Last update: 28 April 2014

This is an excerpt of the paper *Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory* by Ivan Cherednik, in collaboration with Etsuro Date, Kenji Iohara, Michio Jimbo, Masaki Kashiwara, Tetsuji Miwa, Masatoshi Noumi, and Yoshihisa Saito.

Before a systematic exposition, I will try to outline the main directions of the representation theory and harmonic analysis connected with the Macdonald theory.

A couple of remarks about the growth of Mathematics. It can be illustrated (with all buts and ifs) by the following diagram. $$\begin{array}{c}\begin{array}{c}\n\n\n\n\nImaginary axis (conceptual mathematics)\nReal axis (special functions, numbers)\n\n\end{array}\\ \text{Figure. 1. Real and Imaginary}\end{array}$$

It is extremely fast in the imaginary (conceptual) direction but very slow in the real direction. Mainly I mean modern mathematics, but it may be more general. For
instance, ancient Greeks created a highly conceptual axiomatic geometry with a modest 'real output'. I do not think that the ratio *Real*/*Imaginary*
is much higher now. There are many theories and a very limited number of functions which are really special. Let us try to project the representation theory on the
real axis (Fig.1). We focus on Lie groups (algebras) and Kac-Moody algebras, ignoring the arithmetic direction (adàles and automorphic forms). Look at Fig.2.

1): | By this I mean the zonal spherical functions on $K\backslash G/K$ for maximal compact $K$ in a semi-simple Lie group $G\text{.}$ The theory was started by Gelfand et al. in the early 50's and completed by Harish-Chandra and many others. It generalized quite a few classical special functions. Lie groups helped a lot to elaborate a systematic approach, although much can be done without them, as we will see below. |

2): | The characters of Kac-Moody algebras can also be introduced without any representation theory (Looijenga, Saito). They are not too far from the products of classical one-dimensional $\theta \text{-functions.}$ However it is a new and very important class of special functions with various applications. The representation theory explains well some of their properties (but not all). |

3): | This construction gives a lot of remarkable combinatorial formulas, and generating functions. Decomposing tensor products of finite dimensional representations of compact Lie groups was in the focus of representation theory in the 70's and early 80's, as well as various restriction problems. This direction is still very important, but the representation theory moved towards infinite-dimensional objects. $$\begin{array}{c}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIm\nRe\n\n1Spherical functions\n2Characters of KM\nalgebras\n3[V\lambda \u2297V\mu :V\nu ]\n(irreps of dim <\u221e)\n4[M\lambda :L\mu ]\n(induced: irreps)\n\n\nRepresentation theory of Lie groups, Lie algebras, and Kac-Moody algebras\n\n\end{array}\\ \text{Figure. 2. Representation Theory}\end{array}$$ |

4): | Here the problem is to calculate the multiplicities of irreducible representations of Lie algebras in the Verma modules or other induced representations. It is complicated. It took time to realize that these multiplicities are 'real'. |

Let us update the picture adding the results which were obtained in the 80's and 90's. $$\begin{array}{c}\begin{array}{c}\n\n\n\nRepresentation theory\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIm\nRe\n\n1Spherical fns\n2KM characters\n3[V\lambda \u2297V\mu :V\lambda ]\n4[M\lambda :L\mu ]\n\n\n\n\n\n\n\n\n\n\n\n1\u223cGeneralized hypergeom. functions\n2\u223cConformal blocks\n3\u223cVerlinde algebras\n4\u223cModular reps\n\n\n\n\n\n\n\end{array}\\ \text{Figure. 3. New Vintage}\end{array}$$

$\stackrel{\sim}{1}\text{):}$ | These functions will be the subject of my mini-course. We will study them in the differential and difference cases. It was an old question of how to introduce and generalize them using the representation theory. Now we have an answer. |

$\stackrel{\sim}{2}\text{):}$ | Actually conformal blocks belong to the imaginary axis (conceptual mathematics). Only some of them can be considered as 'real' functions. Mostly it happens in the case of KZ-Bernard equation (a sort of elliptic KZ). |

$\stackrel{\sim}{3}\text{):}$ | By Verlinde algebras, we mean the category of integrable representations of Kac-Moody algebras of given level with the fusion instead of tensoring. They can be also defined using quantum groups at roots of unity (Kazhdan-Lusztig). |

$\stackrel{\sim}{4}\text{):}$ | Whatever you think about the 'reality' of $[{M}_{\lambda}:{L}_{\mu}],$ these multiplicities are connected with modular representations including the representations of the symmetric group over fields of finite characteristic. Nothing can be more real. |

