Maximal hypoellipticity and Dolbeault cohomology representations for U(p,q)

TL;DR

This paper investigates an equivariant differential operator on certain flag manifolds related to U(p,q), showing that it is not maximally hypoelliptic, which impacts the understanding of Dolbeault cohomology representations.

Contribution

It introduces a new equivariant differential operator on flag manifolds and demonstrates its non-maximal hypoellipticity in the case of U(p,q).

Findings

The operator acts as an equivariant Dolbeault Laplacian on G/L.

Proves the operator is not maximal hypoelliptic for G=U(p,q).

Provides insights into Dolbeault cohomology representations for these groups.

Abstract

Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormander condition. We prove here that this operator is not maximal hypoelliptic when G=U(p,q).