Classification of the Irreducible Representations of the Affine Hecke algebra of Type B2 with Unequal Parameters

The representation theory of the affine Hecke algebras has two different approaches. One is a geometric approach and the other is a combinatorial one. In the equal parameter case, affine Hecke algebras are constructed using equivariant K-groups, and their irreducible representations are constructed on Borel-Moore homologies. By this method, their irreducible representations are parameterized by the index triples ([CG],[KL]). On the other hand, G. Lusztig classified the irreducible representations in the unequal parameter case. His ideas are to use equivariant cohomologies and graded Hecke algebras ([Lus89],[LusI],[LusII],[LusIII]). Although the geometric approach will give us a powerful method for the classification, but it does not tell us the detailed structure of irreducible representations. Thus it is important to construct them explicitly in combinatorial approach. Using semi-normal representations and the generalized Young tableaux, A. Ram constructed calibrated irreducible representations with equal parameters ([Ram1]). Furthermore C. Kriloff and A. Ram constructed irreducible calibrated representations of graded Hecke algebras ([KR]). However, we cannot always construct irreducible representations by combinatorial manner. A. Ram classified irreducible representations of affine Hecke algebras of type A1, A2, B2, G2 in equal parameter case ([Ram2]). But there are some mistakes in his list of irreducible representations and his construction of induced representation of type B2. In this paper, we will correct his list about type B2 and also classify the irreducible representations in the unequal parameter case. There are three one-parameter families of calibrated irreducible representations and some other irreducible representations. Acknowledgement. I would like to thank Professor M. Kashiwara and Professor S. Ariki for their advices and suggestions, and Mathematica for its power of calculation.