A Note on the Double Affine Hecke Algebra of Type GL₂

Abstract

We express the double affine Hecke algebra ℍ associated to the general linear group GL₂(k) (here, k is a field with char(k) ≠ 2) as an amalgamated free product of quadratic extensions over the three-dimensional quantum torus 𝒪_q((k^×)³). With an eye towards proving ring-theoretic results pertaining to ℍ, a general treatment of amalgamated products of Ore and quadratic extensions is given. We prove an analogue of the Hilbert Basis Theorem for an amalgamated product Q of quadratic extensions and determine conditions for when the one-sided ideals of Q are principal or doubly-generated. Furthermore, we determine sufficient conditions which imply Q is a principal ideal ring. Finally, we construct an explicit isomorphism from ℍ to the amalgamated free product ring of quadratic extensions over 𝒪_q((k^×)³), a ring known to be noetherian. Therefore, it follows that ℍ is noetherian.

Key Words:

• Amalgamated products
• Double affine Hecke algebras
• Ideals
• Ore localization
• Skew-polynomial rings

2010 Mathematics Subject Classification:

• 16S10
• 16S36
• 16S85
• 16D25
• 20C08

ACKNOWLEDGMENTS

This article is based on some work from my Ph.D. dissertation [10] written at the University of California at Santa Barbara. I am grateful to Ken Goodearl and Milen Yakimov for suggesting the problems that motivated this work.