Last update: 30 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.
The subject of this section is to show how the Hecke algebra technique is applied to the Macdonald polynomials. We will concentrate on the duality and the recurrence relations. The key notion will be the double affine Hecke. Let us start with
The corresponding in the differential case reads as follows where is a complex parameter. There are two special values of when the operator is very simple. For we have When Similarly, we can conjugate by for any Sometimes is more convenient to deal with in this form.
Let us now consider the eigenvalue problem for the operator If the solution of this equation is immediate: In this normalization it is symmetric with respect to and Without it generalizes the characters of finite-dimensional representations of
If this operator is the radial part of the Casimir operator for the symmetric space It is the restriction of the Casimir operator on the double coset space which is identified with a domain in If then corresponds in the same way to and for the quaternions
For any one can find a family of even solutions of the form such that for This family of hyperbolic polynomials satisfy the orthogonality relations They are called the ultraspherical polynomials.
We can also consider the rational limit of the operator switching from the Sutherland model to the Calogero model. The solutions of the rational eigenvalue problem are expressed in terms of the Bessel function. In this case, the solutions can be normalized to ensure the symmetry between the variable and the eigenvalue. In the trigonometric case it is possible only for two special values It is one of the main demerits of the harmonic analysis on the symmetric spaces.
In the difference theory, this symmetry holds for any root systems. This discovery is expected to renew the Harish-Chandra theory. The so-called group case is an intersection point of the differential (classical) and difference (new) theories.
We now turn to the difference version. We set and introduce the 'multiplicative difference' acting as and satisfying the commutation relation The Macdonald operator is expressed as follows:
The parameter in the difference setup is determined from the relation When (or the operator is simple: Compare this formula with (5.2) in the differential case and notice that (5.11) is easier to check than (5.2).
The eigenvalue problem always has a self-dual family of solutions. When there exists a unique family of the so-called (or Roger-Askey-Ismail) Laurent polynomials which are symmetric with respect to the transformation and satisfy the equation The following duality theorem is proved in the next section by using the double affine Hecke algebra.
Theorem 5.1 (Duality). for any
If we set the duality can be rewritten as follows:
The Askey-Ismail polynomials are nothing but the Macdonald polynomials of type There are three main Macdonald's conjectures for the Macdonald polynomials associated with root systems (see [Mac0011046, Mac1423624]):
|(1)||the scalar product conjecture,|
|(2)||the evaluation conjecture,|
|(3)||the duality conjecture.|
One may also add the Pieri rules to the list. These conjectures were justified recently using the double affine Hecke algebras in [Che1993, Che1995-2].
The duality from Theorem 5.1 can be rephrased as the symmetry of a certain scalar product. Actually this product is a difference counterpart of the spherical Fourier transform. For any symmetric Laurent polynomials we set where is a polynomial such that So we apply the operator to and then evaluate the result at
Theorem 5.2. for any
Theorem 5.1 follows from Theorem 5.2. Indeed, if and we can compute the scalar product as follows: Use and the normalization Hence Theorem 5.2 implies Actually the theorems are equivalent, since form a basis in the space of all symmetric Laurent polynomials.
Definition 5.1. The double affine Hecke algebra of type is the quotient by the relations for the generators
Here we consider as numbers or parameters. The first point of the theory is the following statement of PBW type.
Any element of can be uniquely expressed in the form The second important fact is the symmetry of with respect to and
Theorem 5.3. There exists an anti-involution such that and
Indeed, transposes the first two relations and leaves the remaining invariant.
Next, we introduce the expectation value of an element by using the expression (5.22). The definitions of and give that Now we can introduce the operator counterpart of the pairing on setting for any
The of the expectation value (5.24) ensures that it is symmetric We also remark that this pairing is non-degenerate for generic
Theorem 5.2 readily follows from
Lemma 5.4. For any symmetric Laurent polynomials
To prove the lemma we need to introduce the basic representation of the double affine Hecke algebra Consider the one-dimensional representation of the Hecke algebra sending and We denote this representation simply by Then take the induced representation where the last isomorphism is mod Under this identification of with the ring of Laurent polynomials, the element acts on as the multiplication by while and act by the operators respectively. Here the equality in means that The latter readily gives the desired formulas for
The expectation value is the composition where is a residue mod and is the evaluation map at Take any Then The last equality follows from (5.28). If and are symmetric and then since the operator acts on symmetric Laurent polynomials as It is straightforward. The duality is established.
This method of proving of the duality theorem can be generalized to any root system.
