## Isogenies of Hecke algebras and a Langlands correspondence for ramified principal series representations

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- by Mark Reeder
- Represent. Theory
**6**(2002), 101-126 - DOI: https://doi.org/10.1090/S1088-4165-02-00167-X
- Published electronically: July 16, 2002
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## Abstract:

This paper gives a Langlands classification of constituents of ramified principal series representations for split $p$-adic groups with connected center.## References

- Dean Alvis,
*Ratios of dual generic degrees of a finite Coxeter group*, Proc. Amer. Math. Soc.**91**(1984), no. 4, 532–536. MR**746083**, DOI 10.1090/S0002-9939-1984-0746083-1 - Armand Borel,
*Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup*, Invent. Math.**35**(1976), 233–259. MR**444849**, DOI 10.1007/BF01390139 - Armand Borel,
*Linear algebraic groups*, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR**1102012**, DOI 10.1007/978-1-4612-0941-6
[BGG]BGG I.N. Bernstein, I.M. Gel′fand, I.S. Gel′fand, - Colin J. Bushnell and Philip C. Kutzko,
*Smooth representations of reductive $p$-adic groups: structure theory via types*, Proc. London Math. Soc. (3)**77**(1998), no. 3, 582–634. MR**1643417**, DOI 10.1112/S0024611598000574 - Roger W. Carter,
*Finite groups of Lie type*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR**794307** - Neil Chriss and Victor Ginzburg,
*Representation theory and complex geometry*, Birkhäuser Boston, Inc., Boston, MA, 1997. MR**1433132** - C. De Concini, G. Lusztig, and C. Procesi,
*Homology of the zero-set of a nilpotent vector field on a flag manifold*, J. Amer. Math. Soc.**1**(1988), no. 1, 15–34. MR**924700**, DOI 10.1090/S0894-0347-1988-0924700-2 - N. Iwahori and H. Matsumoto,
*On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak {p}}$-adic Chevalley groups*, Inst. Hautes Études Sci. Publ. Math.**25**(1965), 5–48. MR**185016**, DOI 10.1007/BF02684396 - David Kazhdan and George Lusztig,
*Proof of the Deligne-Langlands conjecture for Hecke algebras*, Invent. Math.**87**(1987), no. 1, 153–215. MR**862716**, DOI 10.1007/BF01389157 - Victor Ginzburg,
*Geometrical aspects of representation theory*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 840–848. MR**934285**, DOI 10.1090/S0894-0347-1989-0991016-9
[L2]L2 —, - Lawrence Morris,
*Tamely ramified intertwining algebras*, Invent. Math.**114**(1993), no. 1, 1–54. MR**1235019**, DOI 10.1007/BF01232662 - I. G. Macdonald,
*Polynomial functors and wreath products*, J. Pure Appl. Algebra**18**(1980), no. 2, 173–204. MR**585222**, DOI 10.1016/0022-4049(80)90128-0 - Mark Reeder,
*$p$-adic Whittaker functions and vector bundles on flag manifolds*, Compositio Math.**85**(1993), no. 1, 9–36. MR**1199202** - Mark Reeder,
*$p$-adic Whittaker functions and vector bundles on flag manifolds*, Compositio Math.**85**(1993), no. 1, 9–36. MR**1199202** - Mark Reeder,
*Whittaker functions, prehomogeneous vector spaces and standard representations of $p$-adic groups*, J. Reine Angew. Math.**450**(1994), 83–121. MR**1273956**, DOI 10.1515/crll.1994.450.83 - Mark Reeder,
*Nonstandard intertwining operators and the structure of unramified principal series representations*, Forum Math.**9**(1997), no. 4, 457–516. MR**1457135**, DOI 10.1515/form.1997.9.457 - Alan Roche,
*Types and Hecke algebras for principal series representations of split reductive $p$-adic groups*, Ann. Sci. École Norm. Sup. (4)**31**(1998), no. 3, 361–413 (English, with English and French summaries). MR**1621409**, DOI 10.1016/S0012-9593(98)80139-0 - Franz Rádl,
*Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen*, Math. Z.**45**(1939), 429–446 (German). MR**82**, DOI 10.1007/BF01580293 - François Rodier,
*Whittaker models for admissible representations of reductive $p$-adic split groups*, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 425–430. MR**0354942** - Jean-Pierre Serre,
*Local fields*, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR**554237**, DOI 10.1007/978-1-4757-5673-9 - Freydoon Shahidi,
*A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups*, Ann. of Math. (2)**132**(1990), no. 2, 273–330. MR**1070599**, DOI 10.2307/1971524 - T. A. Springer,
*Reductive groups*, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. MR**546587**, DOI 10.1090/pspum/033.1/546587 - Robert Steinberg,
*Torsion in reductive groups*, Advances in Math.**15**(1975), 63–92. MR**354892**, DOI 10.1016/0001-8708(75)90125-5

*Schubert cells and cohomology of the spaces $G/P$*, Russian Math. Surveys

**28**(1973), 1–26.

*Notes on Affine Hecke Algebras*, preprint, 1999.

## Bibliographic Information

**Mark Reeder**- Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
- Email: reederma@bc.edu
- Received by editor(s): April 30, 2001
- Received by editor(s) in revised form: January 30, 2002
- Published electronically: July 16, 2002
- Additional Notes: This work was partially supported by the National Science Foundation
- © Copyright 2002 American Mathematical Society
- Journal: Represent. Theory
**6**(2002), 101-126 - MSC (2000): Primary 22E50
- DOI: https://doi.org/10.1090/S1088-4165-02-00167-X
- MathSciNet review: 1915088