Irreducible unitary representations with non-zero relative Lie algebra cohomology of a Lie group of type f4(4)
Pampa Paul
TL;DR
This work classifies irreducible unitary representations of a real form with Lie algebra $\mathfrak{f}_{4(4)}$ that have non-zero $(\mathfrak{g},K)$-cohomology by applying Vogan–Zuckerman cohomological induction, focusing on representations $A_\mathfrak{q}(0)$. It provides a complete accounting of $\theta$-stable parabolics, computes the associated cohomology via $P_\mathfrak{q}(t)=t^{R(\mathfrak{q})}P(Y_\mathfrak{q},t)$, and lists the compact duals $Y_\mathfrak{q}$ that occur. The paper finds a total of $46$ unitary equivalence classes with non-zero $(\mathfrak{g},K)$-cohomology, including $12$ discrete series with trivial infinitesimal character and exactly one Borel-de Siebenthal discrete series, with explicit Poincaré polynomials tabulated. These results extend the understanding of cohomological representations for exceptional real forms and provide concrete data that can inform automorphic and geometric cycle considerations via Matsushima’s isomorphism.
Abstract
In this article, we have determined the irreducible unitary representations with non-zero relative Lie algebra cohomology and Poincare polynomials of cohomologies of these representations for a connected Lie group G with Lie algebra f4(4). We have also determined a necessary and sufficient condition for these representations to be discrete series representations and identified the discrete series representations and Borel-de Siebenthal discrete series representations among the irreducible unitary representations of G with non-zero relative Lie algebra cohomology.