Self-contragredient supercuspidal representations of 𝐺𝐿_{𝑛}

Adler, Jeffrey

doi:10.1090/S0002-9939-97-03786-6

Self-contragredient supercuspidal representations of $\mathrm {GL}_n$
HTML articles powered by AMS MathViewer

by Jeffrey D. Adler PDF

Proc. Amer. Math. Soc. 125 (1997), 2471-2479 Request permission

Abstract:

Let $F$ be a non-archimedean local field of residual characteristic $p$. Then $\mathrm {GL}_n(F)$ has tamely ramified self-contragredient supercuspidal representations if and only if $n$ or $p$ is even. When such representations exist, they do so in abundance.

References

Colin J. Bushnell and Philip C. Kutzko, The admissible dual of $\textrm {GL}(N)$ via compact open subgroups, Annals of Mathematics Studies, vol. 129, Princeton University Press, Princeton, NJ, 1993. MR 1204652, DOI 10.1515/9781400882496
I. M. Gel′fand and M. I. Graev, Representations of the group of second-order matrices with elements in a locally compact field and special functions on locally compact fields, Uspehi Mat. Nauk 18 (1963), no. 4 (112), 29–99 (Russian). MR 0155931
I. M. Gel′fand and D. A. Kajdan, Representations of the group $\textrm {GL}(n,K)$ where $K$ is a local field, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 95–118. MR 0404534
Roger E. Howe, Tamely ramified supercuspidal representations of $\textrm {Gl}_{n}$, Pacific J. Math. 73 (1977), no. 2, 437–460. MR 492087
George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
L. Morris, Level zero ${G}$-types, preprint.
Allen Moy, Local constants and the tame Langlands correspondence, Amer. J. Math. 108 (1986), no. 4, 863–930. MR 853218, DOI 10.2307/2374518
Allen Moy and Gopal Prasad, Unrefined minimal $K$-types for $p$-adic groups, Invent. Math. 116 (1994), no. 1-3, 393–408. MR 1253198, DOI 10.1007/BF01231566
—, Jacquet functors and unrefined minimal ${K}$-types, Comm. Math. Helv., to appear.
Paul J. Sally Jr., Invariant subspaces and Fourier-Bessel transforms on the ${\mathfrak {P}}$-adic plane, Math. Ann. 174 (1967), 247–264. MR 579179, DOI 10.1007/BF01364271
Jean-Pierre Serre, Corps locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, Hermann, Paris, 1962 (French). Actualités Sci. Indust., No. 1296. MR 0150130
Freydoon Shahidi, Twisted endoscopy and reducibility of induced representations for $p$-adic groups, Duke Math. J. 66 (1992), no. 1, 1–41. MR 1159430, DOI 10.1215/S0012-7094-92-06601-4