Matrix Points on Varieties

Abstract

We study the cohomology of $C_n(X)$, the moduli space of commuting $n$-by-$n$ matrices satisfying the equations defining a quasi-projective scheme $X$. This space can be viewed as a non-commutative Weil restriction from the algebra of $n$-by-$n$ matrices to the ground field. We introduce a semi-simple counterpart $S_n(X)$, defined as the quotient of $X^n \times \mathrm{GL}_n/\mathrm{T}_n$ by the diagonal $S_n$ action. We show that there exists a natural map $σ\colon S_n(X) \to C_n(X)$ inducing isomorphism on $\ell$-adic cohomology under mild restrictions on $X$ or the characteristic of the field. This confirms a heuristic derived from Weil restrictions. Furthermore, we provide explicit combinatorial formulae for the Betti numbers of $C_n(X)$ and prove a Macdonald-type generating series. A version for Hermitian matrix point is also proved in the last section.