. Stud., vol. 141. Princeton University Press, Princeton (1996) · Zbl 0873.14039
[33] Grothendieck, A.: Cohomologie locale des faisceaux cohérents et théoremes de Lefschetz locaux et globaux (SGA 2). Revised reprint of the 1968 French original. Documents Mathématiques (Paris), 4. Soc. Math. Fr., Paris (2005)
[34] Schneider, P.: The cohomology of local systems on p-adically uniformized varieties. Math. Ann. 293(4), 623–650 (1992) · Zbl 0774.14022 · doi:10.1007/BF01444738
[35] Schneider, P.: Nonarchimedean Functional Analysis. Springer Monographs in Mathematics. Springer, Berlin (2002) · Zbl 0998.46044
[36] Schneider, P., Teitelbaum, J.: p-adic boundary values. Cohomologies p-adiques et applications arithmetiques. Astérisque 278, 51–125 (2002)
[37] Schneider, P., Teitelbaum, J.: Algebras of p-adic distributions and admissible representations. Invent. Math. 153(1), 145–196 (2003) · Zbl 1028.11070 · doi:10.1007/s00222-002-0284-1
[38] Schneider, P., Teitelbaum, J.: Locally analytic distributions and p-adic representation theory, with applications to GL2. J. Am. Math. Soc. 15(2), 443–468 (2002) · Zbl 1028.11071 · doi:10.1090/S0894-0347-01-00377-0
[39] Schneider, P., Stuhler, U.: The cohomology of p-adic symmetric spaces. Invent. Math. 105, 47–122 (1991) · Zbl 0751.14016 · doi:10.1007/BF01232257
[40] Teitelbaum, J.: Values of p-adic L-functions and a p-adic Poisson kernel. Invent. Math. 101(2), 395–410 (1990) · Zbl 0731.11065 · doi:10.1007/BF01231508
[41] Teitelbaum, J.: Modular representations of PGL2 and automorphic forms for Shimura curves. Invent. Math. 113(3), 561–580 (1993) · Zbl 0806.11027 · doi:10.1007/BF01244318
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