Type $A_1\times A_1$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 9 March 2013

Type $A_1\times A_1$

We now turn to the rank two affine Hecke algebras. There are four different rank two root systems, each generated by two simple roots. The root systems are distinguished by the angle between the two simple roots - either $\frac{\pi }{2},\frac{2\pi }{3},\frac{3\pi }{4},$ or $\frac{5\pi }{6}\text{.}$ (See [Hum1994] for more details.)

The case where the two simple roots are perpendicular deserves special mention at this point. The root system $R=\{\pm {\alpha }_1,\pm {\alpha }_2\}$ is said to have type $A_1\times A_1,$ since it is the union of two root systems of type $A_1,$ $R_1=\{\pm {\alpha }_1\}$ and $R_2=\{\pm {\alpha }_2\}\text{.}$ Since all the elements of $R_1$ are perpendicular to all the elements of $R_2,$ $R$ is reducible. Algebraically, the reducibility of $R$ is reflected in the fact that the Weyl group of R is isomorphic to ${\mathbb{Z}}_2\times {\mathbb{Z}}_2,$ the direct product of the Weyl groups of $R_1$ and $R_2\text{.}$

At the level of the affine Hecke algebra, the reducibility also has a strong algebraic meaning. Specifically, let ${\stackrel{\sim }{H}}_{\left\{1\right\}}$ and ${\stackrel{\sim }{H}}_{\left\{2\right\}}$ be the affine Hecke algebras built on the root data of $R_1$ and $R_2,$ respectively. Then ${\stackrel{\sim }{H}}_{\left\{1\right\}}$ and ${\stackrel{\sim }{H}}_{\left\{2\right\}}$ are subalgebras of $\stackrel{\sim }{H}$ which centralize each other. In fact, each is the full centralizer of each other in $\stackrel{\sim }{H}\text{.}$ Then a form of Schur-Weyl duality shows that all the irreducible $\stackrel{\sim }{H}$ modules are of the form $L{\otimes }_{ℂ}M,$ where $L$ and $M$ are irreducible ${\stackrel{\sim }{H}}_{{\left\{1\right\}}^{-}}$ and ${\stackrel{\sim }{H}}_{{\left\{2\right\}}^{-}}$ modules, respectively. Hence the irreducible $\stackrel{\sim }{H}\text{-modules}$ are in bijection with pairs of irreducible modules over ${\stackrel{\sim }{H}}_{\left\{1\right\}}$ and ${\stackrel{\sim }{H}}_{\left\{2\right\}}\text{.}$ Since, then, the representation theory of the type $A_1\times A_1$ affine Hecke algebra is completely determined by the representation theory of the type $A_1$ affine Hecke algebra, we will make no further comment about the type $A_1\times A_1$ case.

Notes and References

This is an excerpt from Matt Davis' Ph.D Thesis entitled Representations of Rank Two Affine Hecke Algebras at Roots of Unity, University of Wisconsin, 2010.

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