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  • Print publication year: 2005
  • Online publication date: January 2010

4 - First results on Hn-modules

from PART I - LINEAR REPRESENTATIONS

Summary

The polynomial subalgebra Pn of Hn is a maximal commutative subalgebra, and so we may try to “do Lie Theory” using Pn as an analogue of Cartan subalgebra. The main difference however is that Pn is not semisimple, so we have to consider generalized eigenspaces, rather than usual eigenspaces. We define the formal character of an Hn-module M as the generating function for the dimensions of simultaneous generalized eigenspaces of the elements x1, …, xn on M. In Chapter 5 we will prove that the formal characters of irreducible Hn-modules are linearly independent (as any reasonable formal characters should be). The “Shuffle Lemma”, which is a special case of the Mackey Theorem, gives a transparent description of what induction “does” to the formal characters.

Our knowledge of the center of Hn allows us to develop an easy theory of blocks. The central characters of Hn (and so the blocks too) are labeled by the Sn-orbits on the n-tuples of scalars.

Next we study the properties of what can be considered as one of the main technical tools of the theory: the so-called Kato module (cf. [Kt]). Miraculously, if we take the 1-dimensional module over the polynomial algebra Pn on which every xi acts with the same scalar, and then induce it to Hn, we get an irreducible module. This is not hard to prove once you believe it is true.

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