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Representation Theory
Representation Theory
ISSN 1088-4165

 

Rogawski's conjecture on the Jantzen
filtration for the degenerate affine
Hecke algebra of type $A$


Author: Takeshi Suzuki
Journal: Represent. Theory 2 (1998), 393-409
MSC (1991): Primary 22E50; Secondary 17B10
Published electronically: October 26, 1998
MathSciNet review: 1651408
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Abstract: The functors constructed by Arakawa and the author relate the representation theory of ${\mathfrak{gl}}_n$ and that of the degenerate affine Hecke algebra $H_{\ell}$ of $\mathrm{GL}_{\ell}$. They transform the Verma modules over ${\mathfrak{gl}}_n$ to the standard modules over $H_{\ell}$. In this paper we prove that they transform the simple modules to the simple modules (in more general situations than in the previous paper). We also prove that they transform the Jantzen filtration on the Verma modules to that on the standard modules. We obtain the following results for the representations of $H_{\ell}$ by translating the corresponding results for ${\mathfrak{gl}}_n$ through the functors: (i) the (generalized) Bernstein-Gelfand-Gelfand resolution for a certain class of simple modules, (ii) the multiplicity formula for the composition series of the standard modules, and (iii) its refinement concerning the Jantzen filtration on the standard modules, which was conjectured by Rogawski.


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Additional Information

Takeshi Suzuki
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Japan
Email: takeshi@kurims.kyoto-u.ac.jp

DOI: http://dx.doi.org/10.1090/S1088-4165-98-00043-0
PII: S 1088-4165(98)00043-0
Received by editor(s): January 23, 1998
Received by editor(s) in revised form: August 31, 1998
Published electronically: October 26, 1998
Additional Notes: The author is supported by the JSPS Research Fellowships for Young Scientists.
Article copyright: © Copyright 1998 American Mathematical Society