Last update: 13 November 2012
For simple roots and in and let be the rank-two root subsystem of generated by and A weight is calibratable if, for every pair is a weight of a calibrated representation of the rank-two affine Hecke (sub)algebra generated by and A local region
The classification of irreducible representations of rank-two affine Hecke algebras given in [Ram2002] can be used to state this condition combinatorially. Specifically, a weight is calibratable if
Condition (a) says that is regular for all rank-1 subsystems of generated by simple roots. This condition guarantees that the weight is “calibratable” (i.e., appears as a weight of some calibrated representation) for all rank-1 affine Hecke subalgebras of Condition (b) is an “almost regular” condition on t with respect to rank-2 subsystems generated by simple roots.
Remark. The conversion between the definition of calibratable weight and the combinatorial condition given in (a) and (b) is as follows. Consider a rank-two affine Hecke algebra
From (A) and (B) it follows that the local regions which satisfy (a) and (b) do contribute calibrated weights. The following shows that the other local regions do not contribute calibratable weights.
(Note that to satisfy (b) must be nonempty.) From the tables in [Ram2002] we see that none of these local regions supports a calibrated representation.
Remark. The paper [Ram2002] does not treat roots of unity. However, it is interesting to note that, provided the methods of [Ram2002] go through without change to classify all representations of rank-two affine Hecke algebras even when is a root of unity. This classification can be used (as in the previous remark) to show that (a) and (b) above still characterize calibratable weights when is a root of unity such that The key point is that Lemma 1.19 of [Ram2002] still holds. If then Lemma 1.19 of [Ram2002] breaks down at the next to last line of the proof in the statement “... forces to have Jordan blocks of size 1 ....” When it is possible that has a Jordan block of size 2. If then one can change the definition of the and use similar methods to produce a complete analysis of simple but we shall not do this here, choosing instead to exclude the case for simplicity of exposition.
The following lemma provides fundamental results about the structure of irreducible calibrated We omit the proof since it is accomplished in exactly the same way as in [KRa2002, Lemmas 4.1 and 4.2].
Lemma 3.1 Let be an irreducible calibrated module. Then, for all such that
This lemma together with the classification of irreducible modules for rank-two affine Hecke algebras gives the following fundamental structural result for irreducible calibrated The proof is essentially the same as the proof of Proposition 4.3 in [KRa2002]. We repeat the proof here for continuity.
Theorem 3.2 If is an irreducible calibrated with central character then there is a unique skew local region such that
By Lemma 3.1(b) all nonzero generalized weight spaces of have dimension 1 and by Lemma 3.1(c) 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 generated by and Since is calibrated as an it is calibrated as a and so all factors of a composition series of as an are calibrated. Thus the weights of are calibratable. So is a skew local region.
The following proposition shows that the weight space structure of calibrated representations, as determined in Theorem 3.2, essentially forces the on a weight basis. The proof is quite similar to the proof of Proposition 4.4 in [KRa2002]. However, we include the details since there is a technicality here; to make the conclusion in (3.4) we use the fact that the group corresponds to the weight lattice
Proposition 3.3. Let be a calibrated and assume that for all such that
For each such that let be a nonzero vector in The vectors form a basis of Let and given by
The definition equation for
Comparing coefficients gives
These relations give:
By assumption (A1), for all For each fundamental weight and since Thus we conclude that
This completes the proof of (a) and (b). By the definition of the vector
Using the formula for and we find So, by comparing coefficients of we obtain the equation
Theorem 3.5. Let be a skew local region 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
and we set if
Since is a skew local region for all and all simple roots This implies that the coefficient is 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 2.19. So is irreducible if it is an
It remains to show that the defining relations for are satisfied. This is accomplished as in the proof of [KRa2002, Theorem 4.5]. The only relation which is tricky to check is the braid relation. This can be verified as in [KRa2002] or it can be checked by case by case arguments (as in [Ram1998]).
We summarize the results of this section with the following corollary of Theorem 3.2 and the construction in Theorem 3.5.
Theorem 3.6. Let be an irreducible calibrated Let be (a fixed choice of) the central character of and let for any such that Then is a skew local region and where is the module defined in Theorem 3.5.
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).