## On intertwining operators for $$GL_ N(F)$$, F a nonarchimedean local field.(English)Zbl 0665.22006

Let F be a nonarchimedean local field of residual characteristic p. As a first step toward a construction of the set of irreducible supercuspidal representations of $$GL_ N(F)$$ when $$p| N$$, the authors generalize a theorem of Howe to the wildly ramified case. More precisely let E be a finite-dimensional extension of F, let V be a finite-dimensional vector space over E and let $$\alpha$$ be a primitive element of E/F with “good arithmetic properties”. The main result asserts, roughly speaking, that the g in $$GL_ F(V)$$ such that $$g\alpha g^{-1}=\alpha$$ up to “small terms” lie in $$GL_ E(V)$$ up to equally small terms. The paper is rather difficult to read because of its technical nature and of its numerous notations.
Reviewer: G.Christol

### MSC:

 22E50 Representations of Lie and linear algebraic groups over local fields 11S15 Ramification and extension theory
Full Text:

### References:

 [1] C. Bushnell, Hereditary orders, Gauss sums and supercuspidal representations of $$\mathrm GL_N$$ , J. Reine Angew. Math. 375/376 (1987), 184-210. · Zbl 0601.12025 [2] C. Bushnell and A. Fröhlich, Non-abelian congruence Gauss sums and $$p$$-adic simple algebras , Proc. London Math. Soc. (3) 50 (1985), no. 2, 207-264. · Zbl 0558.12007 [3] H. Carayol, Representations cuspidales du groupe lineaire , Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 2, 191-225. · Zbl 0549.22009 [4] R. Howe, Tamely ramified supercuspidal representations of $$\mathrm Gl_n$$ , Pacific J. Math. 73 (1977), no. 2, 437-460. · Zbl 0404.22019 [5] R. Howe and A. Moy, Harish-Chandra homomorphisms for $$p$$-adic groups , CBMS Regional Conference Series in Mathematics, vol. 59, Amer. Math. Soc., Providence, 1985. · Zbl 0593.22014 [6] G. Hardy and H. Wright, The Theory of Numbers , Oxford University Press, London, 1938. · Zbl 0020.29201 [7] N. Iwahori, Generalized Tits system (Bruhat decompostition) on $$p$$-adic semisimple groups , Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Proc. Symp. Pure Math., vol. 9, Amer. Math. Soc., Providence, R.I., 1966, pp. 71-83. · Zbl 0199.06901 [8] P. Kutzko, On the supercuspidal representations of $$\mathrm GL_ N$$ and other $$p$$-adic groups , Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 853-861. · Zbl 0675.22009 [9] P. Kutzko, Towards a classification of the supercuspidal representations of $$\mathrm GL_N$$ , London J. Math., · Zbl 0678.22009 [10] A. Moy, Local constants and the tame Langlands correspondence , Amer. J. Math. 108 (1986), no. 4, 863-930. JSTOR: · Zbl 0597.12019 [11] J. P. Serre, Local Fields , Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York, 1979. · Zbl 0423.12016 [12] J. L. Waldspurger, Algèbres de Hecke et induites de représentations cuspidales, pour $$\mathrm GL(N)$$ , J. Reine Angew. Math. 370 (1986), 127-191. · Zbl 0586.20020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.