On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras
- 4 Downloads
Let π be an irreducible smooth complex representation of a general linear p-adic group and let σ be an irreducible complex supercuspidal representation of a classical p-adic group of a given type, so that π ⨁ σ is a representation of a standard Levi subgroup of a p-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate p-adic classical group obtained by (normalized) parabolic induction from π ⨁ σ does not depend on σ, if σ is “separated” from the supercuspidal support of π. (Here, “separated” means that, for each factor ρ of a representation in the supercuspidal support of π, the representation parabolically induced from ρ ⨁ σ is irreducible.) This was conjectured by E. Lapid and M. Tadić. (In addition, they proved, using results of C. Jantzen, that this induced representation is always reducible if the supercuspidal support is not separated.)
More generally, we study, for a given set I of inertial orbits of supercuspidal representations of p-adic general linear groups, the category CI,σ of smooth complex finitely generated representations of classical p-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by σ and I, and show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type AB and D and establish functoriality properties, relating categories with disjoint I’s. In this way, we extend results of C. Jantzen who proved a bijection between irreducible representations corresponding to these categories. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato’s exotic geometry.
Unable to display preview. Download preview PDF.
- [BD]J. Bernstein, Le “centre” de Bernstein, in Representations of Reductive Groups Over a Local Field, Travaux en Cours, Hermann, Paris, 1984, pp. 1–32.Google Scholar
- [D]P. Deligne, Catégories tannakiennes, in The Grothendieck Festschrift Vol. II, Progress in Mathematics, Vol. 87, Birkhäuser, Boston, MA, 1990, pp. 111–195.Google Scholar
- [LT]E. Lapid and M. Tadić, Some results on reducibility of parabolic induction for classical groups, American Journal of Mathematics, to appear, arXiv:1703.09475.Google Scholar
- [Lu3]G. Lusztig, Graded Lie algebras and intersection cohomology, in Representation Theory of Algebraic Groups and Quantum Groups, Progress in Mathematics, Vol. 284, Birkhäuser/Springer, New York, 2010, pp. 191–224.Google Scholar
- [P]N. Popescu, Abelian Categories with Applications to Rings and Modules, London Mathematical Society Monographs, Vol. 3, Academic Press, London–New York, 1973.Google Scholar
- [RR]A. Ram and J. Rammage, Affine Hecke algebras, cyclotomic Hecke algebras, and Clifford theory, in A Tribute to C. S. Seshadri (Chennai, 2002), Trends in Mathematics, Birkhäuser, Basel, 2003, pp. 428–466.Google Scholar
- [Ti]J. Tits, Reductive groups over local fields, in Automorphic Forms, Representations and L-functions (Proceedings of Symposium in Pure Mathematics, Oregon State University, Corvallis, OR, 1977), Part 1, Proceedings of Symposia in Pure Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 1979, pp. 29–69.Google Scholar
- [Ze]A. Zelevinsky, A p-adic analogue of the Kazhdan–Lusztig hypothesis, Funktsional’nyĭ Analiz i ego Prilozheniya 15 (1981), 9–21; English translation: Functional Analysis and its Applications 15 (1981), 83–92.Google Scholar