Unrefined minimal \(K\)-types for \(p\)-adic groups. (English) Zbl 0804.22008

Moy, Allen; Prasad, Gopal

Invent. Math. 116, No. 1-3, 393-408 (1994).

Let \(G\) be a reductive group over a \(p\)-adic field of characteristic zero. It is widely believed that a classification of the admissible dual of \(G\) may be achieved via studying the restriction of an admissible representation of \(G\) to certain compact open subgroups. To this end the first author gave [in Contemp. Math. 86, 249-254 (1989; Zbl 0685.22009)] the definition of a minimal \(K\)-type as a certain type of representation of parahoric filtration subgroups. Contrary to the real case the definition of a minimal \(K\)-type is intrinsic in that it does not involve a given admissible representation of \(G\). To use it for classification arguments one thus must prove that every admissible representation of \(G\) contains a minimal \(K\)-type. In the paper under consideration this is established.
The author uses an isomorphism of the graded parts of the parahoric filtration to the graded parts of a corresponding Lie algebra filtration which allows him to switch over to the Lie algebra. Via the minimal \(K\)- types one now can attach to a given admissible representation \(\pi\) a rational number \(\rho(\pi)\), the depth of the minimal \(K\)-type, which should play a key rôle in the classification of the admissible dual similar to the real case.

Reviewer: A.Deitmar (Heidelberg)

MSC:

22E50	Representations of Lie and linear algebraic groups over local fields
22E35	Analysis on \(p\)-adic Lie groups
17B70	Graded Lie (super)algebras
20G25	Linear algebraic groups over local fields and their integers
22E60	Lie algebras of Lie groups

Keywords:

reductive group; \(p\)-adic field; admissible dual; admissible representation; subgroups; minimal \(K\)-type; Lie algebra filtration; depth

Citations:

Zbl 0685.22009

References:

[1]	Borel, A.: Linear Algebraic Groups. Grad. Text Math. vol. 126, New York: Springer, 1991 · Zbl 0726.20030
[2]	Borel, A., Tits, J.: Groupes réductifs. Publ. Math. I. H. E. S.27, 55-150 (1965) · Zbl 0145.17402
[3]	Bruhat, F., Tits, J.: Groupes réductifs sur un corps local I. Publ. Math. I. H. E. S.41 (1972)
[4]	Bruhat, F., Tits, J.: Groupes réductifs sur un corps local II. Publ. Math. I. H. E. S.60 (1984) · Zbl 0565.14028
[5]	Bushnell, C.: Hereditary orders. Gauss sums and supercuspidal representations of GL n . J. Reine Angew. Math.375-376, 184-210 (1987) · Zbl 0601.12025 · doi:10.1515/crll.1987.375-376.184
[6]	Demazure, M., Gabriel, P.: Groupes Algébriques, Tome I. Amsterdam: North Holland, 1970 · Zbl 0203.23401
[7]	Harish-Chandra, Collected Papers, New York: Springer Verlag, 1984 · Zbl 1325.01031
[8]	Howe, R.: Some qualitative results on the representation theory of GL n over a p-adic field. Pac. J. Math.73, 479-538 (1977) · Zbl 0385.22009
[9]	Howe, R., Moy, A.: Minimal K-types for GL n over a p-adic field. Astérisque171-172, 257-271 (1989) · Zbl 0715.22018
[10]	Kempf, G.: Instability in invariant theory. Ann. Math.108, 299-316 (1978) · Zbl 0406.14031 · doi:10.2307/1971168
[11]	Morris, L.: Fundamental G-strata for classical groups. Duke Math. J.64, 501-553 (1991) · Zbl 0799.22009 · doi:10.1215/S0012-7094-91-06426-4
[12]	Moy, A.: A conjecture on minimal K-types for GL n over a p-adic field. Representation Theory and Number Theory in Connection with the Local Langlands Conjecture. Proceedings of a Conference held December 8-14, 1985. Contemp. Math.86, 249-254 (1989)
[13]	Murty, K. G.: Linear Programming. New York: John Wiley, 1983 · Zbl 0521.90071
[14]	Prasad, G., Raghunathan, M. S.: Topological central extensions of semi-simple groups over local fields. Ann. Math.119, 143-268 (1984) · Zbl 0552.20025 · doi:10.2307/2006967
[15]	Tits, J.: Reductive groups over Local Fields. Proceedings of A. M. S. Symposia in Pure Math.33 (Part 1), 29-69 (1979) · Zbl 0415.20035
[16]	Vogan, D.: The algebraic structure of the representations of semisimple Lie groups. Ann. Math.109, 1-60 (1979) · Zbl 0424.22010 · doi:10.2307/1971266