The Relative Lie Algebra Cohomology of the Weil Representation

TL;DR

This paper investigates the relative Lie algebra cohomology of the orthogonal Lie algebra with the Weil representation, constructing a spectral sequence that reveals when cohomology vanishes or is non-zero, without relying on explicit representation theory calculations.

Contribution

It introduces a spectral sequence approach to compute relative Lie algebra cohomology for the Weil representation, identifying conditions for vanishing and non-vanishing cohomology.

Findings

Cohomology vanishes in certain cases when the spectral sequence's E_0 term is a Koszul complex of a regular sequence.

Explicit computation of cohomology groups for SO_0(p,1) in cases where k < p.

The spectral sequence converges to the relative Lie algebra cohomology, providing a new computational framework.

Abstract

We study the relative Lie algebra cohomology of $\mathfrak{so}(p,q)$ with values in the Weil representation $\varpi$ of the dual pair $\mathrm{Sp}(2k, \mathbb{R}) \times \mathrm{O}(p,q)$. Using the Fock model we filter this complex and construct the associated spectral sequence. We then prove that the resulting spectral sequence converges to the relative Lie algebra cohomology and has $E_0$ term, the associated graded complex, isomorphic to a Koszul complex. It is immediate that the construction of the spectral sequence of Chapter 3 can be applied to any reductive subalgebra $\mathfrak{g} \subset \mathfrak{sp}(2k(p+q), \mathbb{R})$. In case the symplectic group is large relative to the orthogonal group ($k \geq pq$), the $E_0$ term is isomorphic to a Koszul complex defined by a regular sequence, see 3.4. Thus, the cohomology vanishes except in top degree. This result is obtained without calculating the space of cochains and hence without using any representation theory. On the other hand, in case $k < p$, we know the Koszul complex is not that of a regular sequence from the existence of the class $\varphi_{kq}$ of Kudla and Millson, see [KM2], a nonzero element of the relative Lie algebra cohomology of degree $kq$. For the case of $\mathrm{SO}_0(p,1)$ we compute the cohomology groups in these remaining cases, namely $k < p$. We do this by first computing a basis for the relative Lie algebra cochains and then splitting the complex into a sum of two complexes, each of whose $E_0$ term is then isomorphic to a Koszul complex defined by a regular sequence. This thesis is adapted from the paper, [BMR], this author wrote with his advisor John Millson and Nicolas Bergeron of the University of Paris.