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Research Summary:
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For
affine Weyl groups of type A, proved Lusztig¡¯s conjecture on based
rings of two-sided cells.
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Determined
the sufficient and necessary condition for the validity of
Deligne-Langlands conjecture for affine Hecke algebras.
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Joint
with Lusztig, discovered that each two-sided cell of an affine Weyl
group contains a unique canonical left cell.
-
Joint
with Tanisaki, proved that for affine Hecke algebras of type A, a
certain algebraic filtration is compatible with a certain geometric
filtration.
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For
quantum groups at roots of 1 and algebraic groups over an
algebraically closed field of characteristic p, realized irreducible
representations in an explicit and concrete
way.
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Systematically
studied bases of quantum groups, the works include: joint with
Chari
, constructed monomial bases of quantum groups; computed certain
canonical bases; study on root vectors and their commutation formulas.
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Representative Research Works:
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(Joint
G. Lusztig)
Canonical left cells in affine Weyl groups,
Adv.
in Math. 72 (1988), 284-288.
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Finite
dimensional modules of some quantum groups over Fp(v),
J.
Rein. Angew. Math.
410 (1990), 109-115.
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The based
ring of the lowest two-sided cell of an affine Weyl group, II,
Ann.
Sci. Éc. Norm. Sup. 27 (1994), 47-61.
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Root vectors
in quantum groups,
Comm.
Math. Helv. 69 (1994), 612-639.
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REPRESENTATIONS
OF AFFINE HECKE ALGEBRAS, Lecture Notes in Mathematics 1587,
Springer-Verlag£¬1994£¬Germany.
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Irreducible
modules of quantized enveloping algebras at roots of 1,
Publ.
RIMS.
Kyoto
Univ.
32 (1996), 235-276.
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(Joint
with V. Chari) Monomial bases of quantized enveloping
algebras,
Contemp.
Math. 248, AMS
Providence
,
RI
, 1999, 69-81.
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THE BASED
RING OF TWO-SIDED CELLS OF AFFINE WEYL GROUPS OF Ãn-1, Mem. of
AMS, Vol. 157, No. 749,
American
Mathematical Society, 2002£¬USA
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(Joint
with T. Tanisaki)
Kazhdan-Lusztig basis and a geometric filtration of an affine Hecke
algebra£¬Nagoya
Math. J. 182 (2006), 285-311.
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Representations
of affine Hecke algebras and based rings of affine Weyl groups ,
J.
Amer. Math. Soc. 20 (2007), 211-217.
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