Degenerations of affine Hecke algebras

Question

Consider an affine Hecke algebra $H$ corresponding to some semisimple algebraic group $G$. Let $H_{deg}$ denote the corresponding degenerate affine Hecke algebra. The algebra $H_{deg}$ can be obtained from $H$ through a process of "degeneration". From what I've read, this process involves formally substituting some exponential terms for the generators of the affine Hecke algebra and taking homogeneous components of minimal degree in the relations defining $H$. Can someone provide a bit more detail regarding this process, for instance an explicit calculation of the degeneration of the relation (I have tried but I failed): $$ T_{s_\alpha} e^{s_\alpha(\lambda)} - e^\lambda T_{s_\alpha} = (1-q) \frac{e^\lambda - e^{s_\alpha(\lambda)}}{1 - e^{-\alpha}}.$$ We can also consider the double affine Hecke algebra. It possesses two degenerations: trigonometric and rational. Can someone give more detail saying what "degeneration" means in this context? I have also read that the degenerate algebras can in some sense be regarded as Lie algebras to the non-degenerate algebras. I'd appreciate an explanation of this analogy as well.

Answer 1

The precise procedure is that one defines a filtration on the affine Hecke algebra, and then the degenerate affine Hecke algebra is the associated graded algebra. This is done in Lusztig's paper "Affine Hecke algebras and their graded version". For a more informal way of getting the degeneration, replace $e$ with $q$ everywhere. Then, dividing both sides of the relation by $1-q$ (and rearranging a bit) gives $$\frac{q^\lambda - 1}{q-1} T_{s_\alpha} - T_{s_\alpha} \frac{q^{s_\alpha(\lambda)}-1}{q-1} = q^\lambda \frac{1-q^{-\langle \alpha^\vee, \lambda \rangle \alpha}}{1-q^{-\alpha}}.$$ (Here we use the fact that $s_\alpha(\lambda) = \lambda - \langle \alpha^\vee, \lambda \rangle \alpha$.) Now taking the limit $q \to 1$ yields $$\lambda T_{s_\alpha} - T_{s_\alpha} s_\alpha(\lambda) = \langle \alpha^\vee, \lambda \rangle,$$ which is the relation for the degenerate affine Hecke algebra.