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).

1. 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?
2. Why we study characters of any Lie algebra? Well, you may want to spectral decompose a huge thing, and character formula is just a way to decompose. A little mind trying to understand a huge-connected thing as many unrelated things (me?)
3. Tensor multiplicities, now out of fashion since we are dealing with infinite dimensional gadgets now.
4. $[M_\mu, L_\lambda]$, the Kazhdan-Lusztig polynomial. (why it is not counted as 'real' math in the beginning?)

Then, we have the updates.

1. Hypergeometric functions. hmm, why do we care such functions? just a differential equation solution, very generic type, and with many many parameters.
2. rational and elliptic KZ equation?
3. Verlinde algebra? Fusion not tensor? aha, I see. previously we have simple module tensor and decompose, now we have 'primary field' in a vertex algebra, fuse and decompose.
4. 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.

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 number $s,x$ are linear operators, such that $$ s^2=1, xs + sx = k. $$ Why we have thse?