I want to understand how human sex-differences come about through
natural selection, and specifically its genetic mechanism.

I understand the basics of natural selection, namely how individuals
with specific advantageous qualities are more likely to survive and
reproduce than individuals who lack these qualities and thereby pass
on the genes responsible for those qualities to their offspring. The
distribution of genes and qualities shifts in a population and adapts
to its environment through this mechanism. (I know the basics of
genetics, but not much more.)

What puzzles me is how differences between the sexes come about
through natural selection, especially differences that rely upon a
combination of genes for their expression. I may be mistaken, but it
seems that any differences between the sexes must ultimately be tied
to the X-chromosome.  This would mean that any possible
sex-differences would be limited to features in the phenotype that are
(at least partially) controlled by the genes on the X-chromosome.

Beyond differences in reproductive organs, pelvic structure, and
relative body size, humans display cognitive and sex-differences that
appear to have a strong biological basis. But we suppose that these
cognitive abilities are influenced by multiple genes.  How can the
genes on the x-chromosome have this effect?

I guess the puzzle comes down to this: if a mother has genes that
contribute to her superior linguistic ability, how is it that her
daughters (on average) will inherit this ability more often (or more
strongly) than her sons will?  Likewise, a father who has superior
spatial skills.  When people speculate about hunter-gatherers and
evolving sex roles, I want
to understand the mechanism. I understand the environmental pressures,
but not the mechanism.

The ideal answer would explain how the evolution of sex-differences is
a special case of natural selection and would focus on the
x-chromosome (or any other known mechanisms).  It would also present
evidence of, or speculation about, the limits of sex-differences,
given that  cognitive differences, which are supposedly governed by
multiple genes. I could find a partial answer useful.

Note: This is not a question about sexual selection or mate selection,
except in so far as that explains sex-differences through genetics.

Thanks!

It's not necessarily true that each gene has a definite corresponding
effect, which seems to be your assumption (thus the conclusion that
all sex differences are based on one chromosome).  Gene expression
relies on how the genes interact as well, so slight differences in one
chromosome (x, say) can have the differences we see in the sexes via
different interactions.  You can read a more detailed (and probably
more accurate) description of this in Dawkins' book "Unweaving the
Rainbow."