Geodesic
On certain eigenvectors in discrete representations of Chevalley groups
Sbornik. Mathematics, Tome 15 (1971) no. 4, pp. 535-548 
M. E. Novodvorskii. On certain eigenvectors in discrete representations of Chevalley groups. Sbornik. Mathematics, Tome 15 (1971) no. 4, pp. 535-548. http://geodesic.mathdoc.fr/item/SM_1971_15_4_a2/
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     title = {On certain eigenvectors in discrete representations of {Chevalley} groups},
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     url = {http://geodesic.mathdoc.fr/item/SM_1971_15_4_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider Chevalley groups over disconnected locally compact fields and subgroups of them which contain their derived groups; we prove that any representation of such a group $\widetilde G$ which is nontrivial on the derived group contains an infinite-dimensional subspace of $\nu$-eigenvectors of the subgroup $\widetilde B_\mathfrak O$, where $\widetilde B_\mathfrak O$ is the intersection of the group $\widetilde G$ with the group of integral points of a Borel subgroup, and $\nu$ is an arbitrary character of it. In passing we prove that any open subgroup of the derived group is compact and is contained in only a finite number of its subgroups. Bibliography: 7 titles.

[1] H. Jacquet, R. P. Langlands, Automorphic forms on $GL(2)$, Springer Verlag, 1970 | MR | Zbl

[2] V. M. Glushkov, “Stroenie lokalno bikompaktnykh grupp i pyataya problema Gilberta”, Uspekhi matem. nauk, XII:2(74) (1957), 3–41

[3] N. Juahori, H. Matsumoto, On some Bruhat decompositions, Math., no. 25, Paris, 1965

[4] K. Shevalle, “O nekotorykh prostykh gruppakh”, Matematika, 2:1 (1958), 3–54 | MR

[5] E. B. Dynkin, “Maksimalnye podgruppy klassicheskikh grupp”, Trudy Mosk. matem. ob-va, I (1952), 39–166 | MR

[6] M. E. Novodvorskii, I. I. Pyatetskii-Shapiro, “Beskonechnomernye abelevy mnogoobraziya i unitarnye predstavleniya grupp”, Matem. sb., 77(119) (1968), 3–20 | MR | Zbl

[7] M. E. Novodvorskii, “O nekotorykh nepodvizhnykh vektorakh v beskonechnomernykh predstavleniyakh grupp Shevalle”, Funkts. analiz, 5:1 (1971), 87–88 | MR | Zbl