### AFFINE HECKE ALGEBRAS AND CANONICAL BASES FOR QUANTUM GROUPS

The grant was used to support two visits of Prof Jean-Yves Thibon
(Université de Marne-la-Vallée, France) to the Univestity of York,
in September 1999 and in May 2000.
The aim of the project was to develop further the representation theory
of the Hecke algebra *H*_{n} of the group
*GL*_{n} over a *p*-adic field,
relative to the Iwahori subgroup. We found an explicit irreducibility
criterion for a wide class of induced representations
of the algebra *H*_{n} . Representations
of this class are parametrised by sequences of pairs consisting
of a Young diagram and a complex number, the total number of boxes
in these diagrams being *n* . In the particular case when each of
these Young diagrams consists of one row only, such a criterion was
given by Zelevinsky in 1980. He used the representation theory of the
*p*-adic group *GL*_{n} itself.
We employed the canonical basis, due to Kashiwara and Lusztig,
in the quantum coordinate ring of the complex group of triangular unipotent
matrices. Our results are available on the e-print archive
*Mathematics*, see

**EPSRC Grant and Duration:**
£ 2380 over 12 months from 13.9.1997,
reference GR/M67155

**For further information please contact:**
Dr ML Nazarov,
Department of Mathematics,
University of York,
York YO10 5DD, England;
`maxim.nazarov@york.ac.uk`