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--- notes:2022-11-08-cherednik-on-hecke [2022/11/08 15:52] – created pzhou
+++ notes:2022-11-08-cherednik-on-hecke [2023/06/25 15:53] (current) – external edit 127.0.0.1
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 ====== 2022-11-08, Cherednik on Hecke ======
-==== Section 1: Hecke algebra in rep theory ====
+===== 1: Hecke algebra in rep theory =====
 very funny comment about real math and imaginary math, one can see he is an concrete guy with feet on ground (i.e. real axis).
   - What is a zonal spherical function? From wiki, it is a //function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G.// But why we study it?
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   - Now, I am really confused and intrigued. These simple-verma coefficients are related to modular representation.
+Most of the remaining discussion is too high for me now. Skip it.
+===== 2: affine KZ =====
+It is a first order differential equation, taking value in an infinite dimensional algebra called ** degenerate affine hecke algebra **
+Here is the equation
+$$ \frac{\d \Phi}{\d u} = \left( k \frac{s}{e^u - 1} + x \right) \Phi(u) $$
+  * There is $e^u$, so the solution is like 1-periodic, or if we use $U = e^u$, then it lives on $\C^*$. But we prefer this way, since $\Phi(U)$ will be multivalued. Wait, around $u=0$, will it also have interesting monodromy?
+  * $k$ is a parameter $s,x$ are linear operators, such that $$ s^2=1, xs + sx = k. $$ Why we have thse?
+  * What?  The above relation among $s,x,k$ generate the $H_{A_1}'$, degenerate affine Hecke of type $A_1$.
+  * Now, if we change variable, let $z = e^{-u}$ (as one should), and repackage the equation, we see poles at $z=0, \infty, 1$. But still, this is a complicated equation. However, if we don't solve for the most general curve in the Hecke algebra (indeed, we are looking at a pair-of-pants in the Hecke algebra parameterized by $k$), but rather we take representation of $H_{A_1}'$, then it become much manageable.
+{{:notes:pasted:20221108-082040.png}}
+OK, too lazy to copy down the formula. A few comments
+  * $s$ should be a permutation matrix, but here we diagonlized it. Not sure what $x$ should be.
+  * When you solve the equation for $\Phi(u)$, or $\Phi(z)$ for $z = e^{-u}$, the first component $\Phi_1(z)$ will be the hypergeometric equation. There are the Poch-hammer symbol $(x)_n$, and series summation. (We should now the series only converge for some range $z$, no?)
+The AKZ equation for $GL_n$. OK, we have
+$$ A_i = \sum_{j \neq i} \frac{\Omega_{ij}}{z_i - z_j} $$
+where $z_i \in \C^*$. So affine means working with $z_i \in \C^*$? why such a big deal?
+One need to make them satisfy
+$$ \d_j A_i - \d_i A_j - [A_i, A_j] = 0 $$
+Nothing but the flat connection property.
+But, then, what is $\Omega_{ij}$? Is it independent of $z_i$? I guess so, they need to satisfy useful commutation relations.
+So, indeed