Last update: 27 April 2013
We first examine some properties that hold for irreducible modules that are calibrated, i.e., can be decomposed into a direct sum of weight spaces (see (2.10)). This section follows closely similar results for affine Hecke algebras in [Ram1998].
Lemma 4.1. Let be an irreducible calibrated module. Then, for all such that
The proof is by contradiction. Assume Let be the subalgebra of generated by and all Then the two-dimensional principal series module is irreducible and there is an homomorphism given by
where is a nonzero element of
is simple, this is an injection and thus, is not calibrated since
is not calibrated. Thus
is a nonzero vector in which is not a multiple of
Assume that the sequence is chosen so that is minimal. Since the in this sequence are all well defined, the elements in the orbit correspond (under the bijection in (2.7)) to a sequence of chambers in on the positive side of all Each chamber in this sequence shares a face with the next chamber in the sequence. Since both and are in this is a sequence which begins and ends at the chamber Since the chambers are in bijection with the elements of it follows that in
This means that there is some such that is not reduced and we can use the braid relations to rewrite this word as By Proposition 2.5(e) the also satisfy the braid relations and so
By Proposition 2.5(c), the operator above will act (on by a constant and so
where the constant is nonzero since is nonzero. But the expression
is shorter than the original expression of and this contradicts the minimality of It follows that
Lemma 4.2. Let be an irreducible calibrated module. Suppose that and are both nonzero. Then the map is a bijection.
By Lemma 4.1(b), and thus it is sufficient to show that is not the zero map. Let be a nonzero vector in Since is irreducible, there must be a sequence of such that
is a nonzero element of Let be minimal such that this is the case. Since it follows, as in the second paragraph of the proof of Lemma 4.1(b), that in For notational convenience let Let be maximal such that is not reduced. If then we can use the braid relations to get
Since acts on by a constant
and since is not 0. But this contradicts the minimality of Thus we must have that and
For simple roots and in let be the rank two root subsystem of generated by and A weight is skew if
Condition (a) says that is regular for all rank 1 subsystems of generated by simple roots. Condition (b) is an “almost regular” condition on with respect to rank 2 subsystems generated by simple roots. By the analysis in Section 3, the weights which appear in calibrated modules for graded Hecke algebras corresponding to rank two root systems are skew.
Recall from Section 2.3 that a pair is a local region if the set
is nonempty. A local region is skew if, for all the weight is skew for all pairs of simple roots in
The following theorem specifies the weight space structure of an irreducible calibrated
Theorem 4.3. If is an irreducible calibrated with central character then there is a unique skew local region such that
By Lemma 4.1 all nonzero generalized weight spaces of have dimension 1 and by Lemma 4.2 all between these weight spaces are bijections. This already guarantees that there is a unique local region which satisfies the condition. It only remains to show that this local region is skew.
Let be the subalgebra of generated by and Since is calibrated as an it is calibrated as an and so all factors of a composition series of as an are calibrated. Thus, by the classification in Section 3, the weights of are skew. So is a skew local region.
The following Proposition shows that the weight structure of calibrated representations as determined in Theorem 4.3 essentially forces the on a weight basis.
Proposition 4.4. Let be a calibrated and for all such that assume that
For each such that let be a nonzero vector in The vectors form a basis of Let and be given by
Comparing coefficients yields
These equations imply that
This completes the proof of (a) and (b). The relation in implies that
Theorem 4.5. Let be skew and let index the chambers in the local region Define
so that the symbols are a labeled basis of the vector space Then the following formulas make into an irreducible For each
where we set if
Since is skew, for all and all simple roots This implies that the coefficients in are well defined for all and
By construction, the nonzero weight spaces of are where Since for any proper submodule of must have and for some with This is a contradiction to Corollary 2.6. So is irreducible if it is an
It remains to show that the defining relations for are satisfied. Let Then
Now let us check the braid relations. Write where
for Then di is a diagonal matrix and is a pseudo-permutation matrix, in the sense that each row and each column contains at most one nonzero entry. For a sequence define a diagonal matrix by the relation
If is generic, then, for all
and all diagonal entries are nonzero, but, in general, some diagonal entries of may be 0. Use this method to expand the expression
and move all the diagonal operators to the right of the and obtain diagonal operators The operators are pseudo-permutation operators that may have some rows and columns without a nonzero entry. By replacing some diagonal entries of the operators by 0, we may "fix the and replace the with operators which have exactly one nonzero entry in each row and each column. This yields the expression
If is generic, then the diagonal entries of are nonzero and have the form where is a rational function in the A similar expansion gives
where the are diagonal operators which, for generic have diagonal entries where is a rational function of the As in the proof of Proposition 2.5(e), for all generic and so it follows that as rational functions.
When is not generic, the operators and may have some diagonal entries equal to zero. From the classification of representations of rank two graded Hecke algebras we know that there exists a calibrated representation of when is skew. This representation has a unique, up to constant multiples, basis of simultaneous eigenvectors for the action of and Proposition 4.4 shows that the action on this basis is forced except for the values of the off diagonal elements of the These values depend on the normalization of the basis. Because we know that this representation exists we know that there are choices of the nonzero entries in the such that (4.2) and (4.3) are equal. If a diagonal entry of is nonzero, then it is equal to and since (as shown above) Thus it follows that nonzero contributions from the terms and are equal and that is equal to
Remark 4.6. The action of on a weight basis of is forced up to the freedom in Proposition 4.4(c). Our choice in Theorem 4.5 and the alternative choice yield isomorphic modules. The change of basis provides the isomorphism.
We summarize the results of this section with the following corollary of Theorem 4.3 and the construction in Theorem 4.5.
Theorem 4.7. Let be an irreducible calibrated Let be (a fixed choice of) the central character of and let for any such that Then is skew and where is the module defined in Theorem 4.5.
This is an excerpt of a paper entitled Representations of graded Hecke algebras, written by Cathy Kriloff (Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085) and Arun Ram.
Research of the first author supported in part by an NSF-AWM Mentoring Travel Grant. Research of the second author supported in part by National Security Agency grant MDA904-01-1-0032 and EPSRC Grant GR K99015.