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In this paper we compute the leading coefficients $\mu (u,w)$ of theKazhdan--Lusztig polynomials $P_{u,w}$ for an affine Weyl group of type$\tilde{B}_2$. By using the \textbf{a}-function of a Coxeter group defined byLusztig (see [L1, \S2]), we compute most $\mu (u,w)$ explicitly. With part ofthese values $\mu (u,w)$, we show that a conjecture of Lusztig on distinguishedinvolutions is true for an affine Weyl group of type $\tilde{B}_2$. We alsoshow that the conjectural formula in [L3, (12)] needs a modification.

Leading coefficients of the Kazhdan-Lusztig polynomials for an Affine Weyl group of type $\widetilde{B_2}$