Last update: 13 November 2012
Recall that the Weyl group acts on
Any element is determined by the values For define the polar decomposition
for all There is a unique and a unique such that
In this way we identify the sets and with and respectively.
For this paragraph (our goal here is (4.3) below) assume that is not a root of unity (we will treat the type root of unity case in detail in Section 7). The representation theory of is “the same” for any which is not a root of unity, i.e. provided is not a root of unity, the classification and construction of simple can be stated uniformly in terms of the parameter Suppose is such that and is such that
For the purposes of representation theory (as in Theorem 3.5) indexes a central character and so we should assume that is chosen nicely in its When
In this case the combinatorics of local regions is a new chapter in the combinatorics of the Shi arrangement defined in (1.16). Other aspects of the combinatorics of the Shi arrangement can be found in [ALi1999,33–35,37–39], and there are several additional places in the literature [Shi1987], [Xi1994, 1.11, 2.6], [Kos2000,KOP2000] which indicate that there is a deep (and not yet completely understood) connection between the structure and representation theory of the affine Hecke algebra and the combinatorics of the Shi arrangement.
Using the formulation in (4.3), Theorem 4.6 will give a complete description of the structure of as a subset of the Weyl group when is not a root of unity. We will treat the type root of unity cases in Section 7.
The weak Bruhat order is the partial order on given by
where denotes the inversion set of as defined in (1.5). This definition of the weak Bruhat order is not the usual definition but is equivalent to the usual one by [Bjo1994, Proposition 2]. A set of positive roots is closed if implies that The closure of a subset is the smallest closed subset of containing A set of positive roots is the inversion set of some permutation if and only if is closed and is closed (see [Bjo1984, Proposition 2] or [KRa2002, Theorem 5.1]).
The following theorem is proved in [KRa2002, Section 5]. The proof of part (b) of the theorem relies crucially on a theorem of J. Losonczy [Los1999].
Theorem 4.6. Let be dominant (i.e., for all and let Let be as given in (4.3).
Assume that is dominant (i.e., for all and Let be as given in (4.3). The conjugate of and of are defined by
where is the minimal length coset representative of and is the longest element of In Section 6.7 we shall show that these maps are generalizations of the classical conjugation operation on partitions.
Theorem 4.9. The conjugation maps defined in (4.8) are well defined involutions.
(a) Since is dominant, is dominant and thus only if Thus the equation gives that
(b) Let such that (By [Bou1968, IV Section 1 Exercise 3], is unique.) Then and it follows that
(c) Let be the set of negative roots in Let such that Then is the longest element of and Thus, since
(d) The weight is dominant and since This shows that is well defined.
(e) Write where is the longest element of
is the minimal length coset representative of
and so the second map in (4.8) is an involution.
(f) Using (e) and (a),
and so the first map in (4.8) is an invoution.
(g) Let and let Since
and thus, by (b),
Thus, by (a),
and so the second map in (4.8) is well defined.
Remark 4.10. In type the conjugation involution coincides with the duality operation for representations of defined by Zelevinsky [Zel1980]. Zelevinsky’s involution has been studied further in [KZe1996,LTV1999,MWa1986] and extended to general Lie type by Kato [Kat1993] and Aubert [Aub1995]. For in type this is the involution on modules induced by the Iwahori–Matsumoto involution of and is detected on the level of characters: it sends an irreducible to the unique irreducible with for each I would like to thank J. Brundan for clarifying this remark and making it precise.
Examples 4.11. (a) If is dominant and is generic (as an element of then and
(b) Let be defined by for all Then
where is the right descent set of The sets which arise here are fundamental to the theory of descent algebras [GRe1989,Reu1993,Sol1976].
(c) This example is a generalization of (b). Suppose that is a local region such that is regular and integral (i.e., for all Then
(d) Let be the root system of type with simple roots where is an orthonormal basis of The positive roots are Let be given by and Then is dominant (i.e., in and integral and
Figure 3 displays the local regions as regions in see the remarks after (2.18).
The solid line is the hyperplane corresponding to the root in and the dashed lines are the hyperplanes corresponding to the roots in
(e) Let be the root system of type as in (d). Let be defined by
If then the unique minimal element of has
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).