Integral formulas for DAHA inner products

TL;DR

This paper derives integral formulas for DAHA coinvariants and inner products applicable to all parameter values, connecting to trace formulas and Plancherel measures in the limit as q approaches 0.

Contribution

It introduces a unified integral formula approach for DAHA coinvariants and inner products, extending existing residue methods to a broader parameter range.

Findings

Integral formulas for DAHA coinvariants valid for all parameters

Connection between DAHA formulas and AHA trace formulas as q→0

Systematic theory of DAHA coinvariants and modules

Abstract

The main aim is to obtain integral formulas for DAHA coinvariants and the corresponding inner products for any values of the DAHA parameters. In the compact case, our approach is similar to the procedure of ``picking up residues" due to Arthur, Heckman, Opdam and others; the resulting formula is a sum of integrals over double affine residual subtori. A single real integral provides the required formula in the noncompact case. As q tends to 0, our integral formulas result in the trace formulas for the corresponding AHA, which calculate the Plancherel measures for the spherical parts of the regular AHA modules. The paper contains a systematic theory of DAHA coinvariants, including various results on the affine symmetrizers and induced DAHA modules.