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Let $M^K_n$ be the moduli space of framed $K$-instantons over $S^4$ withinstanton number $n$ when $K$ is a compact simple Lie group of classical type.Let $U^{K}_{n}$ be the Uhlenbeck partial compactification of $M^{K}_{n}$. Ascheme structure on $U^{K}_{n}$ is endowed by Donaldson as an algebro-geometricHamiltonian reduction of ADHM data. In this paper, for $K=SO(N,R)$, $N\ge5$, weprove that $U^{K}_{n}$ is an irreducible normal variety with smooth locus$M^{K}_{n}$. Hence, together with the author's previous result, the K-theoreticNekrasov partition function for any simple classical group other than$SO(3,R)$, is interpreted as a generating function of Hilbert series of theinstanton moduli spaces.  Using this approach we also study the case $K=SO(4,R)$ which is the uniquesemisimple but non-simple classical group.

Geometry of Uhlenbeck partial compactification of orthogonal instanton spaces and the K-theoretic Nekrasov partition functions