notes:2022-11-08-cherednik-on-hecke
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| notes:2022-11-08-cherednik-on-hecke [2022/11/08 16:07] – 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 ====== | ====== 2022-11-08, Cherednik on Hecke ====== | ||
| - | ==== Section | + | ===== 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). | 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? | - 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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| Most of the remaining discussion is too high for me now. Skip it. | Most of the remaining discussion is too high for me now. Skip it. | ||
| - | ==== 2: affine KZ ==== | + | ===== 2: affine KZ ===== |
| It is a first order differential equation, taking value in an infinite dimensional algebra called ** degenerate affine hecke algebra ** | It is a first order differential equation, taking value in an infinite dimensional algebra called ** degenerate affine hecke algebra ** | ||
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| $$ \frac{\d \Phi}{\d u} = \left( k \frac{s}{e^u - 1} + x \right) \Phi(u) $$ | $$ \frac{\d \Phi}{\d u} = \left( k \frac{s}{e^u - 1} + x \right) \Phi(u) $$ | ||
| * 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? | * 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 number | + | * $k$ is a parameter |
| + | * What? The above relation among $s,x,k$ generate the $H_{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}' | ||
| + | {{: | ||
| + | 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 $\Omega_{ij}$? | ||
| + | |||
| + | So, indeed | ||
notes/2022-11-08-cherednik-on-hecke.1667923661.txt.gz · Last modified: 2023/06/25 15:53 (external edit)