In affine Hecke categories, high tensor powers of a fixed object have a number of indecomposable summands of order n^{-|Phi^+|/2} times an exponential; proved in type A1, and for longest elements in type A2, with coarse bounds in all affine types.
On the Kazhdan-Lusztig basis of a spherical Hecke algebra
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abstract
Lusztig proved that the Kazhdan-Lusztig basis of a spherical Hecke algebra can be essentially identified with the Weyl characters of the Langlands dual group. We generalize this result to the unequal parameter case. The new proof is pretty simple and quite different from Lusztig's.
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Growth in affine Hecke categories
In affine Hecke categories, high tensor powers of a fixed object have a number of indecomposable summands of order n^{-|Phi^+|/2} times an exponential; proved in type A1, and for longest elements in type A2, with coarse bounds in all affine types.