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Let W be an extended affine Weyl group, H be the affine Hecke algebra of the corresponding affine Weyl group over the ring C[q, q −1 ], and J be Lusztig's asymptotic Hecke algebra, viewed as a based ring with basis {tw}. Viewing J as a subalgebra of the (q −1 )-adic completion of H via Lusztig's map φ, we use Harish-Chandra's Plancherel formula for p-adic groups to show that the coefficient of Tx in tw is a rational function of q, depending only on the two-sided cell containing w, with no poles outside of a finite set of roots of unity that depends only on W . In type Ãn and type C2, we show that the denominators all divide a power of the Poincaré polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information bout the failure of the Kazhdan-Lusztig classification at roots of the Poincaré polynomial than is currently known.

Let G G be a split connected reductive algebraic group, let H H be the corresponding affine Hecke algebra, and let J J be the corresponding asymptotic Hecke algebra in the sense of Lusztig. When G = S L 2 G=\mathrm {SL}_2 , and the parameter q q is specialized to a prime power, Braverman and Kazhdan showed recently that for generic values of q q , H H has codimension two as a subalgebra of J J , and described a basis for the quotient in spectral terms. In this note we write these functions explicitly in terms of the basis { t w } \{t_w\} of J J , and further invert the canonical isomorphism between the completions of H H and J J , obtaining explicit formulas for each basis element t w t_w in terms of the basis { T w } \{T_w\} of H H . We conjecture some properties of this expansion for more general groups. We conclude by using our formulas to prove that J J acts on the Schwartz space of the basic affine space of S L 2 \mathrm {SL}_2 , and produce some formulas for this action.

According to a conjecture of Lusztig, the asymptotic affine Hecke algebra should admit a description in terms of the Grothedieck group of sheaves on the square of a finite set equivariant under the action of the centralizer of a nilpotent element in the reductive group. A weaker form of this statement, allowing for possible central extensions of stabilizers of that action, has been proved by the first named author with Ostrik. In the present paper we describe an example showing that nontrivial central extensions do arise, thus the above weaker statement is optimal. We also discuss the relation of these central extensions to the cocenter of the affine Hecke algebra and the asymptotic affine Hecke algebra.

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