LAUSR

For type B̃3 we show that Lusztig’s conjecture on the structure of the based ring of two-sided cell corresponding to the unipotent class in Sp6(C) with 3 equal Jordan blocks needs modified. We are concerned with the based ring of two-sided cells in an affine Weyl group of type B̃3. In this paper we show Lusztig’s conjecture on the structure of the based ring of the two-sided cell corresponding to the nilpotent element in Sp6(C) with 3 equal Jordan blocks needs modified, see section 4. This work was motivated by an inquiry of R. Bezrukanikov, who noted (based on the work of Losev and Panin) that Lusztig’s conjecture describing the summand of J in terms of equivariant sheaves on the square of a finite set does not hold as stated for the 2-sided cell corresponding to the unipotent class in Sp6(C) with 3 equal Jordan blocks and was interested in explicit description of the summand in the asymptotic Hecke algebra J attached to the affine Weyl group of type B̃3. In a subsequent paper, we will discuss based ring of other two-sided cells in the affine Weyl group of type B̃3. The contents of the paper are as follows. Section 1 is devoted to preliminaries, which include some basic facts on (extended) affine Weyl groups and their Hecke algebras and formulation of Lusztig’s conjecture on the structure of based ring of a two-sided cell in an affine Weyl group. In section 2 we recall some results on cells of (extended) affine Weyl group of type B̃3, which are due to J. Du. In section 3 we discuss the based ring JΓ∩Γ−1 for a left cell in the concerned two-sided cell. In section 4 we show that Lusztig’s conjecture on the structure of the based ring of the concerned two-sided cell needs modified. After the paper is complete, we learned from Bezrukavnikov that Stefan Dawydiak also worked out this example. His computation also leads to the same conclusion. N. Xi was partially supported by National Key R&D Program of China, No. 2020YFA0712600, and by National Natural Science Foundation of China, No. 11688101. 1