We now discuss the application of the duality to the Pieri rules, the recurrence formulas for with respect to the index First we will discretize functions and operators.
Recall that the renormalized polynomials are characterized by the conditions where As always, Denote the set of functions on by For any Laurent polynomial or more general rational function, we define by setting Considering as an abstract algebra with the fundamental relation the action of on is as follows: The correspondence whenever it is well-defined (the functions may have denominators), is an
Due to (5.31): The Pieri rules result directly from this equality. Indeed, the duality implies: Here is acting on the indices instead of Explicitly, For generic the mapping is injective. Hence one can pull (5.37) back, removing the hats: This is the Pieri formula in the case of See [AIs1983]. We remark that this formula makes sense when since the coefficient of vanishes at Generally speaking the 'vanishing conditions' are much less obvious.
The Pieri rules obtained above can be used to prove the so-called evaluation conjecture describing the values of at Applying (5.38) repeatedly, we get the formula for each The leading coefficient can be readily calculated: Let us look at (5.39) for Comparing the coefficients of we have since Hence
This value is easy to calculate directly (the formulas for are known). However the method described in this section is applicable to arbitrary root systems. We need only the duality, which is the main advantage of the difference harmonic analysis in contrast to the classical Harish-Chandra theory.
In this last subsection, we will discuss the double affine Hecke algebra and applications for Since we have already clarified the case in full detail, we will try to get concentrated on the main points only.
In the case, the Macdonald operators are as follows: In this normalization, (cf. the differential case). For instance, the so-called group case is for (in contrast to considered above).
The Macdonald polynomials for satisfy the Macdonald eigenvalue problem: where are partitions, i.e., sequences of integers such that Here elementary symmetric function of degree Given is a symmetric polynomial in of degree in the form The lower order terms are understood in the sense of the dominance ordering. Namely, a partition obtained from by subtracting simple roots is lower than For instance. We will use the abbreviation where So Using and
Given a partition we set
Theorem 5.5 (Duality). For any partitions and we have
This duality theorem implies the following Pieri formula.
Theorem 5.6. where (sum of unit vectors).
Here the summation is taken only over subsets such that remain partitions (generally speaking, dominant). It happens automatically, since the coefficient of on the right vanishes unless is dominant.
We can also determine the value of the Macdonald polynomial at exactly by the method used in the case. The formula was conjectured by Macdonald and proved by Koornwinder. The above theorems (for are also due to Macdonald and Koornvvinder. See also [EKi1995]. For arbitrary roots they were established in my recent papers.
Our approach is based on the double Hecke algebras. The operators appear naturally using the operators from (4.32). The latter describe the action of the generators in the induced representation isomorphic to the algebra of Laurent polynomials So the analogy with (5.26) is complete.
The double affine Hecke algebra (DAHA) for is the algebra generated by the following two commutative algebras of Laurent polynomials in variables: and the Hecke algebra of type with the standard braid and quadratic relations. The remaining relations are as follows: Here and They commute with thanks to 5.53 and 5.55.
When we come to the elliptic or 2-extended Weyl group of type due to Saito [Sai1985]. If and there are no quadratic relations, the corresponding group is the elliptic braid group of the product of elliptic curves without the diagonals divided by It was calculated by Birman [Bir1969] and Scott.
Establishing the connection with (4.32), and give the so-called polynomial (or basic) representation of
There is another version of this definition, using the element It is introduced from the formula and has the following commutation relations with and and In the polynomial representation this element coincides with from Lemma 4.3. Note that it acts on the functions through the action of on vectors Considered formally, is the image of the element from Lemma 4.1 with respect to the Kazhdan-Lusztig automorphism, sending
Since we can reduce the list of generators. Namely, or
In terms of the list of defining relations of is as follows:
|(b)||the braid relations and quadratic relations for|
For instance, let us deduce (5.55) from these formulas. Substituting, the left hand side equals: This representation, however, is not convenient from the viewpoint of the symmetry between and which will be discussed next. It is better to use instead of
The algebra contains the following two affine Hecke algebras: They are isomorphic to each other by the correspondence This map can be extended to an anti-involution of It is a general statement which holds for any root systems.
Theorem 5.7. There exists an anti-involution such that for and It preserves
We need to check that the relation (5.55) is self-dual with respect to The other relations are obviously One has: The latter can be rewritten as which is the of (5.55).
Using this involution we can establish the duality theorem for the case in the same way as we did in the case. Generalizing the theory to the case of arbitrary roots we can prove the Macdonald conjectures and much more. It gives a very convincing example of the power of the modern difference-operator methods.