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Twisted strong Macdonald theorems and adjoint orbits

The strong Macdonald theorems state that, for $L$ reductive and $s$ an odd variable, the cohomology algebras $H^*(L[z]/z^N)$ and $H^*(L[z,s])$ are freely generated, and describe the cohomological, $s$-, and $z$-degrees of the generators. The resulting identity for the $z$-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate $H^*(\mathfrak{p} / z^N \mathfrak{p})$ and $H^*(\mathfrak{p}[s])$ for $\mathfrak{p}$ a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist. In particular we show that $H^*(\mathfrak{p} / z^N \mathfrak{p})$ contains a parabolic subalgebra of the coinvariant algebra of the fixed-point subgroup of the Weyl group of $L$, and thus is no longer free. We also prove a strong Macdonald theorem for $H^*(\mathfrak{b}; S^* \mathfrak{n}^*)$ and $H^*(\mathfrak{b} / z^N \mathfrak{n})$ when $\mathfrak{b}$ and $\mathfrak{n}$ are Iwahori and nilpotent subalgebras respectively of a twisted loop algebra. For each strong Macdonald theorem proved, taking $z$-weighted Euler characteristics gives an identity equivalent to Macdonald's constant term identity for the corresponding affine root system. As part of the proof, we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to $L$, proving an analogue of the Kostant slice theorem.