**Conjecture.** *The real projection of the representation theory goes through Hecke-type algebras.*

As to the examples under discussion the picture is as follows: $$\begin{array}{c}\begin{array}{c}\n\n\n\n\n\n\n\nRepresentation theory\n\n\n\n\n\nRepresentation theory of Hecke algebras\n\n\nMacdonald theory, double Hecke algebras\n\n\nKazhdan-Lusztig polynomials\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIm\nRe\n\n1Spherical fns\n2KM characters\n3[V\lambda \u2297V\mu :V\lambda ]\n4[M\lambda :L\mu ]\n\n\n\n\n\n\n\n\n\n\n\n\n1\u223cHypergeom. functions\n2\u223cConformal blocks\n3\u223cVerlinde algebras\n4\u223cModular reps\n\n\n\n\n\n\n\na\na\u223c\nb\u223c\n?\nb\u223c\n?\n!\nc\u223c\nc\n?!\nd\u223c\nd\n\n\end{array}\\ \text{Figure. 4. Hecke Algebras}\end{array}$$

a): | This arrow seems the most recognized now. Several questions in the Harish-Chandra theory (the zonal case) were covered by the representation theory of the degenerate (graded) affine Hecke algebras defined by Lusztig [Lus1989]. For instance, the operators from [Che1991-2, Che1994-3] give a very simple approach to the radial parts of Laplace operators on symmetric spaces and the Harish-Chandra isomorphism. The hypergeometric functions (the arrow $\text{(}\stackrel{\sim}{\text{a}}\text{))}$ appear naturally in this way. Here the main expectations are connected with the difference theory. It was demonstrated in [Che1995-2] that the difference Fourier transform is self-dual (it is not in the differential case). At least it holds for certain classes of functions. It must simplify and generalize the Harish-Chandra theory. The same program was started in the $p\text{-adic}$ representation theory (see [Che1995-3, Che1996]). The coincidence of some difference spherical functions with proper Macdonald polynomials can be established using quantum groups (Noumi and others- see[Nou1992]). However at the moment the Hecke algebra technique is more efficient to deal with these polynomials (especially for arbitrary root systems). |

b): | The double Hecke algebras lead to a certain elliptic generalization of the Macdonald polynomials [Che1995-4, Che1995-5, Che1996]. In the differential case there is also the so-called parabolic operator (see [EKi1993] and [Che1995-4]). Still it is not what one could expect. As to $\text{(}\stackrel{\sim}{\text{b}}\text{),}$ the conformal blocks of type ${GL}_{n}$ (i.e. over the products of curves with the action of the symmetric group) are much more general than the characters. Obviously Hecke algebras are not enough to get all of them. On the other hand, there is almost no theory of the conformal blocks for the configuration spaces connected with other root systems. Double affine Hecke algebras work well for all root systems. |

c): |
Here one can rediscover the same combinatorial formulas (mostly based on the so-called Kostant partition function). I do not expect anything brand new.
However if you switch to the spherical functions (instead of the characters) then the new theory results in the formulas for the products of spherical functions,
which cannot be obtained in the classical theory (they require the difference setting). The multiplicities
$[{V}_{\lambda}\otimes {V}_{\mu}:{V}_{\nu}]$
govern the products of the characters, which are the same in the differential and difference theory. Concerning $\text{(}\stackrel{\sim}{\text{c}}\text{),}$ the Macdonald theory at roots of unity gave a simple approach to the Verlinde algebras. All the results about the inner product and the action of ${SL}_{2}\left(\mathbb{Z}\right)$ were generalized a lot. I mean [Kir1995], and my two papers [Che1995-2, Che1995-3]. A. Kirillov Jr. was the first to find a one-parametric deformation of the Verlinde algebra in the case of ${GL}_{n}\text{.}$ He used quantum groups at roots of unity. My technique is applicable to all root systems. The proofs are much simpler than those based on Kac-Moody algebras or quantum groups. It works even better for the non-symmetric Macdonald polynomials (the conformal blocks and Kac-Moody characters are symmetric in contrast to the main classical elliptic functions). |

d): | This arrow is the Kazhdan-Lusztig conjecture proved by Brylinski-Kashiwara and Beilinson-Bernstein and then generalized to the Kac-Moody case by Kashiwara-Tanisaki. By $\text{(}\stackrel{\sim}{\text{b}}\text{),}$ I mean the modular Lusztig conjecture (partially) proved by Anderson, Jantzen, and Soergel. The arrow from the Macdonald theory to modular representations is marked by '!'. It seems the most challenging now. I hope to continue my results on the Macdonald polynomials at roots of unity from the restricted case (alcove) to arbitrary weights (parallelogram). If might give a one-parametric generalization of the classic theory, formulas for the modular characters (not only those for the multiplicities), and a description of modular representations of arbitrary Weyl groups. However now it looks very difficult. |

To conclude, let me say a little something about the Verlinde algebras. I think now it is the most convincing demonstration of new methods based on double Hecke
algebras. I also have certain personal reasons to be very interested in them. The conformal fusion procedure appeared in my paper 'Functional realization of basic
representations of factorizable groups and Lie algebras' (Funct. Anal. Appl., **19** (1985), 36-52). Given an integrable representation of the
$n\text{-th}$ power of a Kac-Moody algebra and two sets of points on a Riemann surface $\text{(}n$
and $m$ points), I constructed an integrable representation of the $m\text{-th}$ power of the same
Kac-Moody algebra. The central charge here remains fixed. I missed that in the special case when $n=2,$
$m=1$ the multiplicities of irreducibles in the resulting representation are structural constants of a certain commutative
algebra, the Verlinde algebra. It was nice to know that these multiplicities (and much more) can be extracted from the simplest representation of the double affine
Hecke algebra at roots of unity.

I should add one more remark. In fact I borrowed the 'fusion procedure' from arithmetics. I had known Ihara's papers 'On congruence monodromy problem' very well. A similar procedure was the key stone of his theory. Of course I changed something and added something (central charge), but the procedure is basically the same. Can we go back and define Verlinde algebras in arithmetics?