Last update: 20 February 2013
Though we shall never really use the data it is conceptually useful to note that there is an affine Hecke algebra associated to each triple where
An example of this data is when is the subgroup of upper triangular invertible matrices, and is the subgroup of invertible diagonal matrices.
The reason that we can avoid the data is that it is equivalent to different data where
This will be our basic data. In the example where and and are the upper triangular and diagonal matrices, respectively,
where is the symmetric group, acting on by permuting the orthonormal basis This example will be treated in depth in Sections 5–7. We shall show that the labeling sets for weight spaces of affine Hecke algebra representations that are introduced in (2.18) and Corollary 2.19 and used for the classification in Theorem 3.6 are generalizations of standard Young tableaux.
The components and in the data are obtained from by
where is the normalizer of in and is the set of algebraic group homomorphisms from to The notation is designed so that the multiplication in the group is
see [Bou1968, III Section 8]. The reflection (or defining) representation of the group is given by its action on and with respect to a inner product on the group is generated by reflections in the hyperplanes
See the picture which appears just before Theorem 1.17. The chambers are the connected components of and these are the fundamental regions for the action of on Fixing a choice of a fundamental chamber corresponds to the choice of the set of positive roots, which corresponds to the choice of in
In our formulation we may view the set as a labeling set for the reflecting hyperplanes in and
For a root the positive side of the hyperplane is the side towards i.e., and the negative side of is the side away from
For the inversion set of is
where There is a bijection
and the chamber is the unique chamber which is on the positive side of for and on the negative side of for
The simple roots in index the walls of the fundamental chamber and the corresponding reflections generate In fact, can be presented by generators and relations
where the (acute) angle between the hyperplanes and determines the value
Fix with The Iwahori–Hecke algebra associated to is the associative algebra over defined by generators and relations
where are the same as in the presentation of For define where is a reduced expression for By [Bou1968, Chapter IV, Section 2 Exercise 23], the element does not depend on the choice of the reduced expression. The algebra has dimension and the set is a basis of
The affine Hecke algebra associated to algebra given by
where the multiplication of the is as in the Iwahori–Hecke algebra the multiplication of the is as in (1.2) and we impose the relation
This formulation of the definition of is due to Lusztig [Lus1983] following work of Bernstein and Zelevinsky. The elements form a basis of
The group algebra of
is a subalgebra of with a obtained by linearly extending the on
Theorem 1.13 (Bernstein, Zelevinsky, Lusztig [Lus1983, 8.1]). The center of is
Let be maximal in Bruhat order subject to for some If there exists a dominant such that (otherwise for every dominant which is impossible since is a finite linear combination of Since we have
Repeated use of the relation (1.10) yields
where are constants such that for and unless So
and comparing the coefficients of gives Since it follows that which is a contradiction. Hence
The relation (1.10) gives
where Comparing coefficients of on both sides yields Hence and therefore for So
It is often convenient to assume that acts irreducibly on and that the lattice is the weight lattice
where the fundamental weights are the elements of given by
and is the Kronecker delta. Many facts are easier to state in this case and the general case can always be reduced to this one. We will make some further remarks on this reduction at the end of this section.
Consider the connected regions of the negative Shi arrangement [ALi1999,Shi1997,Shi1994,Shi1987,Sta1996,Sta1998], i.e., the arrangement of (affine) hyperplanes given by
Each chamber contains a unique region of which is a cone, and the vertex of this cone is the point which appears in the following theorem.
Theorem 1.17 [Ste1975]. Suppose that acts irreducibly on and that where is the weight lattice. The algebra is a free with
The proof is accomplished by establishing three facts:
(a) For each subtract row from row Then this row is divisible by Since there are pairs of rows the whole determinant is divisible by For the factors and are coprime, and so is divisible by This product and the product in the statement of (a) differ by the unit in
(b) By (a), is divisible by The top coefficient of is equal to
and the top coefficient of is
(c) Assume that are solutions of the equation Act on this equation by the elements of to obtain the system of equations
By (a) the matrix is invertible and so this system has a unique solution with In fact, the can be obtained by Cramer’s rule. Cramer’s rule provides an expression for as a quotient of two determinants. By (a) and (b) the denominator divides the numerator to give an element of Since each determinant is an alternating function, the quotient is an element of
Remark. In [Ste1975] Steinberg proves this type of result in full generality without the assumptions that acts irreducibly on and Note also that the proof given above is sketchy, particularly in the aspect that the top coefficient of the determinant is what we have claimed it is. See [Ste1975] for a proper treatment of this point.
1.18. Deducing the representation theory from
It is often easier to work with the representation theory of in the case when It is important to be able to convert from this case to the case of a general lattice If acts irreducibly on then the lattice satisfies
are the weight lattice and the root lattice, respectively. The group is a finite group (either cyclic or isomorphic to It corresponds to the center of the corresponding complex algebraic group. Let us denote the corresponding affine Hecke algebras by
according which lattice is used to make the group
Theorem 1.19 [RRa2003]. Then there is an action of the finite group on by ring automorphisms, such that
is the subalgebra of fixed points under the action of the group
This theorem is exactly what is needed to apply a (not very well known) version of Clifford theory to completely classify the representations of in terms of the representations of see [RRa2003].
This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.
Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).